%------------------------------------------------------------------------------
% File     : ITP256_1 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Delete 00582_038585
% 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    : 0072_VEBT_Delete_00582_038585 [Des22]

% Status   : Theorem
% Rating   : 0.44 v8.2.0, 0.70 v8.1.0
% Syntax   : Number of formulae    : 13287 (2680 unt;3130 typ;   0 def)
%            Number of atoms       : 28654 (9540 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 22020 (3523   ~; 572   |;2238   &)
%                                         (2897 <=>;12790  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   28 (   2 avg)
%            Number of types       :  668 ( 667 usr)
%            Number of type conns  : 3038 (1970   >;1068   *;   0   +;   0  <<)
%            Number of predicates  :  123 ( 120 usr;   2 prp; 0-4 aty)
%            Number of functors    : 2343 (2343 usr; 493 con; 0-4 aty)
%            Number of variables   : 26395 (25566   !; 829   ?;26395   :)
% 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 05:02:39.956
%------------------------------------------------------------------------------
% Could-be-implicit typings (667)
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tff(ty_n_t__fun_It__Extended____Nat__Oenat_Mt__HOL__Obool_J,type,
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tff(ty_n_t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Extended____Nat__Oenat_J,type,
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tff(ty_n_t__Set__Oset_It__List__Olist_It__HOL__Obool_J_J,type,
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tff(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    list_VEBT_VEBT: $tType ).

tff(ty_n_t__fun_It__Real__Oreal_Mt__Complex__Ocomplex_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Real__Oreal_J,type,
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tff(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    set_list_nat: $tType ).

tff(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
    set_list_int: $tType ).

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

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

tff(ty_n_t__fun_It__Real__Oreal_Mt__Rat__Orat_J,type,
    fun_real_rat: $tType ).

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,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
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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,
    fun_int_bool: $tType ).

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,
    fun_int_rat: $tType ).

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,
    fun_int_nat: $tType ).

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__Option__Ooption_It__HOL__Obool_J,type,
    option_bool: $tType ).

tff(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
    option_num: $tType ).

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__List__Olist_It__Real__Oreal_J,type,
    list_real: $tType ).

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

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

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

% Explicit typings (2463)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_Pr1261947904930325089at_nat > fun_Pr7757163693017731481t_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_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__aab____,type,
    aTP_Lamm_aab: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aac____,type,
    aTP_Lamm_aac: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aad____,type,
    aTP_Lamm_aad: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__aae____,type,
    aTP_Lamm_aae: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__aaf____,type,
    aTP_Lamm_aaf: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__aag____,type,
    aTP_Lamm_aag: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__aah____,type,
    aTP_Lamm_aah: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__aai____,type,
    aTP_Lamm_aai: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aaj____,type,
    aTP_Lamm_aaj: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__aak____,type,
    aTP_Lamm_aak: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__aal____,type,
    aTP_Lamm_aal: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__aam____,type,
    aTP_Lamm_aam: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__aan____,type,
    aTP_Lamm_aan: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__aao____,type,
    aTP_Lamm_aao: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aap____,type,
    aTP_Lamm_aap: fun_real_real ).

tff(sy_c_ATP_058Lamm__aaq____,type,
    aTP_Lamm_aaq: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aar____,type,
    aTP_Lamm_aar: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aas____,type,
    aTP_Lamm_aas: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aat____,type,
    aTP_Lamm_aat: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__aau____,type,
    aTP_Lamm_aau: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aav____,type,
    aTP_Lamm_aav: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aaw____,type,
    aTP_Lamm_aaw: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aax____,type,
    aTP_Lamm_aax: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aay____,type,
    aTP_Lamm_aay: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aaz____,type,
    aTP_Lamm_aaz: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_list_nat > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__aba____,type,
    aTP_Lamm_aba: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__abb____,type,
    aTP_Lamm_abb: real > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__abc____,type,
    aTP_Lamm_abc: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__abd____,type,
    aTP_Lamm_abd: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__abe____,type,
    aTP_Lamm_abe: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__abf____,type,
    aTP_Lamm_abf: real > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__abg____,type,
    aTP_Lamm_abg: fun_re3844104906058862192omplex ).

tff(sy_c_ATP_058Lamm__abh____,type,
    aTP_Lamm_abh: real > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__abi____,type,
    aTP_Lamm_abi: fun_re5512370375968561900l_real ).

tff(sy_c_ATP_058Lamm__abj____,type,
    aTP_Lamm_abj: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abk____,type,
    aTP_Lamm_abk: nat > fun_li8232715659020003717t_bool ).

tff(sy_c_ATP_058Lamm__abl____,type,
    aTP_Lamm_abl: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__abm____,type,
    aTP_Lamm_abm: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__abn____,type,
    aTP_Lamm_abn: fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__abo____,type,
    aTP_Lamm_abo: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__abp____,type,
    aTP_Lamm_abp: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__abq____,type,
    aTP_Lamm_abq: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__abr____,type,
    aTP_Lamm_abr: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__abs____,type,
    aTP_Lamm_abs: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__abt____,type,
    aTP_Lamm_abt: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__abu____,type,
    aTP_Lamm_abu: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abv____,type,
    aTP_Lamm_abv: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abw____,type,
    aTP_Lamm_abw: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abx____,type,
    aTP_Lamm_abx: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aby____,type,
    aTP_Lamm_aby: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__abz____,type,
    aTP_Lamm_abz: fun_real_bool ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: set_set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__aca____,type,
    aTP_Lamm_aca: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acb____,type,
    aTP_Lamm_acb: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acc____,type,
    aTP_Lamm_acc: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__acd____,type,
    aTP_Lamm_acd: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__ace____,type,
    aTP_Lamm_ace: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__acf____,type,
    aTP_Lamm_acf: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__acg____,type,
    aTP_Lamm_acg: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__ach____,type,
    aTP_Lamm_ach: nat > fun_nat_fun_nat_bool ).

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

tff(sy_c_ATP_058Lamm__acj____,type,
    aTP_Lamm_acj: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ack____,type,
    aTP_Lamm_ack: complex > fun_real_bool ).

tff(sy_c_ATP_058Lamm__acl____,type,
    aTP_Lamm_acl: rat > fun_Pr803113274115125473t_bool ).

tff(sy_c_ATP_058Lamm__acm____,type,
    aTP_Lamm_acm: fun_int_int ).

tff(sy_c_ATP_058Lamm__acn____,type,
    aTP_Lamm_acn: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__aco____,type,
    aTP_Lamm_aco: fun_nat_bool > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__acp____,type,
    aTP_Lamm_acp: fun_fun_nat_rat_bool ).

tff(sy_c_ATP_058Lamm__acq____,type,
    aTP_Lamm_acq: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__acr____,type,
    aTP_Lamm_acr: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__acs____,type,
    aTP_Lamm_acs: fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__act____,type,
    aTP_Lamm_act: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__acu____,type,
    aTP_Lamm_acu: fun_Pr803113274115125473t_bool ).

tff(sy_c_ATP_058Lamm__acv____,type,
    aTP_Lamm_acv: fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__acw____,type,
    aTP_Lamm_acw: fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: set_int > fun_int_bool ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: fun_real_bool > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_list_nat_bool > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_set_nat_bool > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_nat_bool > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_int_bool > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_complex_bool > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_Pr7757163693017731481t_bool > fun_fu6897202038601531823t_bool ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_real_bool > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_list_nat_bool > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_set_nat_bool > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_nat_bool > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_int_bool > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_complex_bool > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_Pr7757163693017731481t_bool > fun_fu6897202038601531823t_bool ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: vEBT_VEBT > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: vEBT_VEBT > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: set_complex > fun_set_complex_bool ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: set_Pr1261947904930325089at_nat > fun_se8074476271188067385t_bool ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: set_int > fun_set_int_bool ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: set_Pr1261947904930325089at_nat > fun_se3305628495172992525t_bool ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: set_real > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: set_list_nat > fun_se7035039998181355731t_bool ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: set_set_nat > fun_se1925730536474940935t_bool ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: set_int > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: set_nat > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: set_Pr1261947904930325089at_nat > fun_fu6897202038601531823t_bool ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: set_real > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: set_list_nat > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: set_set_nat > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: set_nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: set_int > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: set_real > fun_fu4470565429337120259l_bool ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: set_real > fun_fu66453571216240387l_bool ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: set_real > fun_fu6440467031773438595l_bool ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: set_nat > fun_fu4970875099359191043t_bool ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: set_nat > fun_fu7165551060251750659t_bool ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: set_nat > fun_fu2548079958729413763t_bool ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: set_int > fun_fu7719728765329718531t_bool ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: set_int > fun_fu691032689367502339t_bool ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: set_int > fun_fu8949065344085950851t_bool ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: set_complex > fun_fu1598310740840436355x_bool ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_Pr7757163693017731481t_bool ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: fun_real_bool ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: fun_int_bool ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: fun_real_real ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_int_int ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: set_Pr958786334691620121nt_int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: set_Pr7592006421409863191nteger > fun_fu6012121808138353003r_bool ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: set_Pr5663546166630115725nteger > fun_fu1674507805476766837r_bool ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: set_Pr233528076387193594nt_int > fun_fu4196931294019812808t_bool ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: set_Pr5033440245338114598nt_int > fun_fu5117548769634986908t_bool ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: fun_nat_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: set_Pr1261947904930325089at_nat > fun_na5265545386895126880t_bool ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: set_bool > fun_na205931550456536010l_bool ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: set_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: set_int > fun_na7919044170365968317t_bool ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: set_Pr1261947904930325089at_nat > fun_na5265545386895126880t_bool ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: set_bool > fun_na205931550456536010l_bool ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: set_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: set_int > fun_na7919044170365968317t_bool ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: nat > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_na6801867027684862029l_bool ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: set_real > fun_fu914841811607101702l_bool ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: set_real > fun_fu7100649987335273288l_bool ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: set_nat > fun_fu4346542433904371390t_bool ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: set_nat > fun_fu8382797250367883996t_bool ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: set_int > fun_fu7152481525987436038t_bool ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: set_int > fun_fu5129045444369776328t_bool ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: set_complex > fun_fu5519326811792979714x_bool ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: set_complex > fun_fu3191020621175015362x_bool ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: set_int > fun_fu549321512867696136t_bool ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: set_int > fun_fu3401574108658714444t_bool ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: set_complex > fun_fu6356453956411242500x_bool ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: set_complex > fun_fu5995210064384465734x_bool ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: set_real > fun_fu2277326744902863452l_bool ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: set_real > fun_fu6750725090257165812l_bool ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: set_nat > fun_fu3052598110916186900t_bool ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: set_nat > fun_fu2971308621445215880t_bool ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: set_real > fun_fu914841811607101702l_bool ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: set_real > fun_fu7100649987335273288l_bool ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: set_nat > fun_fu4346542433904371390t_bool ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: set_nat > fun_fu8382797250367883996t_bool ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: set_int > fun_fu7152481525987436038t_bool ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: set_int > fun_fu5129045444369776328t_bool ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: set_complex > fun_fu5519326811792979714x_bool ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: set_complex > fun_fu3191020621175015362x_bool ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: set_int > fun_fu549321512867696136t_bool ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: set_int > fun_fu3401574108658714444t_bool ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: set_complex > fun_fu6356453956411242500x_bool ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: set_complex > fun_fu5995210064384465734x_bool ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: set_real > fun_fu2277326744902863452l_bool ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: set_real > fun_fu6750725090257165812l_bool ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: set_nat > fun_fu3052598110916186900t_bool ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: set_nat > fun_fu2971308621445215880t_bool ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: fun_Co8310303586393106575r_bool ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: fun_Pr7757163693017731481t_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_Pr2477239776880934807at_nat > fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: fun_Pr248683899324676269t_bool > fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: fun_Pr2478283820680522243nt_int > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: fun_Pr803113274115125473t_bool > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: fun_nat_real ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: produc3843707927480180839at_nat > fun_Pr1607443439640623155t_bool ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: list_nat > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: fun_Pr248683899324676269t_bool ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: fun_real_bool > fun_real_bool ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: fun_list_nat_bool > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: fun_set_nat_bool > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_nat_bool > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: fun_int_bool > fun_int_bool ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: set_Pr1261947904930325089at_nat > fun_Pr7757163693017731481t_bool ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: set_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: set_list_nat > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: set_set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: set_int > fun_int_bool ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: product_prod_nat_nat > fun_fu6897202038601531823t_bool ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: produc3843707927480180839at_nat > fun_fu224428745893945571t_bool ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: real > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: list_nat > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: set_nat > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: int > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: produc3843707927480180839at_nat > fun_se6392513987399461773t_bool ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: product_prod_nat_nat > fun_se3305628495172992525t_bool ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: real > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: list_nat > fun_se7035039998181355731t_bool ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: set_nat > fun_se1925730536474940935t_bool ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: nat > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: int > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: set_nat > fun_fu9073671928554219267t_bool ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: set_nat > fun_fu6463170612142064108al_nat ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: fun_real_fun_nat_nat > fun_nat_fun_real_nat ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: set_real > fun_fu3370445115705697296at_nat ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: set_nat > fun_fu2583778531842586371t_bool ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: set_nat > fun_fu8221675813083619692nt_nat ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_int_fun_nat_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: set_int > fun_fu5002972218919249680at_nat ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: set_nat > fun_fu5088574307552523139t_bool ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: set_nat > fun_fu4637029388598527722ex_nat ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: fun_co636942507509095880at_nat > fun_na8457291176974909128ex_nat ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: set_complex > fun_fu1817986637694864396at_nat ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: set_complex > fun_fu4189668387880940419x_bool ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: set_complex > fun_fu3241646296630235056omplex ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_re7208826350674946562omplex > fun_co286204877037288194omplex ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: set_real > fun_fu7272995576155203506omplex ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: set_complex > fun_fu8281188428872102787x_bool ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: set_complex > fun_fu5276431085538847372omplex ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: fun_na1591161823130732966omplex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: set_nat > fun_fu5286008751397070186omplex ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: set_complex > fun_fu3391973010043818627x_bool ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: set_complex > fun_fu2147946303232500784omplex ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: fun_in1307110748760623106omplex > fun_co7673900812888889602omplex ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: set_int > fun_fu2298032621795249330omplex ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: set_nat > fun_fu1296116514128556524l_real ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: set_real > fun_fu4624264860510622736t_real ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: set_nat > fun_fu3344834855664514412t_real ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: fun_int_fun_nat_real > fun_nat_fun_int_real ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: set_int > fun_fu7689295685718976784t_real ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: set_nat > fun_fu9046435873919503594x_real ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_co8378349014452122532t_real > fun_na5723442673449079972x_real ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: set_complex > fun_fu7575693963208796428t_real ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: set_real > fun_fu4833610055523116803l_bool ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: set_real > fun_fu7766971228835516944at_nat ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_nat_fun_real_nat > fun_real_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: set_nat > fun_fu213344517885440003t_bool ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: set_nat > fun_fu8739491518516473836al_nat ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: fun_Pr1607443439640623155t_bool > fun_Pr124437560471516333t_bool ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_list_nat_bool > fun_li8598321315282354825t_bool ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: fun_set_nat_bool > fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: fun_Pr7757163693017731481t_bool > fun_Pr248683899324676269t_bool ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: fun_real_bool > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: fun_int_bool > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: fun_Pr1607443439640623155t_bool > fun_Pr124437560471516333t_bool ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: fun_list_nat_bool > fun_li8598321315282354825t_bool ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: fun_set_nat_bool > fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_Pr7757163693017731481t_bool > fun_Pr248683899324676269t_bool ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: fun_real_bool > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: fun_int_bool > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: complex > fun_fu709861064203714431x_real ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: complex > fun_fu6638204120336560003ex_rat ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: complex > fun_fu2513798379308225899ex_nat ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: complex > fun_fu4692242605864086655ex_int ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: real > fun_fu9132278742410140537l_real ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: real > fun_fu7187499828406537085al_rat ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: real > fun_fu3768230810189712741al_nat ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: real > fun_fu5645595436833112697al_int ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: int > fun_fu4328765270694171897t_real ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: int > fun_fu1617438229336688381nt_rat ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_real_complex > fun_fu1393723446858488622omplex ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_nat_complex > fun_fu7905158015251895726omplex ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_int_complex > fun_fu6363802315473983150omplex ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: fun_real_real > fun_fu1760894646116856364l_real ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: fun_int_real > fun_fu6185597869953674668t_real ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: set_complex > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: fun_complex_real > fun_fu1911030855436031788x_real ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: fun_real_rat > fun_fu7249360421009165912al_rat ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_nat_rat > fun_fu2739651132436891864at_rat ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: fun_int_rat > fun_fu2671659548940339672nt_rat ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_complex_rat > fun_fu4947971316162376536ex_rat ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_Pr2602117396471556196at_nat > fun_fu7732904369943091395at_nat ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_real_complex > fun_real_bool ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: fun_int_complex > fun_int_bool ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_real_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: fun_int_real > fun_int_bool ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: fun_complex_real > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: fun_real_rat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_int_rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: fun_complex_rat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: fun_real_nat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: fun_int_nat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: code_integer > fun_Co6226381647958292740nteger ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: extended_enat > fun_Ex2441982661509597004d_enat ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: fun_na7542918316964655096d_enat ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: fun_int_int ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: fun_real_real ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: fun_num_fun_num_bool ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: fun_nat_int > fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: fun_nat_complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: fun_nat_rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: fun_nat_int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: fun_nat_real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_num_fun_num_bool ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: nat > fun_nat_fun_nat_bool ).

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

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: fun_nat_fun_nat_nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: fun_nat_fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: fun_nat_complex > fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: fun_nat_real > fun_na6801867027684862029l_bool ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: fun_nat_complex > fun_fu4528473944208963392omplex ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: fun_nat_complex > fun_fu6652553491520710942omplex ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: fun_nat_rat > fun_fu6228432884689642900at_rat ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: fun_nat_rat > fun_fu4710970720072501148at_rat ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: fun_nat_int > fun_fu1567874110255095356at_int ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: fun_nat_int > fun_fu5519318738670804632at_int ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: fun_nat_real > fun_fu8064956510386623256t_real ).

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

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: nat > fun_co6147893885205795665omplex ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: nat > fun_in1521351555827773387at_int ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: nat > fun_ra3995598502166916815at_rat ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: nat > fun_Co8502721031311864414nteger ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: nat > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: fun_nat_complex > fun_co3194742168270691230omplex ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: fun_nat_rat > fun_ra2740371568465858076at_rat ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: fun_nat_int > fun_in281396392327853336at_int ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: fun_nat_real > fun_re7819166435457029784t_real ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: fun_nat_complex ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: fun_nat_int ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: fun_nat_bool > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: fun_nat_bool > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: fun_nat_bool > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: fun_nat_bool > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: set_nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: set_nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: set_nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: nat > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: nat > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: nat > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: fun_nat_real > fun_se2100927751953696370t_real ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: nat > fun_nat_fun_nat_nat ).

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

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

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: fun_real_rat > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: fun_int_rat > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: fun_real_int > fun_fu3737244727311182119al_int ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: set_complex > fun_fu3810065756161129304x_bool ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: set_complex > fun_fu5688335724886546948x_bool ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: set_real > fun_fu5258761731371428616l_bool ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: fun_real_complex > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: fun_nat_complex > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: fun_int_complex > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: fun_real_real > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: fun_nat_real > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: fun_int_real > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: fun_real_rat > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: fun_nat_rat > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: fun_int_rat > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: fun_real_nat > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: fun_complex_real > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_int_rat > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: fun_complex_rat > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: fun_complex_nat > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: fun_real_complex > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: fun_int_complex > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: fun_real_rat > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: set_int > fun_fu337402001079966300t_bool ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: real > fun_fu3477889739401271834t_real ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: real > fun_fu4806068833880117316t_real ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: real > fun_fu2115079063411676732t_real ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: real > fun_fu4540635375566031512t_real ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: complex > fun_na1591161823130732966omplex ).

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

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_nat_complex > fun_co8378349014452122532t_real ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: set_Pr958786334691620121nt_int > fun_Pr803113274115125473t_bool ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: set_Pr4329608150637261639at_nat > fun_Pr1607443439640623155t_bool ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: set_Pr1261947904930325089at_nat > fun_Pr7757163693017731481t_bool ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: set_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: set_int > fun_int_bool ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: real > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: fun_Co6545587727236015215e_term > fun_Pr8569001801775958297r_bool ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: fun_Pr272337284026993017e_term > fun_Pr8004053055373584163r_bool ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: fun_Pr9164807099015279978e_term > fun_Pr8627135653929424660t_bool ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: fun_in6429348457993719614e_term > fun_Pr2579719123553071848t_bool ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: int > fun_in6148020733644367646t_bool ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: fun_re8661203372455305201t_bool ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: set_nat > fun_fu6463170612142064108al_nat ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: set_real > fun_fu3370445115705697296at_nat ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: set_nat > fun_fu8221675813083619692nt_nat ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: set_int > fun_fu5002972218919249680at_nat ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: set_nat > fun_fu4637029388598527722ex_nat ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: set_complex > fun_fu1817986637694864396at_nat ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: set_real > fun_fu7766971228835516944at_nat ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: set_nat > fun_fu8739491518516473836al_nat ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: set_int > fun_fu3078129256884107011t_bool ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: set_int > fun_fu2448115762823361296at_nat ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: fun_nat_fun_int_nat > fun_int_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: set_nat > fun_fu5666819356987731308nt_nat ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: set_complex > fun_fu1598814492381399564at_nat ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: fun_na8457291176974909128ex_nat > fun_co636942507509095880at_nat ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: set_nat > fun_fu1634715295319360746ex_nat ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: set_nat > fun_fu4970875099359191043t_bool ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: set_nat > fun_fu4246887593415901128at_nat ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: set_nat > fun_fu4246887593415901128at_nat ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: set_nat > fun_fu5815484198005934595t_bool ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: set_nat > fun_fu5592874601683281848at_nat ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: fun_li2771310203728655226at_nat > fun_na7671319193681171066at_nat ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: set_list_nat > fun_fu818882951641737987t_bool ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: set_list_nat > fun_fu1488094883385887144at_nat ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: set_nat > fun_fu6120145429055243651t_bool ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: set_nat > fun_fu8609932291560699922at_nat ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: fun_se1179353343636554400at_nat > fun_na106483476154090400at_nat ).

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: set_set_nat > fun_fu6121305534314677379t_bool ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: set_set_nat > fun_fu2948519294046016092at_nat ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: set_list_nat > fun_fu2993381634879179267t_bool ).

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: set_list_nat > fun_fu6251344290300103592at_nat ).

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: fun_na7671319193681171066at_nat > fun_li2771310203728655226at_nat ).

tff(sy_c_ATP_058Lamm__xz____,type,
    aTP_Lamm_xz: set_nat > fun_fu7989982881243375875t_bool ).

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: set_nat > fun_fu8233542310379399608at_nat ).

tff(sy_c_ATP_058Lamm__yb____,type,
    aTP_Lamm_yb: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yc____,type,
    aTP_Lamm_yc: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yd____,type,
    aTP_Lamm_yd: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__yf____,type,
    aTP_Lamm_yf: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yh____,type,
    aTP_Lamm_yh: fun_real_complex > fun_fu1393723446858488622omplex ).

tff(sy_c_ATP_058Lamm__yi____,type,
    aTP_Lamm_yi: fun_nat_complex > fun_fu7905158015251895726omplex ).

tff(sy_c_ATP_058Lamm__yj____,type,
    aTP_Lamm_yj: fun_int_complex > fun_fu6363802315473983150omplex ).

tff(sy_c_ATP_058Lamm__yk____,type,
    aTP_Lamm_yk: fun_complex_complex > fun_fu248857536038387758omplex ).

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: fun_real_real > fun_fu1760894646116856364l_real ).

tff(sy_c_ATP_058Lamm__ym____,type,
    aTP_Lamm_ym: fun_nat_real > fun_fu9152424228179946668t_real ).

tff(sy_c_ATP_058Lamm__yn____,type,
    aTP_Lamm_yn: fun_int_real > fun_fu6185597869953674668t_real ).

tff(sy_c_ATP_058Lamm__yo____,type,
    aTP_Lamm_yo: fun_complex_real > fun_fu1911030855436031788x_real ).

tff(sy_c_ATP_058Lamm__yp____,type,
    aTP_Lamm_yp: fun_real_rat > fun_fu7249360421009165912al_rat ).

tff(sy_c_ATP_058Lamm__yq____,type,
    aTP_Lamm_yq: fun_nat_rat > fun_fu2739651132436891864at_rat ).

tff(sy_c_ATP_058Lamm__yr____,type,
    aTP_Lamm_yr: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ys____,type,
    aTP_Lamm_ys: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__yt____,type,
    aTP_Lamm_yt: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__yu____,type,
    aTP_Lamm_yu: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__yv____,type,
    aTP_Lamm_yv: int > fun_in6148020733644367646t_bool ).

tff(sy_c_ATP_058Lamm__yw____,type,
    aTP_Lamm_yw: rat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__yx____,type,
    aTP_Lamm_yx: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__yy____,type,
    aTP_Lamm_yy: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__yz____,type,
    aTP_Lamm_yz: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__za____,type,
    aTP_Lamm_za: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__zb____,type,
    aTP_Lamm_zb: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__zc____,type,
    aTP_Lamm_zc: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__zd____,type,
    aTP_Lamm_zd: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__ze____,type,
    aTP_Lamm_ze: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__zf____,type,
    aTP_Lamm_zf: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__zg____,type,
    aTP_Lamm_zg: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__zh____,type,
    aTP_Lamm_zh: complex > fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__zi____,type,
    aTP_Lamm_zi: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__zj____,type,
    aTP_Lamm_zj: nat > fun_na5042103099923536755t_bool ).

tff(sy_c_ATP_058Lamm__zk____,type,
    aTP_Lamm_zk: code_integer > fun_Co3025358997185726883r_bool ).

tff(sy_c_ATP_058Lamm__zl____,type,
    aTP_Lamm_zl: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__zm____,type,
    aTP_Lamm_zm: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__zn____,type,
    aTP_Lamm_zn: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__zo____,type,
    aTP_Lamm_zo: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__zp____,type,
    aTP_Lamm_zp: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__zq____,type,
    aTP_Lamm_zq: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__zr____,type,
    aTP_Lamm_zr: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__zs____,type,
    aTP_Lamm_zs: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__zt____,type,
    aTP_Lamm_zt: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__zu____,type,
    aTP_Lamm_zu: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__zv____,type,
    aTP_Lamm_zv: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__zw____,type,
    aTP_Lamm_zw: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__zx____,type,
    aTP_Lamm_zx: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__zy____,type,
    aTP_Lamm_zy: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__zz____,type,
    aTP_Lamm_zz: nat > fun_na4143055713577223196at_nat ).

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_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__HOL__Obool_001t__HOL__Obool,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__HOL__Obool_J_001t__fun_It__Int__Oint_Mt__HOL__Obool_J,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__HOL__Obool_001t__HOL__Obool,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001t__HOL__Obool_001t__HOL__Obool,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__HOL__Obool_001t__HOL__Obool,type,
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tff(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
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tff(sy_c_Binomial_Obinomial,type,
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tff(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
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tff(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
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tff(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
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tff(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
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tff(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
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tff(sy_c_Bit__Operations_Oand__int__rel,type,
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tff(sy_c_Bit__Operations_Oand__not__num,type,
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tff(sy_c_Bit__Operations_Oand__not__num__rel,type,
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tff(sy_c_Bit__Operations_Oconcat__bit,type,
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tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
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tff(sy_c_Code__Numeral_Odivmod__abs,type,
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tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
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    linord2614967742042102400et_nat: set_nat > list_nat ).

tff(sy_c_List_Olist_OCons_001t__Int__Oint,type,
    cons_int: ( int * list_int ) > list_int ).

tff(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
    cons_nat: ( nat * list_nat ) > list_nat ).

tff(sy_c_List_Olist_ONil_001t__Int__Oint,type,
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tff(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
    hd_nat: list_nat > nat ).

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_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__Num__Onum,type,
    set_num2: list_num > set_num ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__HOL__Obool_Mt__HOL__Obool_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__HOL__Obool_Mt__Int__Oint_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__HOL__Obool_Mt__Nat__Onat_J,type,
    set_Pr4953607175514100673ol_nat: list_P18149847613930150ol_nat > set_Pr7141997262790060364ol_nat ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__HOL__Obool_Mt__Real__Oreal_J,type,
    set_Pr3516335181496812957l_real: list_P8975979458798387586l_real > set_Pr8860085461980347688l_real ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__HOL__Obool_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__HOL__Obool_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    set_Pr2267922651014402459nt_int: list_P8883597994762331264nt_int > set_Pr5033440245338114598nt_int ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    set_Pr7767296851990987887nt_int: list_P4447908290634120532nt_int > set_Pr233528076387193594nt_int ).

tff(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
    set_real2: list_real > set_real ).

tff(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
    set_set_nat2: list_set_nat > set_set_nat ).

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_Oset_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    set_fu7445659897732421274e_term: list_f4599320226441621759e_term > set_fu3062369701891493797e_term ).

tff(sy_c_List_Olist_Oset_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
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tff(sy_c_List_Olist_Oset_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
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tff(sy_c_List_Olist_Oset_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    set_fu4765906003373919125e_term: list_f8504123576994025594e_term > set_fu8544203214900410144e_term ).

tff(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
    size_list_VEBT_VEBT: ( fun_VEBT_VEBT_nat * list_VEBT_VEBT ) > nat ).

tff(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
    tl_nat: list_nat > list_nat ).

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__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    list_u2254550707601501961nteger: ( list_P5578671422887162913nteger * nat * produc8923325533196201883nteger ) > list_P5578671422887162913nteger ).

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    list_u3002344382305578791nt_int: ( list_P5707943133018811711nt_int * nat * product_prod_int_int ) > list_P5707943133018811711nt_int ).

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    list_u6180841689913720943at_nat: ( list_P6011104703257516679at_nat * nat * product_prod_nat_nat ) > list_P6011104703257516679at_nat ).

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    list_u4696772448584712917at_nat: ( list_P5464809261938338413at_nat * nat * produc3843707927480180839at_nat ) > list_P5464809261938338413at_nat ).

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    list_u3060434466275295603nteger: ( list_P5176709850099431537nteger * nat * produc5207004643709237345nteger ) > list_P5176709850099431537nteger ).

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    list_u6288444477985464578nt_int: ( list_P8883597994762331264nt_int * nat * produc6768802789813714928nt_int ) > list_P8883597994762331264nt_int ).

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
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tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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tff(sy_c_List_Olist__update_001t__Real__Oreal,type,
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tff(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
    list_update_set_nat: ( list_set_nat * nat * set_nat ) > list_set_nat ).

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_Olist__update_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
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tff(sy_c_List_Olist__update_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    list_u8051947491296383824e_term: ( list_f6829383385585334350e_term * nat * fun_in6429348457993719614e_term ) > list_f6829383385585334350e_term ).

tff(sy_c_List_Olist__update_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    list_u6547189781710427531e_term: ( list_f4295387047331035401e_term * nat * fun_Pr272337284026993017e_term ) > list_f4295387047331035401e_term ).

tff(sy_c_List_Olist__update_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
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tff(sy_c_List_Olistrel1_001t__HOL__Obool,type,
    listrel1_bool: set_Pr8621012493645324487l_bool > set_Pr4016434106243378759t_bool ).

tff(sy_c_List_Olistrel1_001t__Int__Oint,type,
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tff(sy_c_List_Olistrel1_001t__Nat__Onat,type,
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tff(sy_c_List_Olistrel1p_001t__Nat__Onat,type,
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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: fun_li2771310203728655226at_nat ).

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    nth_Pr3440142176431000676at_int: ( list_P3521021558325789923at_int * nat ) > product_prod_nat_int ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    nth_Pr7617993195940197384at_nat: ( list_P6011104703257516679at_nat * nat ) > product_prod_nat_nat ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr744662078594809490T_VEBT: ( list_P5647936690300460905T_VEBT * nat ) > produc8025551001238799321T_VEBT ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J,type,
    nth_Pr1289831921885633937T_bool: ( list_P44034876932920720T_bool * nat ) > produc6448083994118892042T_bool ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
    nth_Pr6837108013167703752BT_int: ( list_P4547456442757143711BT_int * nat ) > produc4894624898956917775BT_int ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    nth_Pr1791586995822124652BT_nat: ( list_P7037539587688870467BT_nat * nat ) > produc9072475918466114483BT_nat ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr4953567300277697838T_VEBT: ( list_P7413028617227757229T_VEBT * nat ) > produc8243902056947475879T_VEBT ).

tff(sy_c_List_Onth_001t__Real__Oreal,type,
    nth_real: list_real > fun_nat_real ).

tff(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
    nth_set_nat: ( list_set_nat * nat ) > set_nat ).

tff(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
    nth_VEBT_VEBT: ( list_VEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__HOL__Obool,type,
    product_bool_bool: ( list_bool * list_bool ) > list_P2722043638750504429l_bool ).

tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__Int__Oint,type,
    product_bool_int: ( list_bool * list_int ) > list_P6751438739536979202ol_int ).

tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__Nat__Onat,type,
    product_bool_nat: ( list_bool * list_nat ) > list_P18149847613930150ol_nat ).

tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__HOL__Obool,type,
    product_nat_bool: ( list_nat * list_bool ) > list_P6781292141273955916t_bool ).

tff(sy_c_List_Oproduct_001t__Nat__Onat_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__Int__Oint,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc7295137177222721919BT_nat: ( list_VEBT_VEBT * list_nat ) > list_P7037539587688870467BT_nat ).

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc4743750530478302277T_VEBT: ( list_VEBT_VEBT * list_VEBT_VEBT ) > list_P7413028617227757229T_VEBT ).

tff(sy_c_List_Oremdups_001t__Nat__Onat,type,
    remdups_nat: list_nat > list_nat ).

tff(sy_c_List_Oreplicate_001t__HOL__Obool,type,
    replicate_bool: ( nat * bool ) > list_bool ).

tff(sy_c_List_Oreplicate_001t__Int__Oint,type,
    replicate_int: ( nat * int ) > list_int ).

tff(sy_c_List_Oreplicate_001t__Nat__Onat,type,
    replicate_nat: ( nat * nat ) > list_nat ).

tff(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    replic4235873036481779905at_nat: ( nat * product_prod_nat_nat ) > list_P6011104703257516679at_nat ).

tff(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    replic2264142908078655527at_nat: ( nat * produc3843707927480180839at_nat ) > list_P5464809261938338413at_nat ).

tff(sy_c_List_Oreplicate_001t__Real__Oreal,type,
    replicate_real: ( nat * real ) > list_real ).

tff(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
    replicate_set_nat: ( nat * set_nat ) > list_set_nat ).

tff(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
    replicate_VEBT_VEBT: ( nat * vEBT_VEBT ) > list_VEBT_VEBT ).

tff(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
    sorted_wrt_int: fun_int_fun_int_bool > fun_list_int_bool ).

tff(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
    sorted_wrt_nat: fun_nat_fun_nat_bool > fun_list_nat_bool ).

tff(sy_c_List_Otake_001t__Nat__Onat,type,
    take_nat: ( nat * list_nat ) > list_nat ).

tff(sy_c_List_Oupt,type,
    upt: ( nat * nat ) > list_nat ).

tff(sy_c_List_Oupto,type,
    upto: ( int * int ) > list_int ).

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list_int ) > list_int ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun_Pr3099820731930151597t_bool ).

tff(sy_c_List_Ozip_001t__HOL__Obool_001t__HOL__Obool,type,
    zip_bool_bool: ( list_bool * list_bool ) > list_P2722043638750504429l_bool ).

tff(sy_c_List_Ozip_001t__HOL__Obool_001t__Int__Oint,type,
    zip_bool_int: ( list_bool * list_int ) > list_P6751438739536979202ol_int ).

tff(sy_c_List_Ozip_001t__HOL__Obool_001t__Nat__Onat,type,
    zip_bool_nat: ( list_bool * list_nat ) > list_P18149847613930150ol_nat ).

tff(sy_c_List_Ozip_001t__HOL__Obool_001t__Real__Oreal,type,
    zip_bool_real: ( list_bool * list_real ) > list_P8975979458798387586l_real ).

tff(sy_c_List_Ozip_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
    zip_bool_VEBT_VEBT: ( list_bool * list_VEBT_VEBT ) > list_P5912569763298001674T_VEBT ).

tff(sy_c_List_Ozip_001t__Int__Oint_001t__Int__Oint,type,
    zip_int_int: ( list_int * list_int ) > list_P5707943133018811711nt_int ).

tff(sy_c_List_Ozip_001t__Int__Oint_001t__Real__Oreal,type,
    zip_int_real: ( list_int * list_real ) > list_P6863124054624500543t_real ).

tff(sy_c_List_Ozip_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    zip_int_VEBT_VEBT: ( list_int * list_VEBT_VEBT ) > list_P7524865323317820941T_VEBT ).

tff(sy_c_List_Ozip_001t__Nat__Onat_001t__HOL__Obool,type,
    zip_nat_bool: ( list_nat * list_bool ) > list_P6781292141273955916t_bool ).

tff(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
    zip_nat_nat: ( list_nat * list_nat ) > list_P6011104703257516679at_nat ).

tff(sy_c_List_Ozip_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    zip_nat_VEBT_VEBT: ( list_nat * list_VEBT_VEBT ) > list_P5647936690300460905T_VEBT ).

tff(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    zip_Pr4664179122662387191at_nat: ( list_P6011104703257516679at_nat * list_P6011104703257516679at_nat ) > list_P8469869581646625389at_nat ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001t__HOL__Obool,type,
    zip_real_bool: ( list_real * list_bool ) > list_P3020538015149218416l_bool ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001t__Int__Oint,type,
    zip_real_int: ( list_real * list_int ) > list_P4344331454722006975al_int ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001t__Nat__Onat,type,
    zip_real_nat: ( list_real * list_nat ) > list_P6834414599653733731al_nat ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001t__Real__Oreal,type,
    zip_real_real: ( list_real * list_real ) > list_P8689742595348180415l_real ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    zip_real_VEBT_VEBT: ( list_real * list_VEBT_VEBT ) > list_P877281246627933069T_VEBT ).

tff(sy_c_List_Ozip_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    zip_set_nat_set_nat: ( list_set_nat * list_set_nat ) > list_P6254988961118846195et_nat ).

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
    zip_VEBT_VEBT_bool: ( list_VEBT_VEBT * list_bool ) > list_P44034876932920720T_bool ).

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    zip_VEBT_VEBT_int: ( list_VEBT_VEBT * list_int ) > list_P4547456442757143711BT_int ).

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    zip_VEBT_VEBT_nat: ( list_VEBT_VEBT * list_nat ) > list_P7037539587688870467BT_nat ).

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    zip_VEBT_VEBT_real: ( list_VEBT_VEBT * list_real ) > list_P2623026923184700063T_real ).

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    zip_VE537291747668921783T_VEBT: ( list_VEBT_VEBT * list_VEBT_VEBT ) > list_P7413028617227757229T_VEBT ).

tff(sy_c_List_Ozip_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    zip_fu734256677615756147nteger: ( list_f4599320226441621759e_term * list_P5578671422887162913nteger ) > list_P5176709850099431537nteger ).

tff(sy_c_List_Ozip_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    zip_fu9102111156773635586nt_int: ( list_f6829383385585334350e_term * list_P5707943133018811711nt_int ) > list_P8883597994762331264nt_int ).

tff(sy_c_List_Ozip_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    zip_fu197485079333405161nteger: ( list_f4295387047331035401e_term * list_P5578671422887162913nteger ) > list_P2496775790676384743nteger ).

tff(sy_c_List_Ozip_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    zip_fu2408773928738089174nt_int: ( list_f8504123576994025594e_term * list_P5707943133018811711nt_int ) > list_P4447908290634120532nt_int ).

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

tff(sy_c_Nat_Ocompow_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    compow_fun_nat_nat: ( nat * fun_nat_nat ) > fun_nat_nat ).

tff(sy_c_Nat_Onat_Ocase__nat_001t__HOL__Obool,type,
    case_nat_bool: ( bool * fun_nat_bool ) > fun_nat_bool ).

tff(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
    case_nat_nat: ( nat * fun_nat_nat ) > fun_nat_nat ).

tff(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
    case_nat_option_num: ( option_num * fun_nat_option_num * nat ) > option_num ).

tff(sy_c_Nat_Onat_Opred,type,
    pred: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: fun_nat_Code_integer ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
    semiri8010041392384452111omplex: fun_nat_complex ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
    semiri4216267220026989637d_enat: fun_na8797796813480665199d_enat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: fun_nat_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: fun_nat_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: fun_nat_real ).

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,
    semiri8420488043553186161ux_int: ( fun_int_int * nat ) > fun_int_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
    semiri8422978514062236437ux_nat: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
    semiri7787848453975740701ux_rat: ( fun_rat_rat * nat ) > fun_rat_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
    semiri7260567687927622513x_real: fun_real_real > fun_na6724076748692969250l_real ).

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__Product____Type__Oprod_It__HOL__Obool_Mt__HOL__Obool_J_J,type,
    size_s6988337448923680857l_bool: list_P2722043638750504429l_bool > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__HOL__Obool_Mt__Int__Oint_J_J,type,
    size_s753689718144262038ol_int: list_P6751438739536979202ol_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__HOL__Obool_Mt__Nat__Onat_J_J,type,
    size_s3243772863075988794ol_nat: list_P18149847613930150ol_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__HOL__Obool_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s7211631430880329078T_VEBT: list_P5912569763298001674T_VEBT > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__HOL__Obool_J_J,type,
    size_s783543119881238752t_bool: list_P6781292141273955916t_bool > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    size_s2970893825323803983at_int: list_P3521021558325789923at_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s4762443039079500285T_VEBT: list_P5647936690300460905T_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,
    size_s1343096544515248124T_bool: list_P44034876932920720T_bool > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
    size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).

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__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__Set__Oset_It__Nat__Onat_J_J,type,
    size_s3254054031482475050et_nat: list_set_nat > 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__Num__Onum_J,type,
    size_size_option_num: option_num > 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__VEBT____Definitions__OVEBT,type,
    size_size_VEBT_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: ( nat * nat ) > product_prod_nat_nat ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun_Pr248683899324676269t_bool ).

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_NthRoot_Oroot,type,
    root: fun_na6724076748692969250l_real ).

tff(sy_c_NthRoot_Osqrt,type,
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tff(sy_c_Num_OBitM,type,
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tff(sy_c_Num_Oinc,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
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tff(sy_c_Num_Onum_OBit0,type,
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tff(sy_c_Num_Onum_OBit1,type,
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tff(sy_c_Num_Onum_OOne,type,
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tff(sy_c_Num_Onum_Osize__num,type,
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tff(sy_c_Num_Onum__of__nat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
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tff(sy_c_Num_Opred__numeral,type,
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tff(sy_c_Option_Ooption_ONone_001t__Int__Oint,type,
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tff(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,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__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_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_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,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
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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_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_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
    power_power_complex: complex > fun_nat_complex ).

tff(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
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tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__Int__Oint,type,
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tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__Real__Oreal,type,
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tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Real__Oreal,type,
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tff(sy_c_Product__Type_OPair_001t__List__Olist_It__HOL__Obool_J_001t__List__Olist_It__HOL__Obool_J,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__HOL__Obool,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
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tff(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
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tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__HOL__Obool,type,
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tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__HOL__Obool,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_Rat_OAbs__Rat,type,
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tff(sy_c_Rat_Onormalize,type,
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tff(sy_c_Rat_Oof__int,type,
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tff(sy_c_Rat_Opcr__rat,type,
    pcr_rat: fun_Pr3145601080792579042t_bool ).

tff(sy_c_Rat_Opositive,type,
    positive: fun_rat_bool ).

tff(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod_int_int ).

tff(sy_c_Rat_Oratrel,type,
    ratrel: fun_Pr3099820731930151597t_bool ).

tff(sy_c_Real_OReal,type,
    real2: fun_nat_rat > real ).

tff(sy_c_Real_Ocauchy,type,
    cauchy: fun_nat_rat > $o ).

tff(sy_c_Real_Opcr__real,type,
    pcr_real: fun_fu990398353789569848l_bool ).

tff(sy_c_Real_Opositive,type,
    positive2: fun_real_bool ).

tff(sy_c_Real_Orealrel,type,
    realrel: fun_fu2636726645394866687t_bool ).

tff(sy_c_Real_Orep__real,type,
    rep_real: real > fun_nat_rat ).

tff(sy_c_Real_Ovanishes,type,
    vanishes: fun_nat_rat > $o ).

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,
    real_V975177566351809787t_real: fun_re7681423839302404990l_real ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
    real_V1022390504157884413omplex: fun_complex_real ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
    real_V7735802525324610683m_real: fun_real_real ).

tff(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
    real_V4546457046886955230omplex: fun_real_complex ).

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
    real_V2046097035970521341omplex: fun_re7208826350674946562omplex ).

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
    real_V1485227260804924795R_real: fun_re7681423839302404990l_real ).

tff(sy_c_Relation_OField_001t__Nat__Onat,type,
    field_nat: set_Pr1261947904930325089at_nat > set_nat ).

tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
    algebr932160517623751201me_int: fun_int_fun_int_bool ).

tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
    algebr934650988132801477me_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
    divide6298287555418463151nteger: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
    divide1717551699836669952omplex: complex > fun_complex_complex ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
    divide_divide_int: int > fun_int_int ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
    divide_divide_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
    divide_divide_rat: rat > fun_rat_rat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
    dvd_dvd_int: fun_int_fun_int_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
    dvd_dvd_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
    dvd_dvd_rat: fun_rat_fun_rat_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
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tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
    modulo_modulo_int: int > fun_int_int ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,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_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
    zero_n2052037380579107095ol_rat: bool > rat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
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tff(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
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tff(sy_c_Series_Osuminf_001t__Int__Oint,type,
    suminf_int: fun_nat_int > int ).

tff(sy_c_Series_Osuminf_001t__Nat__Onat,type,
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tff(sy_c_Series_Osuminf_001t__Real__Oreal,type,
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tff(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
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tff(sy_c_Series_Osummable_001t__Int__Oint,type,
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tff(sy_c_Series_Osummable_001t__Nat__Onat,type,
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tff(sy_c_Series_Osummable_001t__Real__Oreal,type,
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tff(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
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tff(sy_c_Series_Osums_001t__Int__Oint,type,
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tff(sy_c_Series_Osums_001t__Nat__Onat,type,
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tff(sy_c_Series_Osums_001t__Real__Oreal,type,
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tff(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
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tff(sy_c_Set_OCollect_001t__Int__Oint,type,
    collect_int: fun_int_bool > set_int ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
    collect_list_complex: fun_li356976171784545970x_bool > set_list_complex ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__HOL__Obool_J,type,
    collect_list_bool: fun_list_bool_bool > set_list_bool ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
    collect_list_int: fun_list_int_bool > set_list_int ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
    collect_list_nat: fun_list_nat_bool > set_list_nat ).

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tff(sy_c_Set_OCollect_001t__Nat__Onat,type,
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tff(sy_c_Set_OCollect_001t__Num__Onum,type,
    collect_num: fun_num_bool > set_num ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
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tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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tff(sy_c_Set_OCollect_001t__Rat__Orat,type,
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tff(sy_c_Set_OCollect_001t__Real__Oreal,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    collect_set_complex: fun_set_complex_bool > set_set_complex ).

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
    collect_set_int: fun_set_int_bool > set_set_int ).

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
    collect_set_nat: fun_set_nat_bool > set_set_nat ).

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Set_OPow_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
    image_int_int: ( fun_int_int * set_int ) > set_int ).

tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
    image_int_nat: ( fun_int_nat * set_int ) > set_nat ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
    image_nat_nat: ( fun_nat_nat * set_nat ) > set_nat ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    image_nat_set_nat: ( fun_nat_set_nat * set_nat ) > set_set_nat ).

tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,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__Complex__Ocomplex,type,
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tff(sy_c_Set_Oinsert_001t__Int__Oint,type,
    insert_int: ( int * set_int ) > set_int ).

tff(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
    insert_list_nat: ( list_nat * set_list_nat ) > set_list_nat ).

tff(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat: ( nat * set_nat ) > set_nat ).

tff(sy_c_Set_Oinsert_001t__Num__Onum,type,
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tff(sy_c_Set_Oinsert_001t__Rat__Orat,type,
    insert_rat: ( rat * set_rat ) > set_rat ).

tff(sy_c_Set_Oinsert_001t__Real__Oreal,type,
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tff(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_Oinsert_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Set_Ois__singleton_001t__Int__Oint,type,
    is_singleton_int: fun_set_int_bool ).

tff(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
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tff(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
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    set_or4548717258645045905et_nat: ( set_nat * set_nat ) > set_set_nat ).

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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Num__Onum,type,
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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Rat__Orat,type,
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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Int__Oint_J,type,
    set_or58775011639299419et_int: set_int > set_set_int ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or4236626031148496127et_nat: set_nat > set_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
    set_or6656581121297822940st_int: fun_in7040156458970095540et_int ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
    set_or6659071591806873216st_nat: ( nat * nat ) > set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
    set_or5832277885323065728an_int: fun_in7040156458970095540et_int ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
    set_or5834768355832116004an_nat: ( nat * nat ) > set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
    set_or1633881224788618240n_real: ( real * real ) > set_real ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
    set_or1210151606488870762an_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
    set_or5849166863359141190n_real: real > set_real ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
    set_ord_lessThan_int: int > set_int ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: nat > set_nat ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
    set_ord_lessThan_num: num > set_num ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
    set_ord_lessThan_rat: rat > set_rat ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
    set_or5984915006950818249n_real: real > set_real ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or890127255671739683et_nat: set_nat > set_set_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__Complex__Ocomplex,type,
    topolo6517432010174082258omplex: fun_nat_complex > $o ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
    topolo4055970368930404560y_real: fun_nat_real > $o ).

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

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__Complex__Ocomplex,type,
    cos_complex: fun_complex_complex ).

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__Complex__Ocomplex,type,
    cosh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Complex__Ocomplex,type,
    cot_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
    diffs_complex: fun_nat_complex > fun_nat_complex ).

tff(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
    diffs_real: fun_nat_real > fun_nat_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__Complex__Ocomplex,type,
    sin_complex: fun_complex_complex ).

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__Complex__Ocomplex,type,
    sinh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
    tan_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
    tanh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

tff(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
    transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

tff(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
    transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: ( bool * bool ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: ( option4927543243414619207at_nat * nat * list_VEBT_VEBT * vEBT_VEBT ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

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: vEBT_VEBT > fun_nat_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: vEBT_VEBT > fun_nat_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: vEBT_VEBT > fun_nat_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: vEBT_VEBT > fun_nat_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: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_bool ).

tff(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun_Pr817965127378462947t_bool ).

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: vEBT_VEBT > fun_nat_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: fun_se680526843285281907t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: fun_se680526843285281907t_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__Num__Onum,type,
    vEBT_V819420779217536731ft_num: ( fun_num_fun_num_num * option_num * option_num ) > option_num ).

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__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__Nat__Onat_Mt__Nat__Onat_J,type,
    accp_P4275260045618599050at_nat: fun_Pr248683899324676269t_bool > fun_Pr7757163693017731481t_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    accp_P3113834385874906142um_num: ( fun_Pr5837708615769641901m_bool * product_prod_num_num ) > $o ).

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__VEBT____Definitions__OVEBT,type,
    accp_VEBT_VEBT: fun_VE5877677516038742509T_bool > fun_VEBT_VEBT_bool ).

tff(sy_c_Wellfounded_Omax__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    max_ex8135407076693332796at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).

tff(sy_c_Wellfounded_Omin__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    min_ex6901939911449802026at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).

tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

tff(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
    wf_nat: fun_se8074476271188067385t_bool ).

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__HOL__Obool,type,
    aa_Code_integer_bool: ( fun_Co8822979462182099027r_bool * code_integer ) > bool ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    aa_Code_integer_int: ( fun_Code_integer_int * code_integer ) > int ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    aa_Code_integer_nat: ( fun_Code_integer_nat * code_integer ) > nat ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_Cod6977894654539884864nteger: ( fun_Co6597186368970283863nteger * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__HOL__Obool_J,type,
    aa_Cod2825494507867081354r_bool: ( fun_Co7333321365496032713r_bool * code_integer ) > produc7417142470700560111r_bool ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_Cod4161879620478718930nteger: ( fun_Co1981900195452921617nteger * code_integer ) > fun_Co8499238673658658103nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__HOL__Obool_J,type,
    aa_Cod7250484176784686200r_bool: ( fun_Co8310303586393106575r_bool * code_integer ) > fun_Co8822979462182099027r_bool ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Int__Oint_J,type,
    aa_Cod1242632294114111009er_int: ( fun_Co1092983284983208416er_int * code_integer ) > fun_Code_integer_int ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J,type,
    aa_Cod5420483313623307717er_nat: ( fun_Co3583066429914935172er_nat * code_integer ) > fun_Code_integer_nat ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_Cod8380929501330563836nteger: ( fun_Co2470855966376689427nteger * code_integer ) > fun_Co6597186368970283863nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__HOL__Obool_J_J,type,
    aa_Cod4981436349852597092r_bool: ( fun_Co3025358997185726883r_bool * code_integer ) > fun_Co7333321365496032713r_bool ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_Cod6397266357770349983nteger: ( fun_Co8502721031311864414nteger * code_integer ) > fun_Co6226381647958292740nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J,type,
    aa_Cod1578151519121547077nteger: ( fun_Co6226381647958292740nteger * code_integer ) > fun_nat_Code_integer ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Code____Numeral__Ointeger,type,
    aa_com4217951221296830281nteger: ( fun_co8153928105989009288nteger * complex ) > code_integer ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    aa_complex_complex: ( fun_complex_complex * complex ) > complex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__HOL__Obool,type,
    aa_complex_bool: ( fun_complex_bool * complex ) > bool ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Int__Oint,type,
    aa_complex_int: ( fun_complex_int * complex ) > int ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Nat__Onat,type,
    aa_complex_nat: ( fun_complex_nat * complex ) > nat ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Num__Onum,type,
    aa_complex_num: ( fun_complex_num * complex ) > num ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Rat__Orat,type,
    aa_complex_rat: ( fun_complex_rat * complex ) > rat ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    aa_complex_real: ( fun_complex_real * complex ) > real ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Complex__Ocomplex_Mt__HOL__Obool_J,type,
    aa_com1553036987509291606x_bool: ( fun_co1378794544173401837x_bool * complex ) > fun_complex_bool ).

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tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_001t__fun_It__Complex__Ocomplex_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J,type,
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tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_001t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J,type,
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tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J,type,
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tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_001t__fun_It__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J,type,
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tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_001t__fun_It__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
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tff(sy_c_member_001t__Product____Type__Oprod_It__HOL__Obool_Mt__Real__Oreal_J,type,
    member3680586897080482057l_real: ( produc5853794331844391154l_real * set_Pr8860085461980347688l_real ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__HOL__Obool_Mt__VEBT____Definitions__OVEBT_J,type,
    member5465154751058088621T_VEBT: ( produc8408751633324799108T_VEBT * set_Pr8097748915547837924T_VEBT ) > $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__Int__Oint_Mt__Real__Oreal_J,type,
    member2744130022092475746t_real: ( produc679980390762269497t_real * set_Pr3538720872664544793t_real ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
    member2056185340421749780T_VEBT: ( produc1531783533982839933T_VEBT * set_Pr8044002425091019955T_VEBT ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__HOL__Obool_J_Mt__List__Olist_It__HOL__Obool_J_J,type,
    member6300827366518646928t_bool: ( produc7619963554364372839t_bool * set_Pr4016434106243378759t_bool ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
    member6698963635872716290st_int: ( produc1186641810826059865st_int * set_Pr765067013931698361st_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member7340969449405702474st_nat: ( produc1828647624359046049st_nat * set_Pr3451248702717554689st_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
    member4439316823752958928T_VEBT: ( produc9211091688327510695T_VEBT * set_Pr1916528119006554503T_VEBT ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > fun_se8074476271188067385t_bool ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member8206827879206165904at_nat: ( produc859450856879609959at_nat * set_Pr8693737435421807431at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__HOL__Obool_J,type,
    member3295258257541327095l_bool: ( produc5468465692305236192l_bool * set_Pr2904644018331178518l_bool ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
    member1627681773268152802al_int: ( produc8786904178792722361al_int * set_Pr1019928272762051225al_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
    member5805532792777349510al_nat: ( produc3741383161447143261al_nat * set_Pr3510011417693777981al_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    member7849222048561428706l_real: ( produc2422161461964618553l_real * set_Pr6218003697084177305l_real ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
    member7262085504369356948T_VEBT: ( produc3757001726724277373T_VEBT * set_Pr6019664923565264691T_VEBT ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member8277197624267554838et_nat: ( produc7819656566062154093et_nat * set_Pr5488025237498180813et_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member8757157785044589968at_nat: produc3843707927480180839at_nat > fun_se8353504738346895699t_bool ).

tff(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J,type,
    member3504487111852181555T_bool: ( produc6448083994118892042T_bool * set_Pr2229214029182756970T_bool ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
    member5419026705395827622BT_int: ( produc4894624898956917775BT_int * set_Pr5066593544530342725BT_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    member373505688050248522BT_nat: ( produc9072475918466114483BT_nat * set_Pr7556676689462069481BT_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
    member8675245146396747942T_real: ( produc5170161368751668367T_real * set_Pr7765410600122031685T_real ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    member568628332442017744T_VEBT: ( produc8243902056947475879T_VEBT * set_Pr6192946355708809607T_VEBT ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    member4581088367259704696nteger: ( produc5207004643709237345nteger * set_Pr7592006421409863191nteger ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member500491123954006279nt_int: ( produc6768802789813714928nt_int * set_Pr5033440245338114598nt_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    member6423679177046879214nteger: ( produc1616345218596728535nteger * set_Pr5663546166630115725nteger ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member690604328116145115nt_int: ( produc4441460161354545860nt_int * set_Pr233528076387193594nt_int ) > $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__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > fun_set_set_nat_bool ).

tff(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
    member_VEBT_VEBT: fun_VE7346385682144900493T_bool ).

tff(sy_c_member_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    member4312719604404396038e_term: ( fun_Co6545587727236015215e_term * set_fu3062369701891493797e_term ) > $o ).

tff(sy_c_member_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    member7723273712176309973e_term: ( fun_in6429348457993719614e_term * set_fu6991988656872959732e_term ) > $o ).

tff(sy_c_member_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    member4565687819040644368e_term: ( fun_Pr272337284026993017e_term * set_fu2883028153058477999e_term ) > $o ).

tff(sy_c_member_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    member2948136515907314945e_term: ( fun_Pr9164807099015279978e_term * set_fu8544203214900410144e_term ) > $o ).

tff(sy_c_pp,type,
    pp: bool > $o ).

tff(sy_v_deg____,type,
    deg: nat ).

tff(sy_v_m____,type,
    m: nat ).

tff(sy_v_ma____,type,
    ma: nat ).

tff(sy_v_maxi____,type,
    maxi: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

tff(sy_v_xa____,type,
    xa: nat ).

tff(sy_v_ya____,type,
    ya: nat ).

% Relevant facts (10084)
tff(fact_0__C4_Ohyps_C_I3_J,axiom,
    m = na ).

% "4.hyps"(3)
tff(fact_1_xmi,axiom,
    xa != mi ).

% xmi
tff(fact_2_True,axiom,
    xa = ma ).

% True
tff(fact_3_valid__eq,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(T),D))
    <=> pp(aa_nat_bool(vEBT_invar_vebt(T),D)) ) ).

% valid_eq
tff(fact_4_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),D))
     => pp(aa_nat_bool(vEBT_VEBT_valid(T),D)) ) ).

% valid_eq1
tff(fact_5_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(T),D))
     => pp(aa_nat_bool(vEBT_invar_vebt(T),D)) ) ).

% valid_eq2
tff(fact_6_valid__0__not,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(vEBT_invar_vebt(T),zero_zero_nat)) ).

% valid_0_not
tff(fact_7_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(vEBT_invar_vebt(T),zero_zero_nat)) ).

% valid_tree_deg_neq_0
tff(fact_8__092_060open_062high_Ax_An_A_060_Alength_AtreeList_092_060close_062,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),size_s6755466524823107622T_VEBT(treeList))) ).

% \<open>high x n < length treeList\<close>
tff(fact_9__C4_Ohyps_C_I1_J,axiom,
    pp(aa_nat_bool(vEBT_invar_vebt(summary),m)) ).

% "4.hyps"(1)
tff(fact_10_False,axiom,
    ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na)))) ).

% False
tff(fact_11_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),N))
     => ( Deg = N ) ) ).

% deg_deg_n
tff(fact_12_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( vEBT_set_vebt(T) = vEBT_VEBT_set_vebt(T) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_13_notemp,axiom,
    ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na))),X_1)) ).

% notemp
tff(fact_14_hprolist,axiom,
    nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)) = vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na)) ).

% hprolist
tff(fact_15_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X_1)) ) ).

% not_min_Null_member
tff(fact_16_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_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_17_bit__split__inv,axiom,
    ! [X: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),D)),D) = X ).

% bit_split_inv
tff(fact_18__C4_OIH_C_I2_J,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_delete(summary,X)),Y))
    <=> ( ( X != Y )
        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,summary),Y)) ) ) ).

% "4.IH"(2)
tff(fact_19_mimapr,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,mi),ma)) ).

% mimapr
tff(fact_20__092_060open_062length_AtreeList_A_061_Alength_A_ItreeList_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_J_092_060close_062,axiom,
    size_s6755466524823107622T_VEBT(treeList) = size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na)))) ).

% \<open>length treeList = length (treeList[high x n := vebt_delete (treeList ! high x n) (low x n)])\<close>
tff(fact_21__C4_Ohyps_C_I7_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,mi),ma)) ).

% "4.hyps"(7)
tff(fact_22__C4_Ohyps_C_I6_J,axiom,
    ( ( mi = ma )
   => ! [X2: vEBT_VEBT] :
        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(treeList)))
       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_12)) ) ) ).

% "4.hyps"(6)
tff(fact_23__C4_OIH_C_I1_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(treeList)))
     => ( pp(aa_nat_bool(vEBT_invar_vebt(X2),na))
        & ! [Xa: nat,Xb: nat] :
            ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_delete(X2,Xa)),Xb))
          <=> ( ( Xa != Xb )
              & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),Xb)) ) ) ) ) ).

% "4.IH"(1)
tff(fact_24_in__children__def,axiom,
    ! [N: nat,TreeList: list_VEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(N,TreeList),X))
    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N))) ) ).

% in_children_def
tff(fact_25_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),I) = X ) ) ).

% nth_list_update_eq
tff(fact_26_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_bool,X: 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,X)),I))
      <=> pp(X) ) ) ).

% nth_list_update_eq
tff(fact_27_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,list_update_nat(Xs,I,X)),I) = X ) ) ).

% nth_list_update_eq
tff(fact_28_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),I) = X ) ) ).

% nth_list_update_eq
tff(fact_29_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(vEBT_invar_vebt(vEBT_vebt_buildup(N)),N)) ) ).

% buildup_gives_valid
tff(fact_30_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_nat,X: nat] :
      ( ( I != J )
     => ( aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,list_update_nat(Xs,I,X)),J) = aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_31_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_int,X: int] :
      ( ( I != J )
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),J) = aa_nat_int(nth_int(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_32_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( I != J )
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),J) = nth_VEBT_VEBT(Xs,J) ) ) ).

% nth_list_update_neq
tff(fact_33_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] : list_update_nat(Xs,I,aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,Xs),I)) = Xs ).

% list_update_id
tff(fact_34_list__update__id,axiom,
    ! [Xs: list_int,I: nat] : list_update_int(Xs,I,aa_nat_int(nth_int(Xs),I)) = Xs ).

% list_update_id
tff(fact_35_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat] : list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,I)) = Xs ).

% list_update_id
tff(fact_36_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_list_update
tff(fact_37_length__list__update,axiom,
    ! [Xs: list_bool,I: nat,X: bool] : size_size_list_bool(list_update_bool(Xs,I,X)) = size_size_list_bool(Xs) ).

% length_list_update
tff(fact_38_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,X: nat] : size_size_list_nat(list_update_nat(Xs,I,X)) = size_size_list_nat(Xs) ).

% length_list_update
tff(fact_39_length__list__update,axiom,
    ! [Xs: list_int,I: nat,X: int] : size_size_list_int(list_update_int(Xs,I,X)) = size_size_list_int(Xs) ).

% length_list_update
tff(fact_40_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A)) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_41_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_42_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_43_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_44__092_060open_062mi_A_092_060le_062_Ax_A_092_060and_062_Ax_A_092_060le_062_Ama_092_060close_062,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)) ) ).

% \<open>mi \<le> x \<and> x \<le> ma\<close>
tff(fact_45_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( ( ( I = J )
         => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),J) = X ) )
        & ( ( I != J )
         => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),J) = nth_VEBT_VEBT(Xs,J) ) ) ) ) ).

% nth_list_update
tff(fact_46_nth__list__update,axiom,
    ! [I: nat,Xs: list_bool,X: 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,X)),J))
      <=> ( ( ( I = J )
           => pp(X) )
          & ( ( I != J )
           => pp(aa_nat_bool(nth_bool(Xs),J)) ) ) ) ) ).

% nth_list_update
tff(fact_47_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( ( ( I = J )
         => ( aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,list_update_nat(Xs,I,X)),J) = X ) )
        & ( ( I != J )
         => ( aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,list_update_nat(Xs,I,X)),J) = aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_48_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,J: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( ( ( I = J )
         => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),J) = X ) )
        & ( ( I != J )
         => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),J) = aa_nat_int(nth_int(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_49_min__in__set__def,axiom,
    ! [Xs: set_nat,X: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(vEBT_VEBT_min_in_set,Xs),X))
    <=> ( pp(aa_set_nat_bool(member_nat(X),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,X),X3)) ) ) ) ).

% min_in_set_def
tff(fact_50_max__in__set__def,axiom,
    ! [Xs: set_nat,X: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(vEBT_VEBT_max_in_set,Xs),X))
    <=> ( pp(aa_set_nat_bool(member_nat(X),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),X)) ) ) ) ).

% max_in_set_def
tff(fact_51_inthall,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: fun_Pr7757163693017731481t_bool,N: nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),set_Pr5648618587558075414at_nat(Xs)))
         => pp(aa_Pro2232559814647311250t_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s5460976970255530739at_nat(Xs)))
       => pp(aa_Pro2232559814647311250t_bool(P,nth_Pr7617993195940197384at_nat(Xs,N))) ) ) ).

% inthall
tff(fact_52_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_53_inthall,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_bool,N: nat] :
      ( ! [X4: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X4),set_set_nat2(Xs)))
         => pp(aa_set_nat_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
       => pp(aa_set_nat_bool(P,nth_set_nat(Xs,N))) ) ) ).

% inthall
tff(fact_54_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,N: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_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,nth_VEBT_VEBT(Xs,N))) ) ) ).

% inthall
tff(fact_55_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_56_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(aa_lis1299702591413342011at_nat(nth_nat,Xs),N))) ) ) ).

% inthall
tff(fact_57_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_58_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT,Y: vEBT_VEBT] : list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),I,Y) = list_u1324408373059187874T_VEBT(Xs,I,Y) ).

% list_update_overwrite
tff(fact_59_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_60_le0,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% le0
tff(fact_61_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A)) ).

% bot_nat_0.extremum
tff(fact_62_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: 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,X) = Xs ) ) ).

% list_update_beyond
tff(fact_63_list__update__beyond,axiom,
    ! [Xs: list_bool,I: nat,X: 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,X) = Xs ) ) ).

% list_update_beyond
tff(fact_64_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X: 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,X) = Xs ) ) ).

% list_update_beyond
tff(fact_65_list__update__beyond,axiom,
    ! [Xs: list_int,I: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Xs)),I))
     => ( list_update_int(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_66_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: fun_Pr7757163693017731481t_bool] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),collec3392354462482085612at_nat(P)))
    <=> pp(aa_Pro2232559814647311250t_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_67_mem__Collect__eq,axiom,
    ! [A: real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(member_real(A),collect_real(P)))
    <=> pp(aa_real_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_68_mem__Collect__eq,axiom,
    ! [A: list_nat,P: fun_list_nat_bool] :
      ( member_list_nat(A,collect_list_nat(P))
    <=> pp(aa_list_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_69_mem__Collect__eq,axiom,
    ! [A: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),collect_set_nat(P)))
    <=> pp(aa_set_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_70_mem__Collect__eq,axiom,
    ! [A: nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(member_nat(A),collect_nat(P)))
    <=> pp(aa_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_71_mem__Collect__eq,axiom,
    ! [A: int,P: fun_int_bool] :
      ( pp(aa_set_int_bool(member_int(A),collect_int(P)))
    <=> pp(aa_int_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_72_Collect__mem__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : collec3392354462482085612at_nat(aTP_Lamm_a(A2)) = A2 ).

% Collect_mem_eq
tff(fact_73_Collect__mem__eq,axiom,
    ! [A2: set_real] : collect_real(aTP_Lamm_aa(A2)) = A2 ).

% Collect_mem_eq
tff(fact_74_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] : collect_list_nat(aTP_Lamm_ab(A2)) = A2 ).

% Collect_mem_eq
tff(fact_75_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] : collect_set_nat(aTP_Lamm_ac(A2)) = A2 ).

% Collect_mem_eq
tff(fact_76_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_ad,A2)) = A2 ).

% Collect_mem_eq
tff(fact_77_Collect__mem__eq,axiom,
    ! [A2: set_int] : collect_int(aTP_Lamm_ae(A2)) = A2 ).

% Collect_mem_eq
tff(fact_78_Collect__cong,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(P,X4))
        <=> pp(aa_real_bool(Q,X4)) )
     => ( collect_real(P) = collect_real(Q) ) ) ).

% Collect_cong
tff(fact_79_Collect__cong,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ! [X4: list_nat] :
          ( pp(aa_list_nat_bool(P,X4))
        <=> pp(aa_list_nat_bool(Q,X4)) )
     => ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_80_Collect__cong,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ! [X4: set_nat] :
          ( pp(aa_set_nat_bool(P,X4))
        <=> pp(aa_set_nat_bool(Q,X4)) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_81_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_82_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_83_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,nth_VEBT_VEBT(Xs,J)),J,nth_VEBT_VEBT(Xs,I))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_84_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_85_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(aa_lis1299702591413342011at_nat(nth_nat,Xs),J)),J,aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,Xs),I))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_86_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_int(Xs)))
       => ( set_int2(list_update_int(list_update_int(Xs,I,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),I))) = set_int2(Xs) ) ) ) ).

% set_swap
tff(fact_87_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y2: nat] :
            ( pp(aa_nat_bool(P,Y2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),B)) )
       => ? [X4: nat] :
            ( pp(aa_nat_bool(P,X4))
            & ! [Y3: nat] :
                ( pp(aa_nat_bool(P,Y3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X4)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_88_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_89_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_90_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_91_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_92_le__refl,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N)) ).

% le_refl
tff(fact_93_zero__le,axiom,
    ! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X)) ).

% zero_le
tff(fact_94_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_95_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_96_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
    <=> ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_97_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_98_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_99_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_100_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_101_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_102_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_103_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_104_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))
            & ! [I3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K2))
               => ~ pp(aa_nat_bool(P,I3)) )
            & pp(aa_nat_bool(P,K2)) ) ) ) ).

% ex_least_nat_le
tff(fact_105_length__pos__if__in__set,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s5460976970255530739at_nat(Xs))) ) ).

% length_pos_if_in_set
tff(fact_106_length__pos__if__in__set,axiom,
    ! [X: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(X),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_107_length__pos__if__in__set,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s3254054031482475050et_nat(Xs))) ) ).

% length_pos_if_in_set
tff(fact_108_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),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_109_length__pos__if__in__set,axiom,
    ! [X: bool,Xs: list_bool] :
      ( member_bool(X,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_110_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),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_111_length__pos__if__in__set,axiom,
    ! [X: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X),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_112_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_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,nth_VEBT_VEBT(Xs,I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_113_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_114_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(aa_lis1299702591413342011at_nat(nth_nat,Xs),I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_115_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_bool] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),set_int2(Xs)))
         => pp(aa_int_bool(P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_116_all__nth__imp__all__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: fun_Pr7757163693017731481t_bool,X: product_prod_nat_nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s5460976970255530739at_nat(Xs)))
         => pp(aa_Pro2232559814647311250t_bool(P,nth_Pr7617993195940197384at_nat(Xs,I2))) )
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
       => pp(aa_Pro2232559814647311250t_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_117_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_bool,X: 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(X),set_real2(Xs)))
       => pp(aa_real_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_118_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_bool,X: set_nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s3254054031482475050et_nat(Xs)))
         => pp(aa_set_nat_bool(P,nth_set_nat(Xs,I2))) )
     => ( pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(Xs)))
       => pp(aa_set_nat_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_119_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,X: 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,nth_VEBT_VEBT(Xs,I2))) )
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
       => pp(aa_VEBT_VEBT_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_120_all__nth__imp__all__set,axiom,
    ! [Xs: list_bool,P: fun_bool_bool,X: 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(X,set_bool2(Xs))
       => pp(aa_bool_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_121_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,X: 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(aa_lis1299702591413342011at_nat(nth_nat,Xs),I2))) )
     => ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
       => pp(aa_nat_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_122_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_bool,X: 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(X),set_int2(Xs)))
       => pp(aa_int_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_123_in__set__conv__nth,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s5460976970255530739at_nat(Xs)))
          & ( nth_Pr7617993195940197384at_nat(Xs,I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_124_in__set__conv__nth,axiom,
    ! [X: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(X),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) = X ) ) ) ).

% in_set_conv_nth
tff(fact_125_in__set__conv__nth,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s3254054031482475050et_nat(Xs)))
          & ( nth_set_nat(Xs,I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_126_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs)))
          & ( nth_VEBT_VEBT(Xs,I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_127_in__set__conv__nth,axiom,
    ! [X: bool,Xs: list_bool] :
      ( member_bool(X,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(X) ) ) ) ).

% in_set_conv_nth
tff(fact_128_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),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(aa_lis1299702591413342011at_nat(nth_nat,Xs),I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_129_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X),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) = X ) ) ) ).

% in_set_conv_nth
tff(fact_130_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(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Xs)))
           => pp(aa_VEBT_VEBT_bool(P,X4)) )
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,N))) ) ) ).

% list_ball_nth
tff(fact_131_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_132_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(aa_lis1299702591413342011at_nat(nth_nat,Xs),N))) ) ) ).

% list_ball_nth
tff(fact_133_list__ball__nth,axiom,
    ! [N: nat,Xs: list_int,P: fun_int_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs)))
           => pp(aa_int_bool(P,X4)) )
       => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_134_nth__mem,axiom,
    ! [N: nat,Xs: list_P6011104703257516679at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s5460976970255530739at_nat(Xs)))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(nth_Pr7617993195940197384at_nat(Xs,N)),set_Pr5648618587558075414at_nat(Xs))) ) ).

% nth_mem
tff(fact_135_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_136_nth__mem,axiom,
    ! [N: nat,Xs: list_set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
     => pp(aa_set_set_nat_bool(member_set_nat(nth_set_nat(Xs,N)),set_set_nat2(Xs))) ) ).

% nth_mem
tff(fact_137_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(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,nth_VEBT_VEBT(Xs,N)),set_VEBT_VEBT2(Xs))) ) ).

% nth_mem
tff(fact_138_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_139_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(aa_lis1299702591413342011at_nat(nth_nat,Xs),N)),set_nat2(Xs))) ) ).

% nth_mem
tff(fact_140_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_141_set__update__memI,axiom,
    ! [N: nat,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s5460976970255530739at_nat(Xs)))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_142_set__update__memI,axiom,
    ! [N: nat,Xs: list_real,X: 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(X),set_real2(list_update_real(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_143_set__update__memI,axiom,
    ! [N: nat,Xs: list_set_nat,X: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
     => pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(list_update_set_nat(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_144_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_145_set__update__memI,axiom,
    ! [N: nat,Xs: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => member_bool(X,set_bool2(list_update_bool(Xs,N,X))) ) ).

% set_update_memI
tff(fact_146_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X: 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(X),set_nat2(list_update_nat(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_147_set__update__memI,axiom,
    ! [N: nat,Xs: list_int,X: 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(X),set_int2(list_update_int(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_148_zero__reorient,axiom,
    ! [X: complex] :
      ( ( zero_zero_complex = X )
    <=> ( X = zero_zero_complex ) ) ).

% zero_reorient
tff(fact_149_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
    <=> ( X = zero_zero_real ) ) ).

% zero_reorient
tff(fact_150_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
    <=> ( X = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_151_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
    <=> ( X = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_152_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
    <=> ( X = zero_zero_int ) ) ).

% zero_reorient
tff(fact_153_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neqE_nat
tff(fact_154_infinite__descent,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ~ pp(aa_nat_bool(P,N2))
         => ? [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N2))
              & ~ pp(aa_nat_bool(P,M2)) ) )
     => pp(aa_nat_bool(P,N)) ) ).

% infinite_descent
tff(fact_155_nat__less__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N2))
             => pp(aa_nat_bool(P,M2)) )
         => pp(aa_nat_bool(P,N2)) )
     => pp(aa_nat_bool(P,N)) ) ).

% nat_less_induct
tff(fact_156_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_157_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S),T))
     => ( S != T ) ) ).

% less_not_refl3
tff(fact_158_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_159_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_160_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_161_size__neq__size__imp__neq,axiom,
    ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(X) != size_s6755466524823107622T_VEBT(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_162_size__neq__size__imp__neq,axiom,
    ! [X: list_bool,Y: list_bool] :
      ( ( size_size_list_bool(X) != size_size_list_bool(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_163_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( size_size_list_nat(X) != size_size_list_nat(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_164_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( size_size_list_int(X) != size_size_list_int(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_165_size__neq__size__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( aa_num_nat(size_size_num,X) != aa_num_nat(size_size_num,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_166_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_167_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_168_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_169_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) != size_size_list_int(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_170_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(Xs2) = N ).

% Ex_list_of_length
tff(fact_171_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_bool] : size_size_list_bool(Xs2) = N ).

% Ex_list_of_length
tff(fact_172_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] : size_size_list_nat(Xs2) = N ).

% Ex_list_of_length
tff(fact_173_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_int] : size_size_list_int(Xs2) = N ).

% Ex_list_of_length
tff(fact_174_list__update__swap,axiom,
    ! [I: nat,I5: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT,X5: vEBT_VEBT] :
      ( ( I != I5 )
     => ( list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),I5,X5) = list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I5,X5),I,X) ) ) ).

% list_update_swap
tff(fact_175_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_176_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_177_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_178_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_179_infinite__descent0,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( ~ pp(aa_nat_bool(P,N2))
             => ? [M2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N2))
                  & ~ pp(aa_nat_bool(P,M2)) ) ) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% infinite_descent0
tff(fact_180_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_181_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_182_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_183_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_184_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_185_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ).

% bot_nat_0.extremum_strict
tff(fact_186_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_187_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_188_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_189_length__induct,axiom,
    ! [P: fun_list_int_bool,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys2: list_int] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_int(Ys2)),size_size_list_int(Xs2)))
             => pp(aa_list_int_bool(P,Ys2)) )
         => pp(aa_list_int_bool(P,Xs2)) )
     => pp(aa_list_int_bool(P,Xs)) ) ).

% length_induct
tff(fact_190_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)))
           => ( nth_VEBT_VEBT(Xs,I4) = nth_VEBT_VEBT(Ys,I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_191_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_192_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(aa_lis1299702591413342011at_nat(nth_nat,Xs),I4) = aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_193_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
        & ! [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) = aa_nat_int(nth_int(Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_194_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_13: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I4),X_13)) )
    <=> ? [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),nth_VEBT_VEBT(Xs3,I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_195_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_13: bool] : pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I4),X_13)) )
    <=> ? [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_196_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_13: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I4),X_13)) )
    <=> ? [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(aa_lis1299702591413342011at_nat(nth_nat,Xs3),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_197_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_int_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
         => ? [X_13: int] : pp(aa_int_bool(aa_nat_fun_int_bool(P,I4),X_13)) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
             => pp(aa_int_bool(aa_nat_fun_int_bool(P,I4),aa_nat_int(nth_int(Xs3),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_198_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)))
           => ( nth_VEBT_VEBT(Xs,I2) = nth_VEBT_VEBT(Ys,I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_199_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_200_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(aa_lis1299702591413342011at_nat(nth_nat,Xs),I2) = aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_201_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
           => ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_202_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I,X) = Xs )
      <=> ( nth_VEBT_VEBT(Xs,I) = X ) ) ) ).

% list_update_same_conv
tff(fact_203_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( ( list_update_bool(Xs,I,X) = Xs )
      <=> ( pp(aa_nat_bool(nth_bool(Xs),I))
        <=> pp(X) ) ) ) ).

% list_update_same_conv
tff(fact_204_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( ( list_update_nat(Xs,I,X) = Xs )
      <=> ( aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,Xs),I) = X ) ) ) ).

% list_update_same_conv
tff(fact_205_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( ( list_update_int(Xs,I,X) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),I) = X ) ) ) ).

% list_update_same_conv
tff(fact_206_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] : ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_vebt_buildup(N)),X)) ).

% buildup_nothing_in_leaf
tff(fact_207_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(vEBT_vebt_member(T),X))
      <=> pp(aa_set_nat_bool(member_nat(X),vEBT_set_vebt(T))) ) ) ).

% member_correct
tff(fact_208_maxidef,axiom,
    some_nat(maxi) = vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na))) ).

% maxidef
tff(fact_209_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
      <=> pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ) ).

% both_member_options_equiv_member
tff(fact_210_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
       => pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ) ).

% valid_member_both_member_options
tff(fact_211_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] :
      ( ( vEBT_Node(X11,X12,X13,X14) = vEBT_Node(Y11,Y12,Y13,Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_212_buildup__gives__empty,axiom,
    ! [N: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(N)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_213_dual__order_Orefl,axiom,
    ! [A: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A),A)) ).

% dual_order.refl
tff(fact_214_dual__order_Orefl,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),A)) ).

% dual_order.refl
tff(fact_215_dual__order_Orefl,axiom,
    ! [A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),A)) ).

% dual_order.refl
tff(fact_216_dual__order_Orefl,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),A)) ).

% dual_order.refl
tff(fact_217_dual__order_Orefl,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A)) ).

% dual_order.refl
tff(fact_218_order__refl,axiom,
    ! [X: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(X),X)) ).

% order_refl
tff(fact_219_order__refl,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),X)) ).

% order_refl
tff(fact_220_order__refl,axiom,
    ! [X: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),X)) ).

% order_refl
tff(fact_221_order__refl,axiom,
    ! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),X)) ).

% order_refl
tff(fact_222_order__refl,axiom,
    ! [X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),X)) ).

% order_refl
tff(fact_223_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_vebt_buildup(N)),X)) ).

% buildup_nothing_in_min_max
tff(fact_224__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062maxi_O_ASome_Amaxi_A_061_Avebt__maxt_A_ItreeList_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Ahigh_Ax_An_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Maxi: nat] : some_nat(Maxi) != vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na))) ).

% \<open>\<And>thesis. (\<And>maxi. Some maxi = vebt_maxt (treeList[high x n := vebt_delete (treeList ! high x n) (low x n)] ! high x n) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_225_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_set_nat_bool(finite_finite_nat,vEBT_VEBT_set_vebt(T))) ) ).

% set_vebt_finite
tff(fact_226_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ~ pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ).

% min_Null_member
tff(fact_227_maxbmo,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_maxt(T) = some_nat(X) )
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X)) ) ).

% maxbmo
tff(fact_228_both__member__options__def,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
    <=> ( pp(aa_nat_bool(vEBT_V5719532721284313246member(T),X))
        | pp(aa_nat_bool(vEBT_VEBT_membermima(T),X)) ) ) ).

% both_member_options_def
tff(fact_229_maxt__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi2: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi2) )
       => pp(aa_nat_bool(vEBT_vebt_member(T),Maxi2)) ) ) ).

% maxt_member
tff(fact_230_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),N))
     => ( pp(aa_nat_bool(vEBT_vebt_member(Tree),X))
       => ( pp(aa_nat_bool(vEBT_V5719532721284313246member(Tree),X))
          | pp(aa_nat_bool(vEBT_VEBT_membermima(Tree),X)) ) ) ) ).

% member_valid_both_member_options
tff(fact_231_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi2: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi2) )
       => ( pp(aa_nat_bool(vEBT_vebt_member(T),X))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Maxi2)) ) ) ) ).

% maxt_corr_help
tff(fact_232_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(vEBT_VEBT_max_in_set,vEBT_VEBT_set_vebt(T)),X))
       => ( vEBT_vebt_maxt(T) = some_nat(X) ) ) ) ).

% maxt_sound
tff(fact_233_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_maxt(T) = some_nat(X) )
       => pp(aa_nat_bool(aa_set3744802623701456604t_bool(vEBT_VEBT_max_in_set,vEBT_VEBT_set_vebt(T)),X)) ) ) ).

% maxt_corr
tff(fact_234_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : finite5795047828879050333T_VEBT(set_VEBT_VEBT2(Xs)) ).

% List.finite_set
tff(fact_235_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : pp(aa_set_nat_bool(finite_finite_nat,set_nat2(Xs))) ).

% List.finite_set
tff(fact_236_List_Ofinite__set,axiom,
    ! [Xs: list_int] : pp(aa_set_int_bool(finite_finite_int,set_int2(Xs))) ).

% List.finite_set
tff(fact_237_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : pp(aa_set_complex_bool(finite3207457112153483333omplex,set_complex2(Xs))) ).

% List.finite_set
tff(fact_238_List_Ofinite__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat] : pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,set_Pr5648618587558075414at_nat(Xs))) ).

% List.finite_set
tff(fact_239_pred__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(Xs),A),X_1))
     => ( pp(aa_set_nat_bool(finite_finite_nat,Xs))
       => ~ ? [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),Xs))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),A)) ) ) ) ).

% pred_none_empty
tff(fact_240_succ__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(Xs),A),X_1))
     => ( pp(aa_set_nat_bool(finite_finite_nat,Xs))
       => ~ ? [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),Xs))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),X2)) ) ) ) ).

% succ_none_empty
tff(fact_241_obtain__set__pred,axiom,
    ! [Z: nat,X: nat,A2: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X))
     => ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(vEBT_VEBT_min_in_set,A2),Z))
       => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
         => ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(A2),X),X_1)) ) ) ) ).

% obtain_set_pred
tff(fact_242_obtain__set__succ,axiom,
    ! [X: nat,Z: nat,A2: set_nat,B2: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z))
     => ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(vEBT_VEBT_max_in_set,A2),Z))
       => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
         => ( ( A2 = B2 )
           => ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(A2),X),X_1)) ) ) ) ) ).

% obtain_set_succ
tff(fact_243_lesseq__shift,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,some_nat(X)),some_nat(Y))) ) ).

% lesseq_shift
tff(fact_244_pred__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T)),X),Y))
    <=> ( pp(aa_nat_bool(vEBT_vebt_member(T),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
        & ! [Z2: nat] :
            ( ( pp(aa_nat_bool(vEBT_vebt_member(T),Z2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z2),Y)) ) ) ) ).

% pred_member
tff(fact_245_succ__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T)),X),Y))
    <=> ( pp(aa_nat_bool(vEBT_vebt_member(T),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        & ! [Z2: nat] :
            ( ( pp(aa_nat_bool(vEBT_vebt_member(T),Z2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z2)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z2)) ) ) ) ).

% succ_member
tff(fact_246_bot_Oextremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(bot_bo2099793752762293965at_nat),A)) ).

% bot.extremum
tff(fact_247_bot_Oextremum,axiom,
    ! [A: set_real] : pp(aa_set_real_bool(ord_less_eq_set_real(bot_bot_set_real),A)) ).

% bot.extremum
tff(fact_248_bot_Oextremum,axiom,
    ! [A: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,bot_bot_set_nat),A)) ).

% bot.extremum
tff(fact_249_bot_Oextremum,axiom,
    ! [A: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(bot_bot_set_int),A)) ).

% bot.extremum
tff(fact_250_bot_Oextremum,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,bot_bot_nat),A)) ).

% bot.extremum
tff(fact_251_bot_Oextremum__unique,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A),bot_bo2099793752762293965at_nat))
    <=> ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_unique
tff(fact_252_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),bot_bot_set_real))
    <=> ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
tff(fact_253_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),bot_bot_set_nat))
    <=> ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_254_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),bot_bot_set_int))
    <=> ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_255_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),bot_bot_nat))
    <=> ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_256_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A),bot_bo2099793752762293965at_nat))
     => ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_257_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),bot_bot_set_real))
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
tff(fact_258_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),bot_bot_set_nat))
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_259_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),bot_bot_set_int))
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_260_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),bot_bot_nat))
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_261_bot_Oextremum__strict,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ~ pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A),bot_bo2099793752762293965at_nat)) ).

% bot.extremum_strict
tff(fact_262_bot_Oextremum__strict,axiom,
    ! [A: set_real] : ~ pp(aa_set_real_bool(ord_less_set_real(A),bot_bot_set_real)) ).

% bot.extremum_strict
tff(fact_263_bot_Oextremum__strict,axiom,
    ! [A: set_nat] : ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A),bot_bot_set_nat)) ).

% bot.extremum_strict
tff(fact_264_bot_Oextremum__strict,axiom,
    ! [A: set_int] : ~ pp(aa_set_int_bool(ord_less_set_int(A),bot_bot_set_int)) ).

% bot.extremum_strict
tff(fact_265_bot_Oextremum__strict,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),bot_bot_nat)) ).

% bot.extremum_strict
tff(fact_266_bot_Onot__eq__extremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( A != bot_bo2099793752762293965at_nat )
    <=> pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(bot_bo2099793752762293965at_nat),A)) ) ).

% bot.not_eq_extremum
tff(fact_267_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_set_real(bot_bot_set_real),A)) ) ).

% bot.not_eq_extremum
tff(fact_268_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,bot_bot_set_nat),A)) ) ).

% bot.not_eq_extremum
tff(fact_269_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_set_int(bot_bot_set_int),A)) ) ).

% bot.not_eq_extremum
tff(fact_270_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,bot_bot_nat),A)) ) ).

% bot.not_eq_extremum
tff(fact_271_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A2)
     => ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A2 ) ).

% finite_list
tff(fact_272_finite__list,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ? [Xs2: list_nat] : set_nat2(Xs2) = A2 ) ).

% finite_list
tff(fact_273_finite__list,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ? [Xs2: list_int] : set_int2(Xs2) = A2 ) ).

% finite_list
tff(fact_274_finite__list,axiom,
    ! [A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ? [Xs2: list_complex] : set_complex2(Xs2) = A2 ) ).

% finite_list
tff(fact_275_finite__list,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
     => ? [Xs2: list_P6011104703257516679at_nat] : set_Pr5648618587558075414at_nat(Xs2) = A2 ) ).

% finite_list
tff(fact_276_subset__code_I1_J,axiom,
    ! [Xs: list_P6011104703257516679at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Xs)),B2))
    <=> ! [X3: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),set_Pr5648618587558075414at_nat(Xs)))
         => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),B2)) ) ) ).

% subset_code(1)
tff(fact_277_subset__code_I1_J,axiom,
    ! [Xs: list_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(Xs)),B2))
    <=> ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),set_real2(Xs)))
         => pp(aa_set_real_bool(member_real(X3),B2)) ) ) ).

% subset_code(1)
tff(fact_278_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),B2))
    <=> ! [X3: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(Xs)))
         => pp(aa_set_set_nat_bool(member_set_nat(X3),B2)) ) ) ).

% subset_code(1)
tff(fact_279_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),B2)
    <=> ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
         => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),B2)) ) ) ).

% subset_code(1)
tff(fact_280_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Xs)),B2))
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
         => pp(aa_set_nat_bool(member_nat(X3),B2)) ) ) ).

% subset_code(1)
tff(fact_281_subset__code_I1_J,axiom,
    ! [Xs: list_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),B2))
    <=> ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),set_int2(Xs)))
         => pp(aa_set_int_bool(member_int(X3),B2)) ) ) ).

% subset_code(1)
tff(fact_282_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Node(Uu,zero_zero_nat,Uv,Uw)),Ux)) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_283_set__update__subsetI,axiom,
    ! [Xs: list_P6011104703257516679at_nat,A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,I: nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Xs)),A2))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2))
       => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,I,X))),A2)) ) ) ).

% set_update_subsetI
tff(fact_284_set__update__subsetI,axiom,
    ! [Xs: list_real,A2: set_real,X: real,I: nat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(Xs)),A2))
     => ( pp(aa_set_real_bool(member_real(X),A2))
       => pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,X))),A2)) ) ) ).

% set_update_subsetI
tff(fact_285_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A2: set_set_nat,X: set_nat,I: nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),A2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
       => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_set_nat2(list_update_set_nat(Xs,I,X))),A2)) ) ) ).

% set_update_subsetI
tff(fact_286_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,X: nat,I: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Xs)),A2))
     => ( pp(aa_set_nat_bool(member_nat(X),A2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(list_update_nat(Xs,I,X))),A2)) ) ) ).

% set_update_subsetI
tff(fact_287_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),A2)
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
       => ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,X)),A2) ) ) ).

% set_update_subsetI
tff(fact_288_set__update__subsetI,axiom,
    ! [Xs: list_int,A2: set_int,X: int,I: nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),A2))
     => ( pp(aa_set_int_bool(member_int(X),A2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X))),A2)) ) ) ).

% set_update_subsetI
tff(fact_289_nle__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & ( B != A ) ) ) ).

% nle_le
tff(fact_290_nle__le,axiom,
    ! [A: num,B: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & ( B != A ) ) ) ).

% nle_le
tff(fact_291_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & ( B != A ) ) ) ).

% nle_le
tff(fact_292_nle__le,axiom,
    ! [A: int,B: 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_eq_int,B),A))
        & ( B != A ) ) ) ).

% nle_le
tff(fact_293_le__cases3,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
       => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) )
     => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z)) )
       => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z))
           => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y)) )
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y))
             => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) )
           => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z))
               => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X)) )
             => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X))
                 => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_294_le__cases3,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
       => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z)) )
     => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Z)) )
       => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Z))
           => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y)) )
         => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y))
             => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) )
           => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z))
               => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X)) )
             => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X))
                 => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_295_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
       => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z)) )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z)) )
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z))
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y)) )
         => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y))
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) )
           => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
               => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X)) )
             => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X))
                 => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_296_le__cases3,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
       => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z)) )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z)) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y)) )
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y))
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) )
           => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
               => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X)) )
             => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X))
                 => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_297_order__class_Oorder__eq__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
        & pp(aa_set_int_bool(ord_less_eq_set_int(Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_298_order__class_Oorder__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_299_order__class_Oorder__eq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_300_order__class_Oorder__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_301_order__class_Oorder__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_302_ord__eq__le__trans,axiom,
    ! [A: set_int,B: set_int,C: 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)) ) ) ).

% ord_eq_le_trans
tff(fact_303_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_304_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_305_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_306_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_307_ord__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( ( B = C )
       => pp(aa_set_int_bool(ord_less_eq_set_int(A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_308_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( B = C )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_309_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( B = C )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_310_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( B = C )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_311_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( ( B = C )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_312_order__antisym,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_313_order__antisym,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_314_order__antisym,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_315_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_316_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_317_order_Otrans,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)) ) ) ).

% order.trans
tff(fact_318_order_Otrans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C)) ) ) ).

% order.trans
tff(fact_319_order_Otrans,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C)) ) ) ).

% order.trans
tff(fact_320_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C)) ) ) ).

% order.trans
tff(fact_321_order_Otrans,axiom,
    ! [A: int,B: int,C: 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_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C)) ) ) ).

% order.trans
tff(fact_322_order__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),Z))
       => pp(aa_set_int_bool(ord_less_eq_set_int(X),Z)) ) ) ).

% order_trans
tff(fact_323_order__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z)) ) ) ).

% order_trans
tff(fact_324_order__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Z)) ) ) ).

% order_trans
tff(fact_325_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z)) ) ) ).

% order_trans
tff(fact_326_order__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z)) ) ) ).

% order_trans
tff(fact_327_linorder__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,A: rat,B: 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,A),B)) ) ) ).

% linorder_wlog
tff(fact_328_linorder__wlog,axiom,
    ! [P: fun_num_fun_num_bool,A: num,B: 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,A),B)) ) ) ).

% linorder_wlog
tff(fact_329_linorder__wlog,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B: 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,A),B)) ) ) ).

% linorder_wlog
tff(fact_330_linorder__wlog,axiom,
    ! [P: fun_int_fun_int_bool,A: int,B: 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,A),B)) ) ) ).

% linorder_wlog
tff(fact_331_dual__order_Oeq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
        & pp(aa_set_int_bool(ord_less_eq_set_int(A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_332_dual__order_Oeq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_333_dual__order_Oeq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_334_dual__order_Oeq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_335_dual__order_Oeq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_336_dual__order_Oantisym,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_337_dual__order_Oantisym,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_338_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_339_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_340_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_341_dual__order_Otrans,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(C),B))
       => pp(aa_set_int_bool(ord_less_eq_set_int(C),A)) ) ) ).

% dual_order.trans
tff(fact_342_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A)) ) ) ).

% dual_order.trans
tff(fact_343_dual__order_Otrans,axiom,
    ! [B: num,A: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A)) ) ) ).

% dual_order.trans
tff(fact_344_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).

% dual_order.trans
tff(fact_345_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A)) ) ) ).

% dual_order.trans
tff(fact_346_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 ) ) ) ).

% antisym
tff(fact_347_antisym,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_348_antisym,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_349_antisym,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_350_antisym,axiom,
    ! [A: int,B: 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_eq_int,B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_351_Orderings_Oorder__eq__iff,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)) ) ) ).

% Orderings.order_eq_iff
tff(fact_352_Orderings_Oorder__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_353_Orderings_Oorder__eq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_354_Orderings_Oorder__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_355_Orderings_Oorder__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( 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_eq_int,B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_356_order__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_357_order__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_358_order__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_359_order__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => ( ! [X4: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_360_order__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_361_order__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_362_order__subst1,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_363_order__subst1,axiom,
    ! [A: num,F: fun_int_num,B: int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_int_num(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => ( ! [X4: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F,X4)),aa_int_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_int_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_364_order__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% order_subst1
tff(fact_365_order__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F,C))) ) ) ) ).

% order_subst1
tff(fact_366_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,B)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_367_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,B)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_368_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,B)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_369_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,B)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_370_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,B)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_371_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,B)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_372_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,B)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_373_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,B)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_374_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,B)),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_375_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,B)),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_376_order__eq__refl,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
     => pp(aa_set_int_bool(ord_less_eq_set_int(X),Y)) ) ).

% order_eq_refl
tff(fact_377_order__eq__refl,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% order_eq_refl
tff(fact_378_order__eq__refl,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% order_eq_refl
tff(fact_379_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% order_eq_refl
tff(fact_380_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% order_eq_refl
tff(fact_381_linorder__linear,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).

% linorder_linear
tff(fact_382_linorder__linear,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% linorder_linear
tff(fact_383_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_linear
tff(fact_384_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_linear
tff(fact_385_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_386_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_387_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_388_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_rat_int(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_389_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( ( A = aa_num_rat(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_390_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( ( A = aa_num_num(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_391_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( ( A = aa_num_nat(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_392_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_num_int,B: num,C: num] :
      ( ( A = aa_num_int(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_num_int(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_393_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( ( A = aa_nat_rat(F,B) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_394_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C: nat] :
      ( ( A = aa_nat_num(F,B) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_395_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_396_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_397_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_398_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_399_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_rat(F,B) = C )
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_400_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_num(F,B) = C )
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_401_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_nat(F,B) = C )
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_402_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_int(F,B) = C )
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_403_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( aa_nat_rat(F,B) = C )
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_404_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( aa_nat_num(F,B) = C )
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_405_linorder__le__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).

% linorder_le_cases
tff(fact_406_linorder__le__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% linorder_le_cases
tff(fact_407_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_le_cases
tff(fact_408_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_le_cases
tff(fact_409_order__antisym__conv,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_410_order__antisym__conv,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_411_order__antisym__conv,axiom,
    ! [Y: num,X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_412_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_413_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_414_lt__ex,axiom,
    ! [X: real] :
    ? [Y2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),X)) ).

% lt_ex
tff(fact_415_lt__ex,axiom,
    ! [X: rat] :
    ? [Y2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y2),X)) ).

% lt_ex
tff(fact_416_lt__ex,axiom,
    ! [X: int] :
    ? [Y2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y2),X)) ).

% lt_ex
tff(fact_417_gt__ex,axiom,
    ! [X: real] :
    ? [X_1: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X_1)) ).

% gt_ex
tff(fact_418_gt__ex,axiom,
    ! [X: rat] :
    ? [X_1: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),X_1)) ).

% gt_ex
tff(fact_419_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X_1)) ).

% gt_ex
tff(fact_420_gt__ex,axiom,
    ! [X: int] :
    ? [X_1: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),X_1)) ).

% gt_ex
tff(fact_421_dense,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ? [Z3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z3))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),Y)) ) ) ).

% dense
tff(fact_422_dense,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ? [Z3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z3))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),Y)) ) ) ).

% dense
tff(fact_423_less__imp__neq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_424_less__imp__neq,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_425_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_426_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_427_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_428_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ).

% order.asym
tff(fact_429_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).

% order.asym
tff(fact_430_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A)) ) ).

% order.asym
tff(fact_431_order_Oasym,axiom,
    ! [A: nat,B: 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,B),A)) ) ).

% order.asym
tff(fact_432_order_Oasym,axiom,
    ! [A: int,B: 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,B),A)) ) ).

% order.asym
tff(fact_433_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_434_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_435_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_436_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_437_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_438_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( B = C )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_439_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( B = C )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_440_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( ( B = C )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_441_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( ( B = C )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_442_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( ( B = C )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_443_less__induct,axiom,
    ! [P: fun_nat_bool,A: nat] :
      ( ! [X4: nat] :
          ( ! [Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),X4))
             => pp(aa_nat_bool(P,Y3)) )
         => pp(aa_nat_bool(P,X4)) )
     => pp(aa_nat_bool(P,A)) ) ).

% less_induct
tff(fact_444_antisym__conv3,axiom,
    ! [Y: real,X: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_445_antisym__conv3,axiom,
    ! [Y: rat,X: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_446_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_447_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_448_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_449_linorder__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( ( X != Y )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_cases
tff(fact_450_linorder__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( ( X != Y )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_cases
tff(fact_451_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( ( X != Y )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_cases
tff(fact_452_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( ( X != Y )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_cases
tff(fact_453_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( ( X != Y )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_cases
tff(fact_454_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% dual_order.asym
tff(fact_455_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% dual_order.asym
tff(fact_456_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ).

% dual_order.asym
tff(fact_457_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% dual_order.asym
tff(fact_458_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% dual_order.asym
tff(fact_459_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),A)) ).

% dual_order.irrefl
tff(fact_460_dual__order_Oirrefl,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),A)) ).

% dual_order.irrefl
tff(fact_461_dual__order_Oirrefl,axiom,
    ! [A: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),A)) ).

% dual_order.irrefl
tff(fact_462_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A)) ).

% dual_order.irrefl
tff(fact_463_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),A)) ).

% dual_order.irrefl
tff(fact_464_exists__least__iff,axiom,
    ! [P: fun_nat_bool] :
      ( ? [X_13: nat] : pp(aa_nat_bool(P,X_13))
    <=> ? [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)) ) ) ) ).

% exists_least_iff
tff(fact_465_linorder__less__wlog,axiom,
    ! [P: fun_re1255401033344146673l_bool,A: real,B: 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,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_466_linorder__less__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,A: rat,B: 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,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_467_linorder__less__wlog,axiom,
    ! [P: fun_num_fun_num_bool,A: num,B: 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,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_468_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B: 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,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_469_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_bool,A: int,B: 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,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_470_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans
tff(fact_471_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% order.strict_trans
tff(fact_472_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% order.strict_trans
tff(fact_473_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: 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,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans
tff(fact_474_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: 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,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans
tff(fact_475_not__less__iff__gr__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_476_not__less__iff__gr__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_477_not__less__iff__gr__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_478_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_479_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_480_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_481_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_482_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_483_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_484_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_485_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_486_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_487_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_488_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_489_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_490_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_491_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_492_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_493_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_494_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_495_linorder__neqE,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_neqE
tff(fact_496_linorder__neqE,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_neqE
tff(fact_497_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_neqE
tff(fact_498_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neqE
tff(fact_499_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neqE
tff(fact_500_order__less__asym,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% order_less_asym
tff(fact_501_order__less__asym,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% order_less_asym
tff(fact_502_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_asym
tff(fact_503_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_asym
tff(fact_504_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_asym
tff(fact_505_linorder__neq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_506_linorder__neq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_507_linorder__neq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_508_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_509_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_510_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ).

% order_less_asym'
tff(fact_511_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).

% order_less_asym'
tff(fact_512_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A)) ) ).

% order_less_asym'
tff(fact_513_order__less__asym_H,axiom,
    ! [A: nat,B: 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,B),A)) ) ).

% order_less_asym'
tff(fact_514_order__less__asym_H,axiom,
    ! [A: int,B: 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,B),A)) ) ).

% order_less_asym'
tff(fact_515_order__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z)) ) ) ).

% order_less_trans
tff(fact_516_order__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z)) ) ) ).

% order_less_trans
tff(fact_517_order__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).

% order_less_trans
tff(fact_518_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z)) ) ) ).

% order_less_trans
tff(fact_519_order__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z)) ) ) ).

% order_less_trans
tff(fact_520_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( ( A = aa_real_real(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_521_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( ( A = aa_real_rat(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_522_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_real_num,B: real,C: real] :
      ( ( A = aa_real_num(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_real_num(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_523_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_real_nat,B: real,C: real] :
      ( ( A = aa_real_nat(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_real_nat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_524_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_real_int,B: real,C: real] :
      ( ( A = aa_real_int(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_real_int(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_525_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( ( A = aa_rat_real(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_526_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_527_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_528_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_529_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_530_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_real(F,B) = C )
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_531_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_rat(F,B) = C )
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_532_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_num(F,B) = C )
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_533_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_nat(F,B) = C )
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_534_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_int(F,B) = C )
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_535_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_real(F,B) = C )
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_536_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_537_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_538_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_539_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_540_order__less__irrefl,axiom,
    ! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X)) ).

% order_less_irrefl
tff(fact_541_order__less__irrefl,axiom,
    ! [X: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),X)) ).

% order_less_irrefl
tff(fact_542_order__less__irrefl,axiom,
    ! [X: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),X)) ).

% order_less_irrefl
tff(fact_543_order__less__irrefl,axiom,
    ! [X: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X)) ).

% order_less_irrefl
tff(fact_544_order__less__irrefl,axiom,
    ! [X: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),X)) ).

% order_less_irrefl
tff(fact_545_order__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_546_order__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_547_order__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_548_order__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_549_order__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => ( ! [X4: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_550_order__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_551_order__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_552_order__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_553_order__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_554_order__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => ( ! [X4: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_555_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,B)),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_556_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,B)),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_557_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,B)),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_558_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,B)),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_559_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,B)),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_560_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,B)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_561_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,B)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_562_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,B)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_563_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,B)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_564_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,B)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_565_order__less__not__sym,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% order_less_not_sym
tff(fact_566_order__less__not__sym,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% order_less_not_sym
tff(fact_567_order__less__not__sym,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_not_sym
tff(fact_568_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_not_sym
tff(fact_569_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_not_sym
tff(fact_570_order__less__imp__triv,axiom,
    ! [X: real,Y: real,P: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_571_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,P: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_572_order__less__imp__triv,axiom,
    ! [X: num,Y: num,P: bool] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_573_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_574_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_575_linorder__less__linear,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
      | ( X = Y )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% linorder_less_linear
tff(fact_576_linorder__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
      | ( X = Y )
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% linorder_less_linear
tff(fact_577_linorder__less__linear,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      | ( X = Y )
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_less_linear
tff(fact_578_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      | ( X = Y )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_less_linear
tff(fact_579_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      | ( X = Y )
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_less_linear
tff(fact_580_order__less__imp__not__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_581_order__less__imp__not__eq,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_582_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_583_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_584_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_585_order__less__imp__not__eq2,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_586_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_587_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_588_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_589_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_590_order__less__imp__not__less,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_591_order__less__imp__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_592_order__less__imp__not__less,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_593_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_594_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_595_leD,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% leD
tff(fact_596_leD,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X))
     => ~ pp(aa_set_int_bool(ord_less_set_int(X),Y)) ) ).

% leD
tff(fact_597_leD,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% leD
tff(fact_598_leD,axiom,
    ! [Y: num,X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y)) ) ).

% leD
tff(fact_599_leD,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% leD
tff(fact_600_leD,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ).

% leD
tff(fact_601_leI,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ).

% leI
tff(fact_602_leI,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).

% leI
tff(fact_603_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% leI
tff(fact_604_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% leI
tff(fact_605_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% leI
tff(fact_606_nless__le,axiom,
    ! [A: real,B: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_607_nless__le,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 ) ) ) ).

% nless_le
tff(fact_608_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_609_nless__le,axiom,
    ! [A: num,B: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
    <=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_610_nless__le,axiom,
    ! [A: nat,B: 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_eq_nat,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_611_nless__le,axiom,
    ! [A: int,B: 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_eq_int,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_612_antisym__conv1,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_613_antisym__conv1,axiom,
    ! [X: set_int,Y: set_int] :
      ( ~ pp(aa_set_int_bool(ord_less_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_614_antisym__conv1,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_615_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_616_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_617_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_618_antisym__conv2,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_619_antisym__conv2,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( ~ pp(aa_set_int_bool(ord_less_set_int(X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_620_antisym__conv2,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_621_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_622_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_623_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_624_dense__ge,axiom,
    ! [Z: real,Y: 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,Y),X4)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ).

% dense_ge
tff(fact_625_dense__ge,axiom,
    ! [Z: rat,Y: 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,Y),X4)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ).

% dense_ge
tff(fact_626_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Z)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ).

% dense_le
tff(fact_627_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y))
         => 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,Y),Z)) ) ).

% dense_le
tff(fact_628_less__le__not__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ).

% less_le_not_le
tff(fact_629_less__le__not__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(Y),X)) ) ) ).

% less_le_not_le
tff(fact_630_less__le__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ) ).

% less_le_not_le
tff(fact_631_less__le__not__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ) ).

% less_le_not_le
tff(fact_632_less__le__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ) ).

% less_le_not_le
tff(fact_633_less__le__not__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ) ).

% less_le_not_le
tff(fact_634_not__le__imp__less,axiom,
    ! [Y: real,X: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% not_le_imp_less
tff(fact_635_not__le__imp__less,axiom,
    ! [Y: rat,X: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% not_le_imp_less
tff(fact_636_not__le__imp__less,axiom,
    ! [Y: num,X: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y)) ) ).

% not_le_imp_less
tff(fact_637_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% not_le_imp_less
tff(fact_638_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ).

% not_le_imp_less
tff(fact_639_order_Oorder__iff__strict,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_640_order_Oorder__iff__strict,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 ) ) ) ).

% order.order_iff_strict
tff(fact_641_order_Oorder__iff__strict,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_642_order_Oorder__iff__strict,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_643_order_Oorder__iff__strict,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_644_order_Oorder__iff__strict,axiom,
    ! [A: int,B: 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,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_645_order_Ostrict__iff__order,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_646_order_Ostrict__iff__order,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 ) ) ) ).

% order.strict_iff_order
tff(fact_647_order_Ostrict__iff__order,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_648_order_Ostrict__iff__order,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_649_order_Ostrict__iff__order,axiom,
    ! [A: nat,B: 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_eq_nat,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_650_order_Ostrict__iff__order,axiom,
    ! [A: int,B: 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_eq_int,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_651_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans1
tff(fact_652_order_Ostrict__trans1,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)) ) ) ).

% order.strict_trans1
tff(fact_653_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% order.strict_trans1
tff(fact_654_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% order.strict_trans1
tff(fact_655_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans1
tff(fact_656_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: 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,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans1
tff(fact_657_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans2
tff(fact_658_order_Ostrict__trans2,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)) ) ) ).

% order.strict_trans2
tff(fact_659_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% order.strict_trans2
tff(fact_660_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% order.strict_trans2
tff(fact_661_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: 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_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans2
tff(fact_662_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: 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_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans2
tff(fact_663_order_Ostrict__iff__not,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_664_order_Ostrict__iff__not,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)) ) ) ).

% order.strict_iff_not
tff(fact_665_order_Ostrict__iff__not,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_666_order_Ostrict__iff__not,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_667_order_Ostrict__iff__not,axiom,
    ! [A: nat,B: 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_eq_nat,A),B))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_668_order_Ostrict__iff__not,axiom,
    ! [A: int,B: 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_eq_int,A),B))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_669_dense__ge__bounded,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X))
     => ( ! [W: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),W))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W),X))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),W)) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ) ).

% dense_ge_bounded
tff(fact_670_dense__ge__bounded,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X))
     => ( ! [W: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),W))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W),X))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),W)) ) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ) ).

% dense_ge_bounded
tff(fact_671_dense__le__bounded,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( ! [W: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),W))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W),Y))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W),Z)) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ) ).

% dense_le_bounded
tff(fact_672_dense__le__bounded,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( ! [W: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),W))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W),Y))
             => 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,Y),Z)) ) ) ).

% dense_le_bounded
tff(fact_673_dual__order_Oorder__iff__strict,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_674_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_675_dual__order_Oorder__iff__strict,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_676_dual__order_Oorder__iff__strict,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_677_dual__order_Oorder__iff__strict,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_678_dual__order_Oorder__iff__strict,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_679_dual__order_Ostrict__iff__order,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_680_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B),A))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_681_dual__order_Ostrict__iff__order,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_682_dual__order_Ostrict__iff__order,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_683_dual__order_Ostrict__iff__order,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_684_dual__order_Ostrict__iff__order,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_685_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_686_dual__order_Ostrict__trans1,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
     => ( pp(aa_set_int_bool(ord_less_set_int(C),B))
       => pp(aa_set_int_bool(ord_less_set_int(C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_687_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_688_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_689_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_690_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_691_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_692_dual__order_Ostrict__trans2,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B),A))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(C),B))
       => pp(aa_set_int_bool(ord_less_set_int(C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_693_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_694_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_695_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_696_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_697_dual__order_Ostrict__iff__not,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_698_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B),A))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_699_dual__order_Ostrict__iff__not,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_700_dual__order_Ostrict__iff__not,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_701_dual__order_Ostrict__iff__not,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_702_dual__order_Ostrict__iff__not,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_703_order_Ostrict__implies__order,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% order.strict_implies_order
tff(fact_704_order_Ostrict__implies__order,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)) ) ).

% order.strict_implies_order
tff(fact_705_order_Ostrict__implies__order,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% order.strict_implies_order
tff(fact_706_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ).

% order.strict_implies_order
tff(fact_707_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: 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_eq_nat,A),B)) ) ).

% order.strict_implies_order
tff(fact_708_order_Ostrict__implies__order,axiom,
    ! [A: int,B: 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_eq_int,A),B)) ) ).

% order.strict_implies_order
tff(fact_709_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_710_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B),A))
     => pp(aa_set_int_bool(ord_less_eq_set_int(B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_711_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_712_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_713_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_714_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_715_order__le__less,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_716_order__le__less,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_717_order__le__less,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_718_order__le__less,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_719_order__le__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_720_order__le__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_721_order__less__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_722_order__less__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_723_order__less__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_724_order__less__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_725_order__less__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_726_order__less__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_727_linorder__not__le,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% linorder_not_le
tff(fact_728_linorder__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% linorder_not_le
tff(fact_729_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_not_le
tff(fact_730_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_not_le
tff(fact_731_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_not_le
tff(fact_732_linorder__not__less,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ).

% linorder_not_less
tff(fact_733_linorder__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).

% linorder_not_less
tff(fact_734_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% linorder_not_less
tff(fact_735_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_not_less
tff(fact_736_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_not_less
tff(fact_737_order__less__imp__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% order_less_imp_le
tff(fact_738_order__less__imp__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
     => pp(aa_set_int_bool(ord_less_eq_set_int(X),Y)) ) ).

% order_less_imp_le
tff(fact_739_order__less__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% order_less_imp_le
tff(fact_740_order__less__imp__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% order_less_imp_le
tff(fact_741_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% order_less_imp_le
tff(fact_742_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% order_less_imp_le
tff(fact_743_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ( A != B )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_744_order__le__neq__trans,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)) ) ) ).

% order_le_neq_trans
tff(fact_745_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( A != B )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_746_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( A != B )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_747_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( A != B )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_748_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( ( A != B )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_749_order__neq__le__trans,axiom,
    ! [A: real,B: real] :
      ( ( A != B )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_750_order__neq__le__trans,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_set_int(A),B)) ) ) ).

% order_neq_le_trans
tff(fact_751_order__neq__le__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( A != B )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_752_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_753_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_754_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( 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,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_755_order__le__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_756_order__le__less__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_set_int(Y),Z))
       => pp(aa_set_int_bool(ord_less_set_int(X),Z)) ) ) ).

% order_le_less_trans
tff(fact_757_order__le__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_758_order__le__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_759_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_760_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_761_order__less__le__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_762_order__less__le__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),Z))
       => pp(aa_set_int_bool(ord_less_set_int(X),Z)) ) ) ).

% order_less_le_trans
tff(fact_763_order__less__le__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_764_order__less__le__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_765_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_766_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_767_order__le__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_768_order__le__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_rat_real(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_769_order__le__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_num_real(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_770_order__le__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_nat_real(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_771_order__le__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_int_real(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => ( ! [X4: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_772_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_real_rat(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_773_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_774_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_775_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_776_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => ( ! [X4: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_777_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,B)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,X4)),aa_rat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_778_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,B)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_779_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,B)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_780_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,B)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_781_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,B)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_782_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,B)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,X4)),aa_num_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_783_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,B)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_784_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F,B)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_785_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(F,B)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_786_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(F,B)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_787_order__less__le__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,X4)),aa_rat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_788_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_789_order__less__le__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_790_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_791_order__less__le__subst1,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_792_order__less__le__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,X4)),aa_num_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_793_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_794_order__less__le__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_num_num(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_num_num(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_795_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_num_nat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_num_nat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_796_order__less__le__subst1,axiom,
    ! [A: int,F: fun_num_int,B: num,C: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_num_int(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_num_int(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_797_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,B)),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_798_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,B)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_799_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,B)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_800_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,B)),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_801_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_real,C: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,B)),C))
       => ( ! [X4: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_802_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,B)),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_803_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,B)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_804_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,B)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_805_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,B)),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_806_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_rat,C: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,B)),C))
       => ( ! [X4: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_807_linorder__le__less__linear,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_808_linorder__le__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_809_linorder__le__less__linear,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_810_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_811_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_812_order__le__imp__less__or__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_813_order__le__imp__less__or__eq,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_814_order__le__imp__less__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_815_order__le__imp__less__or__eq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_816_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_817_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_818_greater__shift,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(X)),some_nat(Y))) ) ).

% greater_shift
tff(fact_819_less__shift,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(X)),some_nat(Y))) ) ).

% less_shift
tff(fact_820_mint__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(vEBT_VEBT_min_in_set,vEBT_VEBT_set_vebt(T)),X))
       => ( vEBT_vebt_mint(T) = some_nat(X) ) ) ) ).

% mint_sound
tff(fact_821_mint__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_mint(T) = some_nat(X) )
       => pp(aa_nat_bool(aa_set3744802623701456604t_bool(vEBT_VEBT_min_in_set,vEBT_VEBT_set_vebt(T)),X)) ) ) ).

% mint_corr
tff(fact_822_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_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_823_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_mint(T) = some_nat(Mini) )
       => ( pp(aa_nat_bool(vEBT_vebt_member(T),X))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mini),X)) ) ) ) ).

% mint_corr_help
tff(fact_824_infinite__growing,axiom,
    ! [X6: set_real] :
      ( ( X6 != bot_bot_set_real )
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),X6))
           => ? [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),X6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Xa)) ) )
       => ~ pp(aa_set_real_bool(finite_finite_real,X6)) ) ) ).

% infinite_growing
tff(fact_825_infinite__growing,axiom,
    ! [X6: set_rat] :
      ( ( X6 != bot_bot_set_rat )
     => ( ! [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),X6))
           => ? [Xa: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa),X6))
                & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Xa)) ) )
       => ~ pp(aa_set_rat_bool(finite_finite_rat,X6)) ) ) ).

% infinite_growing
tff(fact_826_infinite__growing,axiom,
    ! [X6: set_num] :
      ( ( X6 != bot_bot_set_num )
     => ( ! [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),X6))
           => ? [Xa: num] :
                ( pp(aa_set_num_bool(member_num(Xa),X6))
                & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Xa)) ) )
       => ~ pp(aa_set_num_bool(finite_finite_num,X6)) ) ) ).

% infinite_growing
tff(fact_827_infinite__growing,axiom,
    ! [X6: set_nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),X6))
           => ? [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),X6))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Xa)) ) )
       => ~ pp(aa_set_nat_bool(finite_finite_nat,X6)) ) ) ).

% infinite_growing
tff(fact_828_infinite__growing,axiom,
    ! [X6: set_int] :
      ( ( X6 != bot_bot_set_int )
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),X6))
           => ? [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),X6))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Xa)) ) )
       => ~ pp(aa_set_int_bool(finite_finite_int,X6)) ) ) ).

% infinite_growing
tff(fact_829_ex__min__if__finite,axiom,
    ! [S2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),S2))
            & ~ ? [Xa: real] :
                  ( pp(aa_set_real_bool(member_real(Xa),S2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Xa),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_830_ex__min__if__finite,axiom,
    ! [S2: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,S2))
     => ( ( S2 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),S2))
            & ~ ? [Xa: rat] :
                  ( pp(aa_set_rat_bool(member_rat(Xa),S2))
                  & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Xa),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_831_ex__min__if__finite,axiom,
    ! [S2: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,S2))
     => ( ( S2 != bot_bot_set_num )
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),S2))
            & ~ ? [Xa: num] :
                  ( pp(aa_set_num_bool(member_num(Xa),S2))
                  & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Xa),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_832_ex__min__if__finite,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( S2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),S2))
            & ~ ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),S2))
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_833_ex__min__if__finite,axiom,
    ! [S2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( S2 != bot_bot_set_int )
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),S2))
            & ~ ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),S2))
                  & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Xa),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_834_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_835_finite__has__minimal,axiom,
    ! [A2: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X4: set_int] :
            ( member_set_int(X4,A2)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_836_finite__has__minimal,axiom,
    ! [A2: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( ( A2 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),A2))
            & ! [Xa: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa),A2))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_837_finite__has__minimal,axiom,
    ! [A2: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( ( A2 != bot_bot_set_num )
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),A2))
            & ! [Xa: num] :
                ( pp(aa_set_num_bool(member_num(Xa),A2))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_838_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_839_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_840_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_841_finite__has__maximal,axiom,
    ! [A2: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X4: set_int] :
            ( member_set_int(X4,A2)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_842_finite__has__maximal,axiom,
    ! [A2: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( ( A2 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),A2))
            & ! [Xa: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa),A2))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_843_finite__has__maximal,axiom,
    ! [A2: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( ( A2 != bot_bot_set_num )
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),A2))
            & ! [Xa: num] :
                ( pp(aa_set_num_bool(member_num(Xa),A2))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_844_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_845_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_846_mint__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi2: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_mint(T) = some_nat(Maxi2) )
       => pp(aa_nat_bool(vEBT_vebt_member(T),Maxi2)) ) ) ).

% mint_member
tff(fact_847_option_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( some_nat(X22) = some_nat(Y22) )
    <=> ( X22 = Y22 ) ) ).

% option.inject
tff(fact_848_option_Oinject,axiom,
    ! [X22: product_prod_nat_nat,Y22: product_prod_nat_nat] :
      ( ( some_P7363390416028606310at_nat(X22) = some_P7363390416028606310at_nat(Y22) )
    <=> ( X22 = Y22 ) ) ).

% option.inject
tff(fact_849_option_Oinject,axiom,
    ! [X22: num,Y22: num] :
      ( ( some_num(X22) = some_num(Y22) )
    <=> ( X22 = Y22 ) ) ).

% option.inject
tff(fact_850_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(T) = none_nat )
     => pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T)) ) ).

% minminNull
tff(fact_851_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ( vEBT_vebt_mint(T) = none_nat ) ) ).

% minNullmin
tff(fact_852_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_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_853_not__Some__eq,axiom,
    ! [X: option_nat] :
      ( ! [Y4: nat] : X != some_nat(Y4)
    <=> ( X = none_nat ) ) ).

% not_Some_eq
tff(fact_854_not__Some__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ! [Y4: product_prod_nat_nat] : X != some_P7363390416028606310at_nat(Y4)
    <=> ( X = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_855_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ! [Y4: num] : X != some_num(Y4)
    <=> ( X = none_num ) ) ).

% not_Some_eq
tff(fact_856_not__None__eq,axiom,
    ! [X: option_nat] :
      ( ( X != none_nat )
    <=> ? [Y4: nat] : X = some_nat(Y4) ) ).

% not_None_eq
tff(fact_857_not__None__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( X != none_P5556105721700978146at_nat )
    <=> ? [Y4: product_prod_nat_nat] : X = some_P7363390416028606310at_nat(Y4) ) ).

% not_None_eq
tff(fact_858_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
    <=> ? [Y4: num] : X = some_num(Y4) ) ).

% not_None_eq
tff(fact_859_combine__options__cases,axiom,
    ! [X: option_nat,P: fun_op7792301606653096457t_bool,Y: option_nat] :
      ( ( ( X = none_nat )
       => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X),Y)) )
     => ( ( ( Y = none_nat )
         => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X),Y)) )
       => ( ! [A3: nat,B3: nat] :
              ( ( X = some_nat(A3) )
             => ( ( Y = some_nat(B3) )
               => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X),Y)) ) )
         => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_860_combine__options__cases,axiom,
    ! [X: option_nat,P: fun_op5698422665285572208t_bool,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_nat )
       => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X),Y)) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X),Y)) )
       => ( ! [A3: nat,B3: product_prod_nat_nat] :
              ( ( X = some_nat(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X),Y)) ) )
         => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_861_combine__options__cases,axiom,
    ! [X: option_nat,P: fun_op8690601775248757823m_bool,Y: option_num] :
      ( ( ( X = none_nat )
       => pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P,X),Y)) )
     => ( ( ( Y = none_num )
         => pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P,X),Y)) )
       => ( ! [A3: nat,B3: num] :
              ( ( X = some_nat(A3) )
             => ( ( Y = some_num(B3) )
               => pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P,X),Y)) ) )
         => pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_862_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: fun_op9100744243108201484t_bool,Y: option_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X),Y)) )
     => ( ( ( Y = none_nat )
         => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X),Y)) )
       => ( ! [A3: product_prod_nat_nat,B3: nat] :
              ( ( X = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_nat(B3) )
               => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X),Y)) ) )
         => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_863_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: fun_op5682630191948829869t_bool,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X),Y)) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X),Y)) )
       => ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( ( X = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X),Y)) ) )
         => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_864_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: fun_op775672374849087042m_bool,Y: option_num] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P,X),Y)) )
     => ( ( ( Y = none_num )
         => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P,X),Y)) )
       => ( ! [A3: product_prod_nat_nat,B3: num] :
              ( ( X = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_num(B3) )
               => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P,X),Y)) ) )
         => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_865_combine__options__cases,axiom,
    ! [X: option_num,P: fun_op1064277265382982207t_bool,Y: option_nat] :
      ( ( ( X = none_num )
       => pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P,X),Y)) )
     => ( ( ( Y = none_nat )
         => pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P,X),Y)) )
       => ( ! [A3: num,B3: nat] :
              ( ( X = some_num(A3) )
             => ( ( Y = some_nat(B3) )
               => pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P,X),Y)) ) )
         => pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_866_combine__options__cases,axiom,
    ! [X: option_num,P: fun_op7732454363416259258t_bool,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_num )
       => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P,X),Y)) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P,X),Y)) )
       => ( ! [A3: num,B3: product_prod_nat_nat] :
              ( ( X = some_num(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P,X),Y)) ) )
         => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_867_combine__options__cases,axiom,
    ! [X: option_num,P: fun_op1962577433978643573m_bool,Y: option_num] :
      ( ( ( X = none_num )
       => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P,X),Y)) )
     => ( ( ( Y = none_num )
         => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P,X),Y)) )
       => ( ! [A3: num,B3: num] :
              ( ( X = some_num(A3) )
             => ( ( Y = some_num(B3) )
               => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P,X),Y)) ) )
         => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_868_split__option__all,axiom,
    ! [P: fun_option_nat_bool] :
      ( ! [X_13: option_nat] : pp(aa_option_nat_bool(P,X_13))
    <=> ( pp(aa_option_nat_bool(P,none_nat))
        & ! [X3: nat] : pp(aa_option_nat_bool(P,some_nat(X3))) ) ) ).

% split_option_all
tff(fact_869_split__option__all,axiom,
    ! [P: fun_op2627489687811228755t_bool] :
      ( ! [X_13: option4927543243414619207at_nat] : pp(aa_opt6800557945374683724t_bool(P,X_13))
    <=> ( 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_870_split__option__all,axiom,
    ! [P: fun_option_num_bool] :
      ( ! [X_13: option_num] : pp(aa_option_num_bool(P,X_13))
    <=> ( pp(aa_option_num_bool(P,none_num))
        & ! [X3: num] : pp(aa_option_num_bool(P,some_num(X3))) ) ) ).

% split_option_all
tff(fact_871_split__option__ex,axiom,
    ! [P: fun_option_nat_bool] :
      ( ? [X_13: option_nat] : pp(aa_option_nat_bool(P,X_13))
    <=> ( pp(aa_option_nat_bool(P,none_nat))
        | ? [X3: nat] : pp(aa_option_nat_bool(P,some_nat(X3))) ) ) ).

% split_option_ex
tff(fact_872_split__option__ex,axiom,
    ! [P: fun_op2627489687811228755t_bool] :
      ( ? [X_13: option4927543243414619207at_nat] : pp(aa_opt6800557945374683724t_bool(P,X_13))
    <=> ( 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_873_split__option__ex,axiom,
    ! [P: fun_option_num_bool] :
      ( ? [X_13: option_num] : pp(aa_option_num_bool(P,X_13))
    <=> ( pp(aa_option_num_bool(P,none_num))
        | ? [X3: num] : pp(aa_option_num_bool(P,some_num(X3))) ) ) ).

% split_option_ex
tff(fact_874_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X23: nat] : Y != some_nat(X23) ) ).

% option.exhaust
tff(fact_875_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] : Y != some_P7363390416028606310at_nat(X23) ) ).

% option.exhaust
tff(fact_876_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X23: num] : Y != some_num(X23) ) ).

% option.exhaust
tff(fact_877_option_OdiscI,axiom,
    ! [Option: option_nat,X22: nat] :
      ( ( Option = some_nat(X22) )
     => ( Option != none_nat ) ) ).

% option.discI
tff(fact_878_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X22) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_879_option_OdiscI,axiom,
    ! [Option: option_num,X22: num] :
      ( ( Option = some_num(X22) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_880_option_Odistinct_I1_J,axiom,
    ! [X22: nat] : none_nat != some_nat(X22) ).

% option.distinct(1)
tff(fact_881_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X22) ).

% option.distinct(1)
tff(fact_882_option_Odistinct_I1_J,axiom,
    ! [X22: num] : none_num != some_num(X22) ).

% option.distinct(1)
tff(fact_883_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_884_finite__maxlen,axiom,
    ! [M4: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M4)
     => ? [N2: nat] :
        ! [X2: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X2,M4)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(X2)),N2)) ) ) ).

% finite_maxlen
tff(fact_885_finite__maxlen,axiom,
    ! [M4: set_list_bool] :
      ( finite7866827932528546246t_bool(M4)
     => ? [N2: nat] :
        ! [X2: list_bool] :
          ( member_list_bool(X2,M4)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(X2)),N2)) ) ) ).

% finite_maxlen
tff(fact_886_finite__maxlen,axiom,
    ! [M4: set_list_nat] :
      ( finite8100373058378681591st_nat(M4)
     => ? [N2: nat] :
        ! [X2: list_nat] :
          ( member_list_nat(X2,M4)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(X2)),N2)) ) ) ).

% finite_maxlen
tff(fact_887_finite__maxlen,axiom,
    ! [M4: set_list_int] :
      ( finite3922522038869484883st_int(M4)
     => ? [N2: nat] :
        ! [X2: list_int] :
          ( member_list_int(X2,M4)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_int(X2)),N2)) ) ) ).

% finite_maxlen
tff(fact_888_finite__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ) ).

% finite_subset
tff(fact_889_finite__subset,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ) ).

% finite_subset
tff(fact_890_finite__subset,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B2))
     => ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
       => pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2)) ) ) ).

% finite_subset
tff(fact_891_finite__subset,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => pp(aa_set_int_bool(finite_finite_int,A2)) ) ) ).

% finite_subset
tff(fact_892_infinite__super,axiom,
    ! [S2: set_nat,T2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
       => ~ pp(aa_set_nat_bool(finite_finite_nat,T2)) ) ) ).

% infinite_super
tff(fact_893_infinite__super,axiom,
    ! [S2: set_complex,T2: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
     => ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,T2)) ) ) ).

% infinite_super
tff(fact_894_infinite__super,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(S2),T2))
     => ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,S2))
       => ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,T2)) ) ) ).

% infinite_super
tff(fact_895_infinite__super,axiom,
    ! [S2: set_int,T2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
     => ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
       => ~ pp(aa_set_int_bool(finite_finite_int,T2)) ) ) ).

% infinite_super
tff(fact_896_rev__finite__subset,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
       => pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ) ).

% rev_finite_subset
tff(fact_897_rev__finite__subset,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ) ).

% rev_finite_subset
tff(fact_898_rev__finite__subset,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B2))
       => pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2)) ) ) ).

% rev_finite_subset
tff(fact_899_rev__finite__subset,axiom,
    ! [B2: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => pp(aa_set_int_bool(finite_finite_int,A2)) ) ) ).

% rev_finite_subset
tff(fact_900_finite__psubset__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [A4: set_nat] :
            ( pp(aa_set_nat_bool(finite_finite_nat,A4))
           => ( ! [B4: set_nat] :
                  ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B4),A4))
                 => pp(aa_set_nat_bool(P,B4)) )
             => pp(aa_set_nat_bool(P,A4)) ) )
       => pp(aa_set_nat_bool(P,A2)) ) ) ).

% finite_psubset_induct
tff(fact_901_finite__psubset__induct,axiom,
    ! [A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [A4: set_int] :
            ( pp(aa_set_int_bool(finite_finite_int,A4))
           => ( ! [B4: set_int] :
                  ( pp(aa_set_int_bool(ord_less_set_int(B4),A4))
                 => pp(aa_set_int_bool(P,B4)) )
             => pp(aa_set_int_bool(P,A4)) ) )
       => pp(aa_set_int_bool(P,A2)) ) ) ).

% finite_psubset_induct
tff(fact_902_finite__psubset__induct,axiom,
    ! [A2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [A4: set_complex] :
            ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
           => ( ! [B4: set_complex] :
                  ( pp(aa_set_complex_bool(ord_less_set_complex(B4),A4))
                 => pp(aa_set_complex_bool(P,B4)) )
             => pp(aa_set_complex_bool(P,A4)) ) )
       => pp(aa_set_complex_bool(P,A2)) ) ) ).

% finite_psubset_induct
tff(fact_903_finite__psubset__induct,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
     => ( ! [A4: set_Pr1261947904930325089at_nat] :
            ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
           => ( ! [B4: set_Pr1261947904930325089at_nat] :
                  ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(B4),A4))
                 => pp(aa_set3135850010578026034t_bool(P,B4)) )
             => pp(aa_set3135850010578026034t_bool(P,A4)) ) )
       => pp(aa_set3135850010578026034t_bool(P,A2)) ) ) ).

% finite_psubset_induct
tff(fact_904_finite__has__minimal2,axiom,
    ! [A2: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),A))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_905_finite__has__minimal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
       => ? [X4: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
            & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X4),A))
            & ! [Xa: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(Xa),A2))
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_906_finite__has__minimal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(A,A2)
       => ? [X4: set_int] :
            ( member_set_int(X4,A2)
            & pp(aa_set_int_bool(ord_less_eq_set_int(X4),A))
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_907_finite__has__minimal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( pp(aa_set_rat_bool(member_rat(A),A2))
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),A2))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),A))
            & ! [Xa: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa),A2))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_908_finite__has__minimal2,axiom,
    ! [A2: set_num,A: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( pp(aa_set_num_bool(member_num(A),A2))
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),A2))
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),A))
            & ! [Xa: num] :
                ( pp(aa_set_num_bool(member_num(Xa),A2))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_909_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),A))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_910_finite__has__minimal2,axiom,
    ! [A2: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),A))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_911_finite__has__maximal2,axiom,
    ! [A2: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_912_finite__has__maximal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
       => ? [X4: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
            & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),X4))
            & ! [Xa: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(Xa),A2))
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_913_finite__has__maximal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(A,A2)
       => ? [X4: set_int] :
            ( member_set_int(X4,A2)
            & pp(aa_set_int_bool(ord_less_eq_set_int(A),X4))
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_914_finite__has__maximal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( pp(aa_set_rat_bool(member_rat(A),A2))
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),A2))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),X4))
            & ! [Xa: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa),A2))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_915_finite__has__maximal2,axiom,
    ! [A2: set_num,A: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( pp(aa_set_num_bool(member_num(A),A2))
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),A2))
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),X4))
            & ! [Xa: num] :
                ( pp(aa_set_num_bool(member_num(Xa),A2))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_916_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X4))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_917_finite__has__maximal2,axiom,
    ! [A2: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X4))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_918_infinite__imp__nonempty,axiom,
    ! [S2: set_complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( S2 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
tff(fact_919_infinite__imp__nonempty,axiom,
    ! [S2: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,S2))
     => ( S2 != bot_bo2099793752762293965at_nat ) ) ).

% infinite_imp_nonempty
tff(fact_920_infinite__imp__nonempty,axiom,
    ! [S2: set_real] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,S2))
     => ( S2 != bot_bot_set_real ) ) ).

% infinite_imp_nonempty
tff(fact_921_infinite__imp__nonempty,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( S2 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
tff(fact_922_infinite__imp__nonempty,axiom,
    ! [S2: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
     => ( S2 != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
tff(fact_923_finite_OemptyI,axiom,
    pp(aa_set_complex_bool(finite3207457112153483333omplex,bot_bot_set_complex)) ).

% finite.emptyI
tff(fact_924_finite_OemptyI,axiom,
    pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,bot_bo2099793752762293965at_nat)) ).

% finite.emptyI
tff(fact_925_finite_OemptyI,axiom,
    pp(aa_set_real_bool(finite_finite_real,bot_bot_set_real)) ).

% finite.emptyI
tff(fact_926_finite_OemptyI,axiom,
    pp(aa_set_nat_bool(finite_finite_nat,bot_bot_set_nat)) ).

% finite.emptyI
tff(fact_927_finite_OemptyI,axiom,
    pp(aa_set_int_bool(finite_finite_int,bot_bot_set_int)) ).

% finite.emptyI
tff(fact_928_pred__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_pred(T,X) = some_nat(Sx) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(vEBT_set_vebt(T)),X),Sx)) ) ) ).

% pred_correct
tff(fact_929_succ__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_succ(T,X) = some_nat(Sx) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_set_vebt(T)),X),Sx)) ) ) ).

% succ_correct
tff(fact_930_pred__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Px: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_pred(T,X) = some_nat(Px) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T)),X),Px)) ) ) ).

% pred_corr
tff(fact_931_succ__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_succ(T,X) = some_nat(Sx) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T)),X),Sx)) ) ) ).

% succ_corr
tff(fact_932_subset__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),bot_bo2099793752762293965at_nat))
    <=> ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% subset_empty
tff(fact_933_subset__empty,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),bot_bot_set_real))
    <=> ( A2 = bot_bot_set_real ) ) ).

% subset_empty
tff(fact_934_subset__empty,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),bot_bot_set_nat))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_935_subset__empty,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),bot_bot_set_int))
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_936_empty__subsetI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(bot_bo2099793752762293965at_nat),A2)) ).

% empty_subsetI
tff(fact_937_empty__subsetI,axiom,
    ! [A2: set_real] : pp(aa_set_real_bool(ord_less_eq_set_real(bot_bot_set_real),A2)) ).

% empty_subsetI
tff(fact_938_empty__subsetI,axiom,
    ! [A2: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,bot_bot_set_nat),A2)) ).

% empty_subsetI
tff(fact_939_empty__subsetI,axiom,
    ! [A2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(bot_bot_set_int),A2)) ).

% empty_subsetI
tff(fact_940_is__succ__in__set__def,axiom,
    ! [Xs: set_nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(Xs),X),Y))
    <=> ( pp(aa_set_nat_bool(member_nat(Y),Xs))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),Xs))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3)) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_941_is__pred__in__set__def,axiom,
    ! [Xs: set_nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(Xs),X),Y))
    <=> ( pp(aa_set_nat_bool(member_nat(Y),Xs))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),Xs))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),X))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y)) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_942_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Deg))
     => ( ( Mi = Ma )
       => ( ! [X2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
             => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_12)) )
          & ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_12)) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_943_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),X)) ).

% vebt_member.simps(3)
tff(fact_944_psubsetI,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)) ) ) ).

% psubsetI
tff(fact_945_empty__iff,axiom,
    ! [C: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(C),bot_bot_set_set_nat)) ).

% empty_iff
tff(fact_946_empty__iff,axiom,
    ! [C: product_prod_nat_nat] : ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),bot_bo2099793752762293965at_nat)) ).

% empty_iff
tff(fact_947_empty__iff,axiom,
    ! [C: real] : ~ pp(aa_set_real_bool(member_real(C),bot_bot_set_real)) ).

% empty_iff
tff(fact_948_empty__iff,axiom,
    ! [C: nat] : ~ pp(aa_set_nat_bool(member_nat(C),bot_bot_set_nat)) ).

% empty_iff
tff(fact_949_empty__iff,axiom,
    ! [C: int] : ~ pp(aa_set_int_bool(member_int(C),bot_bot_set_int)) ).

% empty_iff
tff(fact_950_all__not__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ! [X3: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(X3),A2))
    <=> ( A2 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
tff(fact_951_all__not__in__conv,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] : ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),A2))
    <=> ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% all_not_in_conv
tff(fact_952_all__not__in__conv,axiom,
    ! [A2: set_real] :
      ( ! [X3: real] : ~ pp(aa_set_real_bool(member_real(X3),A2))
    <=> ( A2 = bot_bot_set_real ) ) ).

% all_not_in_conv
tff(fact_953_all__not__in__conv,axiom,
    ! [A2: set_nat] :
      ( ! [X3: nat] : ~ pp(aa_set_nat_bool(member_nat(X3),A2))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_954_all__not__in__conv,axiom,
    ! [A2: set_int] :
      ( ! [X3: int] : ~ pp(aa_set_int_bool(member_int(X3),A2))
    <=> ( A2 = bot_bot_set_int ) ) ).

% all_not_in_conv
tff(fact_955_Collect__empty__eq,axiom,
    ! [P: fun_list_nat_bool] :
      ( ( collect_list_nat(P) = bot_bot_set_list_nat )
    <=> ! [X3: list_nat] : ~ pp(aa_list_nat_bool(P,X3)) ) ).

% Collect_empty_eq
tff(fact_956_Collect__empty__eq,axiom,
    ! [P: fun_set_nat_bool] :
      ( ( collect_set_nat(P) = bot_bot_set_set_nat )
    <=> ! [X3: set_nat] : ~ pp(aa_set_nat_bool(P,X3)) ) ).

% Collect_empty_eq
tff(fact_957_Collect__empty__eq,axiom,
    ! [P: fun_Pr7757163693017731481t_bool] :
      ( ( collec3392354462482085612at_nat(P) = bot_bo2099793752762293965at_nat )
    <=> ! [X3: product_prod_nat_nat] : ~ pp(aa_Pro2232559814647311250t_bool(P,X3)) ) ).

% Collect_empty_eq
tff(fact_958_Collect__empty__eq,axiom,
    ! [P: fun_real_bool] :
      ( ( collect_real(P) = bot_bot_set_real )
    <=> ! [X3: real] : ~ pp(aa_real_bool(P,X3)) ) ).

% Collect_empty_eq
tff(fact_959_Collect__empty__eq,axiom,
    ! [P: fun_nat_bool] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ! [X3: nat] : ~ pp(aa_nat_bool(P,X3)) ) ).

% Collect_empty_eq
tff(fact_960_Collect__empty__eq,axiom,
    ! [P: fun_int_bool] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ! [X3: int] : ~ pp(aa_int_bool(P,X3)) ) ).

% Collect_empty_eq
tff(fact_961_empty__Collect__eq,axiom,
    ! [P: fun_list_nat_bool] :
      ( ( bot_bot_set_list_nat = collect_list_nat(P) )
    <=> ! [X3: list_nat] : ~ pp(aa_list_nat_bool(P,X3)) ) ).

% empty_Collect_eq
tff(fact_962_empty__Collect__eq,axiom,
    ! [P: fun_set_nat_bool] :
      ( ( bot_bot_set_set_nat = collect_set_nat(P) )
    <=> ! [X3: set_nat] : ~ pp(aa_set_nat_bool(P,X3)) ) ).

% empty_Collect_eq
tff(fact_963_empty__Collect__eq,axiom,
    ! [P: fun_Pr7757163693017731481t_bool] :
      ( ( bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(P) )
    <=> ! [X3: product_prod_nat_nat] : ~ pp(aa_Pro2232559814647311250t_bool(P,X3)) ) ).

% empty_Collect_eq
tff(fact_964_empty__Collect__eq,axiom,
    ! [P: fun_real_bool] :
      ( ( bot_bot_set_real = collect_real(P) )
    <=> ! [X3: real] : ~ pp(aa_real_bool(P,X3)) ) ).

% empty_Collect_eq
tff(fact_965_empty__Collect__eq,axiom,
    ! [P: fun_nat_bool] :
      ( ( bot_bot_set_nat = collect_nat(P) )
    <=> ! [X3: nat] : ~ pp(aa_nat_bool(P,X3)) ) ).

% empty_Collect_eq
tff(fact_966_empty__Collect__eq,axiom,
    ! [P: fun_int_bool] :
      ( ( bot_bot_set_int = collect_int(P) )
    <=> ! [X3: int] : ~ pp(aa_int_bool(P,X3)) ) ).

% empty_Collect_eq
tff(fact_967_subsetI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),A2))
         => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),B2)) )
     => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B2)) ) ).

% subsetI
tff(fact_968_subsetI,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => pp(aa_set_real_bool(member_real(X4),B2)) )
     => pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2)) ) ).

% subsetI
tff(fact_969_subsetI,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ! [X4: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
         => pp(aa_set_set_nat_bool(member_set_nat(X4),B2)) )
     => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B2)) ) ).

% subsetI
tff(fact_970_subsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_set_nat_bool(member_nat(X4),B2)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2)) ) ).

% subsetI
tff(fact_971_subsetI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_set_int_bool(member_int(X4),B2)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ).

% subsetI
tff(fact_972_subset__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 ) ) ) ).

% subset_antisym
tff(fact_973_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_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),X))
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = none_nat ) ) ) ).

% geqmaxNone
tff(fact_974_psubsetD,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A2),B2))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A2))
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B2)) ) ) ).

% psubsetD
tff(fact_975_psubsetD,axiom,
    ! [A2: set_real,B2: set_real,C: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A2),B2))
     => ( pp(aa_set_real_bool(member_real(C),A2))
       => pp(aa_set_real_bool(member_real(C),B2)) ) ) ).

% psubsetD
tff(fact_976_psubsetD,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A2),B2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(C),A2))
       => pp(aa_set_set_nat_bool(member_set_nat(C),B2)) ) ) ).

% psubsetD
tff(fact_977_psubsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B2))
     => ( pp(aa_set_nat_bool(member_nat(C),A2))
       => pp(aa_set_nat_bool(member_nat(C),B2)) ) ) ).

% psubsetD
tff(fact_978_psubsetD,axiom,
    ! [A2: set_int,B2: set_int,C: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
     => ( pp(aa_set_int_bool(member_int(C),A2))
       => pp(aa_set_int_bool(member_int(C),B2)) ) ) ).

% psubsetD
tff(fact_979_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] : vEBT_vebt_succ(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy,Uz),Va) = none_nat ).

% vebt_succ.simps(3)
tff(fact_980_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] : vEBT_vebt_pred(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz,Va),Vb) = none_nat ).

% vebt_pred.simps(4)
tff(fact_981_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)),X)) ).

% vebt_member.simps(2)
tff(fact_982_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy))) ).

% VEBT_internal.minNull.simps(4)
tff(fact_983_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] : vEBT_vebt_delete(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary),Uu) = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) ).

% vebt_delete.simps(4)
tff(fact_984_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd,Ve),Vf) = none_nat ).

% vebt_pred.simps(5)
tff(fact_985_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc,Vd),Ve) = none_nat ).

% vebt_succ.simps(4)
tff(fact_986_not__psubset__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ~ pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A2),bot_bo2099793752762293965at_nat)) ).

% not_psubset_empty
tff(fact_987_not__psubset__empty,axiom,
    ! [A2: set_real] : ~ pp(aa_set_real_bool(ord_less_set_real(A2),bot_bot_set_real)) ).

% not_psubset_empty
tff(fact_988_not__psubset__empty,axiom,
    ! [A2: set_nat] : ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),bot_bot_set_nat)) ).

% not_psubset_empty
tff(fact_989_not__psubset__empty,axiom,
    ! [A2: set_int] : ~ pp(aa_set_int_bool(ord_less_set_int(A2),bot_bot_set_int)) ).

% not_psubset_empty
tff(fact_990_emptyE,axiom,
    ! [A: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(A),bot_bot_set_set_nat)) ).

% emptyE
tff(fact_991_emptyE,axiom,
    ! [A: product_prod_nat_nat] : ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),bot_bo2099793752762293965at_nat)) ).

% emptyE
tff(fact_992_emptyE,axiom,
    ! [A: real] : ~ pp(aa_set_real_bool(member_real(A),bot_bot_set_real)) ).

% emptyE
tff(fact_993_emptyE,axiom,
    ! [A: nat] : ~ pp(aa_set_nat_bool(member_nat(A),bot_bot_set_nat)) ).

% emptyE
tff(fact_994_emptyE,axiom,
    ! [A: int] : ~ pp(aa_set_int_bool(member_int(A),bot_bot_set_int)) ).

% emptyE
tff(fact_995_equals0D,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( A2 = bot_bot_set_set_nat )
     => ~ pp(aa_set_set_nat_bool(member_set_nat(A),A2)) ) ).

% equals0D
tff(fact_996_equals0D,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( A2 = bot_bo2099793752762293965at_nat )
     => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),A2)) ) ).

% equals0D
tff(fact_997_equals0D,axiom,
    ! [A2: set_real,A: real] :
      ( ( A2 = bot_bot_set_real )
     => ~ pp(aa_set_real_bool(member_real(A),A2)) ) ).

% equals0D
tff(fact_998_equals0D,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( A2 = bot_bot_set_nat )
     => ~ pp(aa_set_nat_bool(member_nat(A),A2)) ) ).

% equals0D
tff(fact_999_equals0D,axiom,
    ! [A2: set_int,A: int] :
      ( ( A2 = bot_bot_set_int )
     => ~ pp(aa_set_int_bool(member_int(A),A2)) ) ).

% equals0D
tff(fact_1000_equals0I,axiom,
    ! [A2: set_set_nat] :
      ( ! [Y2: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(Y2),A2))
     => ( A2 = bot_bot_set_set_nat ) ) ).

% equals0I
tff(fact_1001_equals0I,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ! [Y2: product_prod_nat_nat] : ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y2),A2))
     => ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% equals0I
tff(fact_1002_equals0I,axiom,
    ! [A2: set_real] :
      ( ! [Y2: real] : ~ pp(aa_set_real_bool(member_real(Y2),A2))
     => ( A2 = bot_bot_set_real ) ) ).

% equals0I
tff(fact_1003_equals0I,axiom,
    ! [A2: set_nat] :
      ( ! [Y2: nat] : ~ pp(aa_set_nat_bool(member_nat(Y2),A2))
     => ( A2 = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_1004_equals0I,axiom,
    ! [A2: set_int] :
      ( ! [Y2: int] : ~ pp(aa_set_int_bool(member_int(Y2),A2))
     => ( A2 = bot_bot_set_int ) ) ).

% equals0I
tff(fact_1005_ex__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ? [X3: set_nat] : pp(aa_set_set_nat_bool(member_set_nat(X3),A2))
    <=> ( A2 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
tff(fact_1006_ex__in__conv,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ? [X3: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),A2))
    <=> ( A2 != bot_bo2099793752762293965at_nat ) ) ).

% ex_in_conv
tff(fact_1007_ex__in__conv,axiom,
    ! [A2: set_real] :
      ( ? [X3: real] : pp(aa_set_real_bool(member_real(X3),A2))
    <=> ( A2 != bot_bot_set_real ) ) ).

% ex_in_conv
tff(fact_1008_ex__in__conv,axiom,
    ! [A2: set_nat] :
      ( ? [X3: nat] : pp(aa_set_nat_bool(member_nat(X3),A2))
    <=> ( A2 != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_1009_ex__in__conv,axiom,
    ! [A2: set_int] :
      ( ? [X3: int] : pp(aa_set_int_bool(member_int(X3),A2))
    <=> ( A2 != bot_bot_set_int ) ) ).

% ex_in_conv
tff(fact_1010_bot__set__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(bot_bo6469247547558333820t_bool) ).

% bot_set_def
tff(fact_1011_bot__set__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(bot_bo8343409700046576086t_bool) ).

% bot_set_def
tff(fact_1012_bot__set__def,axiom,
    bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(bot_bo3782518310583830573t_bool) ).

% bot_set_def
tff(fact_1013_bot__set__def,axiom,
    bot_bot_set_real = collect_real(bot_bo4181092155253064880l_bool) ).

% bot_set_def
tff(fact_1014_bot__set__def,axiom,
    bot_bot_set_nat = collect_nat(bot_bot_fun_nat_bool) ).

% bot_set_def
tff(fact_1015_bot__set__def,axiom,
    bot_bot_set_int = collect_int(bot_bot_fun_int_bool) ).

% bot_set_def
tff(fact_1016_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X: nat] : vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst,Smry),X) = 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_1017_in__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B2))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2))
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B2)) ) ) ).

% in_mono
tff(fact_1018_in__mono,axiom,
    ! [A2: set_real,B2: set_real,X: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
     => ( pp(aa_set_real_bool(member_real(X),A2))
       => pp(aa_set_real_bool(member_real(X),B2)) ) ) ).

% in_mono
tff(fact_1019_in__mono,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,X: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
       => pp(aa_set_set_nat_bool(member_set_nat(X),B2)) ) ) ).

% in_mono
tff(fact_1020_in__mono,axiom,
    ! [A2: set_nat,B2: set_nat,X: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
     => ( pp(aa_set_nat_bool(member_nat(X),A2))
       => pp(aa_set_nat_bool(member_nat(X),B2)) ) ) ).

% in_mono
tff(fact_1021_in__mono,axiom,
    ! [A2: set_int,B2: set_int,X: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
     => ( pp(aa_set_int_bool(member_int(X),A2))
       => pp(aa_set_int_bool(member_int(X),B2)) ) ) ).

% in_mono
tff(fact_1022_subsetD,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B2))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A2))
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B2)) ) ) ).

% subsetD
tff(fact_1023_subsetD,axiom,
    ! [A2: set_real,B2: set_real,C: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
     => ( pp(aa_set_real_bool(member_real(C),A2))
       => pp(aa_set_real_bool(member_real(C),B2)) ) ) ).

% subsetD
tff(fact_1024_subsetD,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(C),A2))
       => pp(aa_set_set_nat_bool(member_set_nat(C),B2)) ) ) ).

% subsetD
tff(fact_1025_subsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
     => ( pp(aa_set_nat_bool(member_nat(C),A2))
       => pp(aa_set_nat_bool(member_nat(C),B2)) ) ) ).

% subsetD
tff(fact_1026_subsetD,axiom,
    ! [A2: set_int,B2: set_int,C: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
     => ( pp(aa_set_int_bool(member_int(C),A2))
       => pp(aa_set_int_bool(member_int(C),B2)) ) ) ).

% subsetD
tff(fact_1027_equalityE,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)) ) ) ).

% equalityE
tff(fact_1028_subset__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B2))
    <=> ! [X3: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),A2))
         => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),B2)) ) ) ).

% subset_eq
tff(fact_1029_subset__eq,axiom,
    ! [A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
    <=> ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),A2))
         => pp(aa_set_real_bool(member_real(X3),B2)) ) ) ).

% subset_eq
tff(fact_1030_subset__eq,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B2))
    <=> ! [X3: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X3),A2))
         => pp(aa_set_set_nat_bool(member_set_nat(X3),B2)) ) ) ).

% subset_eq
tff(fact_1031_subset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => pp(aa_set_nat_bool(member_nat(X3),B2)) ) ) ).

% subset_eq
tff(fact_1032_subset__eq,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
    <=> ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),A2))
         => pp(aa_set_int_bool(member_int(X3),B2)) ) ) ).

% subset_eq
tff(fact_1033_equalityD1,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( A2 = B2 )
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ).

% equalityD1
tff(fact_1034_equalityD2,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( A2 = B2 )
     => pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2)) ) ).

% equalityD2
tff(fact_1035_subset__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B2))
    <=> ! [T3: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(T3),A2))
         => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(T3),B2)) ) ) ).

% subset_iff
tff(fact_1036_subset__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
    <=> ! [T3: real] :
          ( pp(aa_set_real_bool(member_real(T3),A2))
         => pp(aa_set_real_bool(member_real(T3),B2)) ) ) ).

% subset_iff
tff(fact_1037_subset__iff,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B2))
    <=> ! [T3: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(T3),A2))
         => pp(aa_set_set_nat_bool(member_set_nat(T3),B2)) ) ) ).

% subset_iff
tff(fact_1038_subset__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
    <=> ! [T3: nat] :
          ( pp(aa_set_nat_bool(member_nat(T3),A2))
         => pp(aa_set_nat_bool(member_nat(T3),B2)) ) ) ).

% subset_iff
tff(fact_1039_subset__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
    <=> ! [T3: int] :
          ( pp(aa_set_int_bool(member_int(T3),A2))
         => pp(aa_set_int_bool(member_int(T3),B2)) ) ) ).

% subset_iff
tff(fact_1040_subset__refl,axiom,
    ! [A2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A2),A2)) ).

% subset_refl
tff(fact_1041_Collect__mono,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(P,X4))
         => pp(aa_real_bool(Q,X4)) )
     => pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(P)),collect_real(Q))) ) ).

% Collect_mono
tff(fact_1042_Collect__mono,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ! [X4: list_nat] :
          ( pp(aa_list_nat_bool(P,X4))
         => pp(aa_list_nat_bool(Q,X4)) )
     => ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q)) ) ).

% Collect_mono
tff(fact_1043_Collect__mono,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ! [X4: set_nat] :
          ( pp(aa_set_nat_bool(P,X4))
         => pp(aa_set_nat_bool(Q,X4)) )
     => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(P)),collect_set_nat(Q))) ) ).

% Collect_mono
tff(fact_1044_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(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(P)),collect_nat(Q))) ) ).

% Collect_mono
tff(fact_1045_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_1046_subset__trans,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)) ) ) ).

% subset_trans
tff(fact_1047_set__eq__subset,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)) ) ) ).

% set_eq_subset
tff(fact_1048_Collect__mono__iff,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(P)),collect_real(Q)))
    <=> ! [X3: real] :
          ( pp(aa_real_bool(P,X3))
         => pp(aa_real_bool(Q,X3)) ) ) ).

% Collect_mono_iff
tff(fact_1049_Collect__mono__iff,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q))
    <=> ! [X3: list_nat] :
          ( pp(aa_list_nat_bool(P,X3))
         => pp(aa_list_nat_bool(Q,X3)) ) ) ).

% Collect_mono_iff
tff(fact_1050_Collect__mono__iff,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(P)),collect_set_nat(Q)))
    <=> ! [X3: set_nat] :
          ( pp(aa_set_nat_bool(P,X3))
         => pp(aa_set_nat_bool(Q,X3)) ) ) ).

% Collect_mono_iff
tff(fact_1051_Collect__mono__iff,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_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_1052_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_1053_subset__iff__psubset__eq,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 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1054_subset__psubset__trans,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)) ) ) ).

% subset_psubset_trans
tff(fact_1055_subset__not__subset__eq,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)) ) ) ).

% subset_not_subset_eq
tff(fact_1056_psubset__subset__trans,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)) ) ) ).

% psubset_subset_trans
tff(fact_1057_psubset__imp__subset,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)) ) ).

% psubset_imp_subset
tff(fact_1058_psubset__eq,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 ) ) ) ).

% psubset_eq
tff(fact_1059_psubsetE,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)) ) ) ).

% psubsetE
tff(fact_1060_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va,Vb)),X))
    <=> ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_1061_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy)),Uz)) ).

% VEBT_internal.membermima.simps(2)
tff(fact_1062_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,vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb,Vc))) ).

% VEBT_internal.minNull.simps(5)
tff(fact_1063_tvalid,axiom,
    pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary)),deg)) ).

% tvalid
tff(fact_1064_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_mint(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_1065_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_maxt(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_1066_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) = none_nat ).

% vebt_mint.simps(2)
tff(fact_1067_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) = none_nat ).

% vebt_maxt.simps(2)
tff(fact_1068_calculation,axiom,
    ( ( ya = mi )
   => ( ( xa != ya )
      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary)),ya)) ) ) ).

% calculation
tff(fact_1069_prod_Oinject,axiom,
    ! [X1: int,X22: int,Y1: int,Y22: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_1070_prod_Oinject,axiom,
    ! [X1: fun_Co6545587727236015215e_term,X22: produc8923325533196201883nteger,Y1: fun_Co6545587727236015215e_term,Y22: produc8923325533196201883nteger] :
      ( ( aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,X1),X22) = aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_1071_prod_Oinject,axiom,
    ! [X1: fun_Pr272337284026993017e_term,X22: produc8923325533196201883nteger,Y1: fun_Pr272337284026993017e_term,Y22: produc8923325533196201883nteger] :
      ( ( aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,X1),X22) = aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_1072_prod_Oinject,axiom,
    ! [X1: fun_Pr9164807099015279978e_term,X22: product_prod_int_int,Y1: fun_Pr9164807099015279978e_term,Y22: product_prod_int_int] :
      ( ( aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,X1),X22) = aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_1073_prod_Oinject,axiom,
    ! [X1: fun_in6429348457993719614e_term,X22: product_prod_int_int,Y1: fun_in6429348457993719614e_term,Y22: product_prod_int_int] :
      ( ( aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,X1),X22) = aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_1074_old_Oprod_Oinject,axiom,
    ! [A: int,B: int,A5: int,B5: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A5),B5) )
    <=> ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
tff(fact_1075_old_Oprod_Oinject,axiom,
    ! [A: fun_Co6545587727236015215e_term,B: produc8923325533196201883nteger,A5: fun_Co6545587727236015215e_term,B5: produc8923325533196201883nteger] :
      ( ( aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,A),B) = aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,A5),B5) )
    <=> ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
tff(fact_1076_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr272337284026993017e_term,B: produc8923325533196201883nteger,A5: fun_Pr272337284026993017e_term,B5: produc8923325533196201883nteger] :
      ( ( aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,A),B) = aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,A5),B5) )
    <=> ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
tff(fact_1077_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr9164807099015279978e_term,B: product_prod_int_int,A5: fun_Pr9164807099015279978e_term,B5: product_prod_int_int] :
      ( ( aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,A),B) = aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,A5),B5) )
    <=> ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
tff(fact_1078_old_Oprod_Oinject,axiom,
    ! [A: fun_in6429348457993719614e_term,B: product_prod_int_int,A5: fun_in6429348457993719614e_term,B5: product_prod_int_int] :
      ( ( aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,A),B) = aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,A5),B5) )
    <=> ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
tff(fact_1079_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
    <=> ! [M3: nat] :
        ? [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N3))
          & pp(aa_set_nat_bool(member_nat(N3),S2)) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_1080_finite__nat__set__iff__bounded__le,axiom,
    ! [N4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
    <=> ? [M3: nat] :
        ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),M3)) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_1081_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set_nat] :
      ( ! [M5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),M5))
         => ? [N5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N5))
              & pp(aa_set_nat_bool(member_nat(N5),S2)) ) )
     => ~ pp(aa_set_nat_bool(finite_finite_nat,S2)) ) ).

% unbounded_k_infinite
tff(fact_1082__C4_Ohyps_C_I4_J,axiom,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

% "4.hyps"(4)
tff(fact_1083_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc7004362771917097900on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_nat,Uv2: option_nat] : X != produc6699827698519632094on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw2: fun_nat_fun_nat_nat,V2: nat] : X != produc6699827698519632094on_nat(Uw2,produc5098337634421038937on_nat(some_nat(V2),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_nat,A3: nat,B3: nat] : X != produc6699827698519632094on_nat(F2,produc5098337634421038937on_nat(some_nat(A3),some_nat(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1084_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc1260151622200656695at_nat] :
      ( ! [Uu2: fun_Pr2477239776880934807at_nat,Uv2: option4927543243414619207at_nat] : X != produc7814092408141220007at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw2: fun_Pr2477239776880934807at_nat,V2: product_prod_nat_nat] : X != 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] : X != produc7814092408141220007at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A3),some_P7363390416028606310at_nat(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1085_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc2703417854250991842on_num] :
      ( ! [Uu2: fun_num_fun_num_num,Uv2: option_num] : X != produc6360833703471310868on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
     => ( ! [Uw2: fun_num_fun_num_num,V2: num] : X != produc6360833703471310868on_num(Uw2,produc8585076106096196333on_num(some_num(V2),none_num))
       => ~ ! [F2: fun_num_fun_num_num,A3: num,B3: num] : X != produc6360833703471310868on_num(F2,produc8585076106096196333on_num(some_num(A3),some_num(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1086_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc3700678068921409119on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_bool,Uv2: option_nat] : X != produc2004789812759622991on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw2: fun_nat_fun_nat_bool,V2: nat] : X != produc2004789812759622991on_nat(Uw2,produc5098337634421038937on_nat(some_nat(V2),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_bool,X4: nat,Y2: nat] : X != produc2004789812759622991on_nat(F2,produc5098337634421038937on_nat(some_nat(X4),some_nat(Y2))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1087_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc533602882674032673at_nat] :
      ( ! [Uu2: fun_Pr248683899324676269t_bool,Uv2: option4927543243414619207at_nat] : X != produc5961576194287291153at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw2: fun_Pr248683899324676269t_bool,V2: product_prod_nat_nat] : X != produc5961576194287291153at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr248683899324676269t_bool,X4: product_prod_nat_nat,Y2: product_prod_nat_nat] : X != produc5961576194287291153at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(X4),some_P7363390416028606310at_nat(Y2))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1088_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc6167219322466932359on_num] :
      ( ! [Uu2: fun_num_fun_num_bool,Uv2: option_num] : X != produc8433281988922931063on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
     => ( ! [Uw2: fun_num_fun_num_bool,V2: num] : X != produc8433281988922931063on_num(Uw2,produc8585076106096196333on_num(some_num(V2),none_num))
       => ~ ! [F2: fun_num_fun_num_bool,X4: num,Y2: num] : X != produc8433281988922931063on_num(F2,produc8585076106096196333on_num(some_num(X4),some_num(Y2))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1089_prod__induct3,axiom,
    ! [P: fun_Pr5900652808213522383r_bool,X: produc5207004643709237345nteger] :
      ( ! [A3: fun_Co6545587727236015215e_term,B3: code_integer,C3: code_integer] : pp(aa_Pro7150874649148492216r_bool(P,aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,A3),produc1086072967326762835nteger(B3,C3))))
     => pp(aa_Pro7150874649148492216r_bool(P,X)) ) ).

% prod_induct3
tff(fact_1090_prod__induct3,axiom,
    ! [P: fun_Pr5991498150630465241r_bool,X: produc1616345218596728535nteger] :
      ( ! [A3: fun_Pr272337284026993017e_term,B3: code_integer,C3: code_integer] : pp(aa_Pro3640759179027349698r_bool(P,aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,A3),produc1086072967326762835nteger(B3,C3))))
     => pp(aa_Pro3640759179027349698r_bool(P,X)) ) ).

% prod_induct3
tff(fact_1091_prod__induct3,axiom,
    ! [P: fun_Pr6019674992625747500t_bool,X: produc4441460161354545860nt_int] :
      ( ! [A3: fun_Pr9164807099015279978e_term,B3: int,C3: int] : pp(aa_Pro5031108791230793749t_bool(P,aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,A3),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B3),C3))))
     => pp(aa_Pro5031108791230793749t_bool(P,X)) ) ).

% prod_induct3
tff(fact_1092_prod__induct3,axiom,
    ! [P: fun_Pr5504123824928244736t_bool,X: produc6768802789813714928nt_int] :
      ( ! [A3: fun_in6429348457993719614e_term,B3: int,C3: int] : pp(aa_Pro7591400126937138665t_bool(P,aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,A3),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B3),C3))))
     => pp(aa_Pro7591400126937138665t_bool(P,X)) ) ).

% prod_induct3
tff(fact_1093_prod__cases3,axiom,
    ! [Y: produc5207004643709237345nteger] :
      ~ ! [A3: fun_Co6545587727236015215e_term,B3: code_integer,C3: code_integer] : Y != aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,A3),produc1086072967326762835nteger(B3,C3)) ).

% prod_cases3
tff(fact_1094_prod__cases3,axiom,
    ! [Y: produc1616345218596728535nteger] :
      ~ ! [A3: fun_Pr272337284026993017e_term,B3: code_integer,C3: code_integer] : Y != aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,A3),produc1086072967326762835nteger(B3,C3)) ).

% prod_cases3
tff(fact_1095_prod__cases3,axiom,
    ! [Y: produc4441460161354545860nt_int] :
      ~ ! [A3: fun_Pr9164807099015279978e_term,B3: int,C3: int] : Y != aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,A3),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B3),C3)) ).

% prod_cases3
tff(fact_1096_prod__cases3,axiom,
    ! [Y: produc6768802789813714928nt_int] :
      ~ ! [A3: fun_in6429348457993719614e_term,B3: int,C3: int] : Y != aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,A3),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B3),C3)) ).

% prod_cases3
tff(fact_1097_Pair__inject,axiom,
    ! [A: int,B: int,A5: int,B5: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A5),B5) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
tff(fact_1098_Pair__inject,axiom,
    ! [A: fun_Co6545587727236015215e_term,B: produc8923325533196201883nteger,A5: fun_Co6545587727236015215e_term,B5: produc8923325533196201883nteger] :
      ( ( aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,A),B) = aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,A5),B5) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
tff(fact_1099_Pair__inject,axiom,
    ! [A: fun_Pr272337284026993017e_term,B: produc8923325533196201883nteger,A5: fun_Pr272337284026993017e_term,B5: produc8923325533196201883nteger] :
      ( ( aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,A),B) = aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,A5),B5) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
tff(fact_1100_Pair__inject,axiom,
    ! [A: fun_Pr9164807099015279978e_term,B: product_prod_int_int,A5: fun_Pr9164807099015279978e_term,B5: product_prod_int_int] :
      ( ( aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,A),B) = aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,A5),B5) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
tff(fact_1101_Pair__inject,axiom,
    ! [A: fun_in6429348457993719614e_term,B: product_prod_int_int,A5: fun_in6429348457993719614e_term,B5: product_prod_int_int] :
      ( ( aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,A),B) = aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,A5),B5) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
tff(fact_1102_prod__cases,axiom,
    ! [P: fun_Pr803113274115125473t_bool,P2: product_prod_int_int] :
      ( ! [A3: int,B3: int] : pp(aa_Pro4501881432599481050t_bool(P,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3)))
     => pp(aa_Pro4501881432599481050t_bool(P,P2)) ) ).

% prod_cases
tff(fact_1103_prod__cases,axiom,
    ! [P: fun_Pr5900652808213522383r_bool,P2: produc5207004643709237345nteger] :
      ( ! [A3: fun_Co6545587727236015215e_term,B3: produc8923325533196201883nteger] : pp(aa_Pro7150874649148492216r_bool(P,aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,A3),B3)))
     => pp(aa_Pro7150874649148492216r_bool(P,P2)) ) ).

% prod_cases
tff(fact_1104_prod__cases,axiom,
    ! [P: fun_Pr5991498150630465241r_bool,P2: produc1616345218596728535nteger] :
      ( ! [A3: fun_Pr272337284026993017e_term,B3: produc8923325533196201883nteger] : pp(aa_Pro3640759179027349698r_bool(P,aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,A3),B3)))
     => pp(aa_Pro3640759179027349698r_bool(P,P2)) ) ).

% prod_cases
tff(fact_1105_prod__cases,axiom,
    ! [P: fun_Pr6019674992625747500t_bool,P2: produc4441460161354545860nt_int] :
      ( ! [A3: fun_Pr9164807099015279978e_term,B3: product_prod_int_int] : pp(aa_Pro5031108791230793749t_bool(P,aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,A3),B3)))
     => pp(aa_Pro5031108791230793749t_bool(P,P2)) ) ).

% prod_cases
tff(fact_1106_prod__cases,axiom,
    ! [P: fun_Pr5504123824928244736t_bool,P2: produc6768802789813714928nt_int] :
      ( ! [A3: fun_in6429348457993719614e_term,B3: product_prod_int_int] : pp(aa_Pro7591400126937138665t_bool(P,aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,A3),B3)))
     => pp(aa_Pro7591400126937138665t_bool(P,P2)) ) ).

% prod_cases
tff(fact_1107_surj__pair,axiom,
    ! [P2: product_prod_int_int] :
    ? [X4: int,Y2: int] : P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y2) ).

% surj_pair
tff(fact_1108_surj__pair,axiom,
    ! [P2: produc5207004643709237345nteger] :
    ? [X4: fun_Co6545587727236015215e_term,Y2: produc8923325533196201883nteger] : P2 = aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,X4),Y2) ).

% surj_pair
tff(fact_1109_surj__pair,axiom,
    ! [P2: produc1616345218596728535nteger] :
    ? [X4: fun_Pr272337284026993017e_term,Y2: produc8923325533196201883nteger] : P2 = aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,X4),Y2) ).

% surj_pair
tff(fact_1110_surj__pair,axiom,
    ! [P2: produc4441460161354545860nt_int] :
    ? [X4: fun_Pr9164807099015279978e_term,Y2: product_prod_int_int] : P2 = aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,X4),Y2) ).

% surj_pair
tff(fact_1111_surj__pair,axiom,
    ! [P2: produc6768802789813714928nt_int] :
    ? [X4: fun_in6429348457993719614e_term,Y2: product_prod_int_int] : P2 = aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,X4),Y2) ).

% surj_pair
tff(fact_1112_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_int_int] :
      ~ ! [A3: int,B3: int] : Y != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) ).

% old.prod.exhaust
tff(fact_1113_old_Oprod_Oexhaust,axiom,
    ! [Y: produc5207004643709237345nteger] :
      ~ ! [A3: fun_Co6545587727236015215e_term,B3: produc8923325533196201883nteger] : Y != aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,A3),B3) ).

% old.prod.exhaust
tff(fact_1114_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1616345218596728535nteger] :
      ~ ! [A3: fun_Pr272337284026993017e_term,B3: produc8923325533196201883nteger] : Y != aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,A3),B3) ).

% old.prod.exhaust
tff(fact_1115_old_Oprod_Oexhaust,axiom,
    ! [Y: produc4441460161354545860nt_int] :
      ~ ! [A3: fun_Pr9164807099015279978e_term,B3: product_prod_int_int] : Y != aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,A3),B3) ).

% old.prod.exhaust
tff(fact_1116_old_Oprod_Oexhaust,axiom,
    ! [Y: produc6768802789813714928nt_int] :
      ~ ! [A3: fun_in6429348457993719614e_term,B3: product_prod_int_int] : Y != aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,A3),B3) ).

% old.prod.exhaust
tff(fact_1117_bounded__Max__nat,axiom,
    ! [P: fun_nat_bool,X: nat,M4: nat] :
      ( pp(aa_nat_bool(P,X))
     => ( ! [X4: nat] :
            ( pp(aa_nat_bool(P,X4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),M4)) )
       => ~ ! [M5: nat] :
              ( pp(aa_nat_bool(P,M5))
             => ~ ! [X2: nat] :
                    ( pp(aa_nat_bool(P,X2))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),M5)) ) ) ) ) ).

% bounded_Max_nat
tff(fact_1118_finite__transitivity__chain,axiom,
    ! [A2: set_set_nat,R: fun_se707272970763642173t_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( ! [X4: set_nat] : ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X4),X4))
       => ( ! [X4: set_nat,Y2: set_nat,Z3: set_nat] :
              ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X4),Y2))
             => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,Y2),Z3))
               => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X4),Z3)) ) )
         => ( ! [X4: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
               => ? [Y3: set_nat] :
                    ( pp(aa_set_set_nat_bool(member_set_nat(Y3),A2))
                    & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X4),Y3)) ) )
           => ( A2 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1119_finite__transitivity__chain,axiom,
    ! [A2: set_complex,R: fun_co1378794544173401837x_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] : ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X4),X4))
       => ( ! [X4: complex,Y2: complex,Z3: complex] :
              ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X4),Y2))
             => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y2),Z3))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X4),Z3)) ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),A2))
               => ? [Y3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y3),A2))
                    & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X4),Y3)) ) )
           => ( A2 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1120_finite__transitivity__chain,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,R: fun_Pr248683899324676269t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
     => ( ! [X4: product_prod_nat_nat] : ~ pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,X4),X4))
       => ( ! [X4: product_prod_nat_nat,Y2: product_prod_nat_nat,Z3: product_prod_nat_nat] :
              ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,X4),Y2))
             => ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,Y2),Z3))
               => pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,X4),Z3)) ) )
         => ( ! [X4: product_prod_nat_nat] :
                ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),A2))
               => ? [Y3: product_prod_nat_nat] :
                    ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y3),A2))
                    & pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,X4),Y3)) ) )
           => ( A2 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1121_finite__transitivity__chain,axiom,
    ! [A2: set_real,R: fun_re1255401033344146673l_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [X4: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X4),X4))
       => ( ! [X4: real,Y2: real,Z3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X4),Y2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y2),Z3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X4),Z3)) ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),A2))
               => ? [Y3: real] :
                    ( pp(aa_set_real_bool(member_real(Y3),A2))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X4),Y3)) ) )
           => ( A2 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1122_finite__transitivity__chain,axiom,
    ! [A2: set_nat,R: fun_nat_fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [X4: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),X4))
       => ( ! [X4: nat,Y2: nat,Z3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Y2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y2),Z3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Z3)) ) )
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),A2))
               => ? [Y3: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y3),A2))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Y3)) ) )
           => ( A2 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1123_finite__transitivity__chain,axiom,
    ! [A2: set_int,R: fun_int_fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(R,X4),X4))
       => ( ! [X4: int,Y2: int,Z3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(R,X4),Y2))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(R,Y2),Z3))
               => pp(aa_int_bool(aa_int_fun_int_bool(R,X4),Z3)) ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),A2))
               => ? [Y3: int] :
                    ( pp(aa_set_int_bool(member_int(Y3),A2))
                    & pp(aa_int_bool(aa_int_fun_int_bool(R,X4),Y3)) ) )
           => ( A2 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1124_finite__nat__set__iff__bounded,axiom,
    ! [N4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
    <=> ? [M3: nat] :
        ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),M3)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_1125_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
    <=> ! [M3: nat] :
        ? [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N3))
          & pp(aa_set_nat_bool(member_nat(N3),S2)) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_1126_bounded__nat__set__is__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),N)) )
     => pp(aa_set_nat_bool(finite_finite_nat,N4)) ) ).

% bounded_nat_set_is_finite
tff(fact_1127_Collect__empty__eq__bot,axiom,
    ! [P: fun_list_nat_bool] :
      ( ( collect_list_nat(P) = bot_bot_set_list_nat )
    <=> ( P = bot_bo6469247547558333820t_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1128_Collect__empty__eq__bot,axiom,
    ! [P: fun_set_nat_bool] :
      ( ( collect_set_nat(P) = bot_bot_set_set_nat )
    <=> ( P = bot_bo8343409700046576086t_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1129_Collect__empty__eq__bot,axiom,
    ! [P: fun_Pr7757163693017731481t_bool] :
      ( ( collec3392354462482085612at_nat(P) = bot_bo2099793752762293965at_nat )
    <=> ( P = bot_bo3782518310583830573t_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1130_Collect__empty__eq__bot,axiom,
    ! [P: fun_real_bool] :
      ( ( collect_real(P) = bot_bot_set_real )
    <=> ( P = bot_bo4181092155253064880l_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1131_Collect__empty__eq__bot,axiom,
    ! [P: fun_nat_bool] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ( P = bot_bot_fun_nat_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1132_Collect__empty__eq__bot,axiom,
    ! [P: fun_int_bool] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ( P = bot_bot_fun_int_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1133_bot__empty__eq,axiom,
    ! [X2: set_nat] :
      ( pp(aa_set_nat_bool(bot_bo8343409700046576086t_bool,X2))
    <=> pp(aa_set_set_nat_bool(member_set_nat(X2),bot_bot_set_set_nat)) ) ).

% bot_empty_eq
tff(fact_1134_bot__empty__eq,axiom,
    ! [X2: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(bot_bo3782518310583830573t_bool,X2))
    <=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),bot_bo2099793752762293965at_nat)) ) ).

% bot_empty_eq
tff(fact_1135_bot__empty__eq,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(bot_bo4181092155253064880l_bool,X2))
    <=> pp(aa_set_real_bool(member_real(X2),bot_bot_set_real)) ) ).

% bot_empty_eq
tff(fact_1136_bot__empty__eq,axiom,
    ! [X2: nat] :
      ( pp(aa_nat_bool(bot_bot_fun_nat_bool,X2))
    <=> pp(aa_set_nat_bool(member_nat(X2),bot_bot_set_nat)) ) ).

% bot_empty_eq
tff(fact_1137_bot__empty__eq,axiom,
    ! [X2: int] :
      ( pp(aa_int_bool(bot_bot_fun_int_bool,X2))
    <=> pp(aa_set_int_bool(member_int(X2),bot_bot_set_int)) ) ).

% bot_empty_eq
tff(fact_1138_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( S2 != bot_bot_set_complex )
       => ~ ? [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),S2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X2)),aa_complex_real(F,lattic8794016678065449205x_real(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1139_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ~ ? [X2: real] :
              ( pp(aa_set_real_bool(member_real(X2),S2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,lattic8440615504127631091l_real(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1140_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( S2 != bot_bot_set_nat )
       => ~ ? [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),S2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X2)),aa_nat_real(F,aa_set_nat_nat(lattic488527866317076247t_real(F),S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1141_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( S2 != bot_bot_set_int )
       => ~ ? [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),S2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X2)),aa_int_real(F,lattic2675449441010098035t_real(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1142_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( S2 != bot_bot_set_complex )
       => ~ ? [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),S2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X2)),aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1143_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ~ ? [X2: real] :
              ( pp(aa_set_real_bool(member_real(X2),S2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,lattic4420706379359479199al_rat(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1144_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( S2 != bot_bot_set_nat )
       => ~ ? [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),S2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,aa_set_nat_nat(lattic6811802900495863747at_rat(F),S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1145_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( S2 != bot_bot_set_int )
       => ~ ? [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),S2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(F,lattic7811156612396918303nt_rat(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1146_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_complex,F: fun_complex_num] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( S2 != bot_bot_set_complex )
       => ~ ? [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),S2))
              & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_complex_num(F,X2)),aa_complex_num(F,lattic1922116423962787043ex_num(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1147_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_real,F: fun_real_num] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ~ ? [X2: real] :
              ( pp(aa_set_real_bool(member_real(X2),S2))
              & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X2)),aa_real_num(F,lattic1613168225601753569al_num(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1148_arg__min__least,axiom,
    ! [S2: set_complex,Y: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( S2 != bot_bot_set_complex )
       => ( pp(aa_set_complex_bool(member_complex(Y),S2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S2))),aa_complex_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1149_arg__min__least,axiom,
    ! [S2: set_real,Y: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ( pp(aa_set_real_bool(member_real(Y),S2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,lattic4420706379359479199al_rat(F,S2))),aa_real_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1150_arg__min__least,axiom,
    ! [S2: set_nat,Y: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( S2 != bot_bot_set_nat )
       => ( pp(aa_set_nat_bool(member_nat(Y),S2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,aa_set_nat_nat(lattic6811802900495863747at_rat(F),S2))),aa_nat_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1151_arg__min__least,axiom,
    ! [S2: set_int,Y: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( S2 != bot_bot_set_int )
       => ( pp(aa_set_int_bool(member_int(Y),S2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,lattic7811156612396918303nt_rat(F,S2))),aa_int_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1152_arg__min__least,axiom,
    ! [S2: set_complex,Y: complex,F: fun_complex_num] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( S2 != bot_bot_set_complex )
       => ( pp(aa_set_complex_bool(member_complex(Y),S2))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_complex_num(F,lattic1922116423962787043ex_num(F,S2))),aa_complex_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1153_arg__min__least,axiom,
    ! [S2: set_real,Y: real,F: fun_real_num] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ( pp(aa_set_real_bool(member_real(Y),S2))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_real_num(F,lattic1613168225601753569al_num(F,S2))),aa_real_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1154_arg__min__least,axiom,
    ! [S2: set_nat,Y: nat,F: fun_nat_num] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( S2 != bot_bot_set_nat )
       => ( pp(aa_set_nat_bool(member_nat(Y),S2))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,aa_set_nat_nat(lattic4004264746738138117at_num(F),S2))),aa_nat_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1155_arg__min__least,axiom,
    ! [S2: set_int,Y: int,F: fun_int_num] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( S2 != bot_bot_set_int )
       => ( pp(aa_set_int_bool(member_int(Y),S2))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F,lattic5003618458639192673nt_num(F,S2))),aa_int_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1156_arg__min__least,axiom,
    ! [S2: set_complex,Y: complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( S2 != bot_bot_set_complex )
       => ( pp(aa_set_complex_bool(member_complex(Y),S2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,lattic5364784637807008409ex_nat(F,S2))),aa_complex_nat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1157_arg__min__least,axiom,
    ! [S2: set_real,Y: real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ( pp(aa_set_real_bool(member_real(Y),S2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,lattic5055836439445974935al_nat(F,S2))),aa_real_nat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1158_nat__descend__induct,axiom,
    ! [N: nat,P: fun_nat_bool,M: nat] :
      ( ! [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2))
         => pp(aa_nat_bool(P,K2)) )
     => ( ! [K2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
           => ( ! [I3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),I3))
                 => pp(aa_nat_bool(P,I3)) )
             => pp(aa_nat_bool(P,K2)) ) )
       => pp(aa_nat_bool(P,M)) ) ) ).

% nat_descend_induct
tff(fact_1159_subset__emptyI,axiom,
    ! [A2: set_set_nat] :
      ( ! [X4: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
     => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),bot_bot_set_set_nat)) ) ).

% subset_emptyI
tff(fact_1160_subset__emptyI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ! [X4: product_prod_nat_nat] : ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),A2))
     => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),bot_bo2099793752762293965at_nat)) ) ).

% subset_emptyI
tff(fact_1161_subset__emptyI,axiom,
    ! [A2: set_real] :
      ( ! [X4: real] : ~ pp(aa_set_real_bool(member_real(X4),A2))
     => pp(aa_set_real_bool(ord_less_eq_set_real(A2),bot_bot_set_real)) ) ).

% subset_emptyI
tff(fact_1162_subset__emptyI,axiom,
    ! [A2: set_nat] :
      ( ! [X4: nat] : ~ pp(aa_set_nat_bool(member_nat(X4),A2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),bot_bot_set_nat)) ) ).

% subset_emptyI
tff(fact_1163_subset__emptyI,axiom,
    ! [A2: set_int] :
      ( ! [X4: int] : ~ pp(aa_set_int_bool(member_int(X4),A2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),bot_bot_set_int)) ) ).

% subset_emptyI
tff(fact_1164_field__lbound__gt__zero,axiom,
    ! [D1: real,D2: 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),D2))
       => ? [E: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E),D1))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E),D2)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1165_field__lbound__gt__zero,axiom,
    ! [D1: rat,D2: 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),D2))
       => ? [E: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E),D1))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E),D2)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1166_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_1167_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_1168_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_1169_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_1170_complete__interval,axiom,
    ! [A: real,B: real,P: fun_real_bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(P,A))
       => ( ~ pp(aa_real_bool(P,B))
         => ? [C3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C3),B))
              & ! [X2: real] :
                  ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),C3)) )
                 => pp(aa_real_bool(P,X2)) )
              & ! [D3: real] :
                  ( ! [X4: real] :
                      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
                        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),D3)) )
                     => pp(aa_real_bool(P,X4)) )
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,D3),C3)) ) ) ) ) ) ).

% complete_interval
tff(fact_1171_complete__interval,axiom,
    ! [A: nat,B: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(P,A))
       => ( ~ pp(aa_nat_bool(P,B))
         => ? [C3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C3))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C3),B))
              & ! [X2: nat] :
                  ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X2))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),C3)) )
                 => pp(aa_nat_bool(P,X2)) )
              & ! [D3: nat] :
                  ( ! [X4: nat] :
                      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X4))
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),D3)) )
                     => pp(aa_nat_bool(P,X4)) )
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D3),C3)) ) ) ) ) ) ).

% complete_interval
tff(fact_1172_complete__interval,axiom,
    ! [A: int,B: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(P,A))
       => ( ~ pp(aa_int_bool(P,B))
         => ? [C3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C3))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C3),B))
              & ! [X2: int] :
                  ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X2))
                    & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),C3)) )
                 => pp(aa_int_bool(P,X2)) )
              & ! [D3: int] :
                  ( ! [X4: int] :
                      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X4))
                        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),D3)) )
                     => pp(aa_int_bool(P,X4)) )
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D3),C3)) ) ) ) ) ) ).

% complete_interval
tff(fact_1173_verit__comp__simplify1_I3_J,axiom,
    ! [B5: real,A5: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B5),A5))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_1174_verit__comp__simplify1_I3_J,axiom,
    ! [B5: rat,A5: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B5),A5))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_1175_verit__comp__simplify1_I3_J,axiom,
    ! [B5: num,A5: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B5),A5))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_1176_verit__comp__simplify1_I3_J,axiom,
    ! [B5: nat,A5: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B5),A5))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_1177_verit__comp__simplify1_I3_J,axiom,
    ! [B5: int,A5: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),A5))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_1178_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_1179_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_1180_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_1181_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_1182_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_1183_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_1184_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_1185_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_1186_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_cancel_left
tff(fact_1187_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_cancel_left
tff(fact_1188_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: 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,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_cancel_left
tff(fact_1189_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: 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,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_cancel_left
tff(fact_1190_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_cancel_right
tff(fact_1191_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_cancel_right
tff(fact_1192_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: 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,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_cancel_right
tff(fact_1193_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: 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,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_cancel_right
tff(fact_1194_add_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.right_neutral
tff(fact_1195_add_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% add.right_neutral
tff(fact_1196_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_1197_add_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% add.right_neutral
tff(fact_1198_add_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% add.right_neutral
tff(fact_1199_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) )
    <=> ( A = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_1200_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_1201_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) )
    <=> ( A = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_1202_add__cancel__left__left,axiom,
    ! [B: complex,A: complex] :
      ( ( aa_complex_complex(plus_plus_complex(B),A) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_left
tff(fact_1203_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_1204_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_1205_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_1206_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_1207_add__cancel__left__right,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_right
tff(fact_1208_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_1209_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_1210_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_1211_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_1212_add__cancel__right__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(B),A) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_left
tff(fact_1213_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_1214_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_1215_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_1216_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_1217_add__cancel__right__right,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(A),B) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_right
tff(fact_1218_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_1219_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_1220_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_1221_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_1222_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_1223_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_1224_add__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add_0
tff(fact_1225_add__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add_0
tff(fact_1226_add__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% add_0
tff(fact_1227_add__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).

% add_0
tff(fact_1228_add__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% add_0
tff(fact_1229_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_cancel_left
tff(fact_1230_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_cancel_left
tff(fact_1231_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_cancel_left
tff(fact_1232_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_cancel_left
tff(fact_1233_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_cancel_right
tff(fact_1234_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_cancel_right
tff(fact_1235_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_cancel_right
tff(fact_1236_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_cancel_right
tff(fact_1237_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_1238_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_1239_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_1240_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_1241_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1242_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1243_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: 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,A),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1244_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1245_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1246_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: 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,A),A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1247_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ).

% le_add_same_cancel2
tff(fact_1248_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ).

% le_add_same_cancel2
tff(fact_1249_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) ).

% le_add_same_cancel2
tff(fact_1250_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ).

% le_add_same_cancel2
tff(fact_1251_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ).

% le_add_same_cancel1
tff(fact_1252_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ).

% le_add_same_cancel1
tff(fact_1253_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) ).

% le_add_same_cancel1
tff(fact_1254_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ).

% le_add_same_cancel1
tff(fact_1255_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% add_le_same_cancel2
tff(fact_1256_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% add_le_same_cancel2
tff(fact_1257_add__le__same__cancel2,axiom,
    ! [A: nat,B: 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,A),B)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat)) ) ).

% add_le_same_cancel2
tff(fact_1258_add__le__same__cancel2,axiom,
    ! [A: int,B: 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,A),B)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% add_le_same_cancel2
tff(fact_1259_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% add_le_same_cancel1
tff(fact_1260_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% add_le_same_cancel1
tff(fact_1261_add__le__same__cancel1,axiom,
    ! [B: nat,A: 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,B),A)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat)) ) ).

% add_le_same_cancel1
tff(fact_1262_add__le__same__cancel1,axiom,
    ! [B: int,A: 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,B),A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% add_le_same_cancel1
tff(fact_1263_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% add_less_same_cancel1
tff(fact_1264_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% add_less_same_cancel1
tff(fact_1265_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ) ).

% add_less_same_cancel1
tff(fact_1266_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% add_less_same_cancel1
tff(fact_1267_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% add_less_same_cancel2
tff(fact_1268_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% add_less_same_cancel2
tff(fact_1269_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ) ).

% add_less_same_cancel2
tff(fact_1270_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% add_less_same_cancel2
tff(fact_1271_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ).

% less_add_same_cancel1
tff(fact_1272_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ).

% less_add_same_cancel1
tff(fact_1273_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ).

% less_add_same_cancel1
tff(fact_1274_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ).

% less_add_same_cancel1
tff(fact_1275_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ).

% less_add_same_cancel2
tff(fact_1276_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ).

% less_add_same_cancel2
tff(fact_1277_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ).

% less_add_same_cancel2
tff(fact_1278_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ).

% less_add_same_cancel2
tff(fact_1279_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1280_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1281_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1282_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1283_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1284_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: 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,A),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1285_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_1286_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_1287_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_1288_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_1289_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_1290_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_1291_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_1292_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_1293_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_1294_add_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% add.left_commute
tff(fact_1295_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% add.left_commute
tff(fact_1296_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.left_commute
tff(fact_1297_add_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.left_commute
tff(fact_1298_add_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) ).

% add.commute
tff(fact_1299_add_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) ).

% add.commute
tff(fact_1300_add_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) ).

% add.commute
tff(fact_1301_add_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) ).

% add.commute
tff(fact_1302_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_1303_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_1304_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_1305_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_1306_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_1307_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_1308_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% add.assoc
tff(fact_1309_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% add.assoc
tff(fact_1310_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.assoc
tff(fact_1311_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.assoc
tff(fact_1312_group__cancel_Oadd2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( 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,A),B)) ) ) ).

% group_cancel.add2
tff(fact_1313_group__cancel_Oadd2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_1314_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B) )
     => ( 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,A),B)) ) ) ).

% group_cancel.add2
tff(fact_1315_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( 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,A),B)) ) ) ).

% group_cancel.add2
tff(fact_1316_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( 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,A),B)) ) ) ).

% group_cancel.add1
tff(fact_1317_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_1318_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( 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,A),B)) ) ) ).

% group_cancel.add1
tff(fact_1319_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( 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,A),B)) ) ) ).

% group_cancel.add1
tff(fact_1320_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_1321_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_1322_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_1323_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_1324_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_1325_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_1326_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_1327_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_1328_verit__sum__simplify,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% verit_sum_simplify
tff(fact_1329_verit__sum__simplify,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% verit_sum_simplify
tff(fact_1330_verit__sum__simplify,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_1331_verit__sum__simplify,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% verit_sum_simplify
tff(fact_1332_verit__sum__simplify,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% verit_sum_simplify
tff(fact_1333_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_1334_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(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1335_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_1336_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_1337_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_1338_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(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1339_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_1340_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_1341_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_1342_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(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1343_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_1344_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_1345_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_mono
tff(fact_1346_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D))) ) ) ).

% add_mono
tff(fact_1347_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_mono
tff(fact_1348_add__mono,axiom,
    ! [A: int,B: int,C: int,D: 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_eq_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_mono
tff(fact_1349_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% add_left_mono
tff(fact_1350_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))) ) ).

% add_left_mono
tff(fact_1351_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))) ) ).

% add_left_mono
tff(fact_1352_add__left__mono,axiom,
    ! [A: int,B: int,C: 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_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% add_left_mono
tff(fact_1353_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ~ ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).

% less_eqE
tff(fact_1354_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))) ) ).

% add_right_mono
tff(fact_1355_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))) ) ).

% add_right_mono
tff(fact_1356_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))) ) ).

% add_right_mono
tff(fact_1357_add__right__mono,axiom,
    ! [A: int,B: int,C: 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_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))) ) ).

% add_right_mono
tff(fact_1358_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ? [C4: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C4) ) ).

% le_iff_add
tff(fact_1359_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_imp_le_left
tff(fact_1360_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_imp_le_left
tff(fact_1361_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: 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,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_imp_le_left
tff(fact_1362_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: 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,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_imp_le_left
tff(fact_1363_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_imp_le_right
tff(fact_1364_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_imp_le_right
tff(fact_1365_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: 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,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_imp_le_right
tff(fact_1366_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: 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,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_imp_le_right
tff(fact_1367_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1368_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1369_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1370_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1371_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1372_add_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.comm_neutral
tff(fact_1373_add_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% add.comm_neutral
tff(fact_1374_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_1375_add_Ocomm__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% add.comm_neutral
tff(fact_1376_add_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% add.comm_neutral
tff(fact_1377_add_Ogroup__left__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add.group_left_neutral
tff(fact_1378_add_Ogroup__left__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add.group_left_neutral
tff(fact_1379_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_1380_add_Ogroup__left__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% add.group_left_neutral
tff(fact_1381_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_1382_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(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1383_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_1384_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_1385_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_1386_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(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1387_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_1388_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_1389_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_1390_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(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1391_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_1392_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_1393_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_strict_mono
tff(fact_1394_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D))) ) ) ).

% add_strict_mono
tff(fact_1395_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: 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,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_strict_mono
tff(fact_1396_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: 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,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_strict_mono
tff(fact_1397_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% add_strict_left_mono
tff(fact_1398_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))) ) ).

% add_strict_left_mono
tff(fact_1399_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: 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,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))) ) ).

% add_strict_left_mono
tff(fact_1400_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: 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,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% add_strict_left_mono
tff(fact_1401_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))) ) ).

% add_strict_right_mono
tff(fact_1402_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))) ) ).

% add_strict_right_mono
tff(fact_1403_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: 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,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))) ) ).

% add_strict_right_mono
tff(fact_1404_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: 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,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))) ) ).

% add_strict_right_mono
tff(fact_1405_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_imp_less_left
tff(fact_1406_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_imp_less_left
tff(fact_1407_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_imp_less_left
tff(fact_1408_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_imp_less_left
tff(fact_1409_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_imp_less_right
tff(fact_1410_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_imp_less_right
tff(fact_1411_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_imp_less_right
tff(fact_1412_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_imp_less_right
tff(fact_1413_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_1414_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_1415_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_1416_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_1417_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_1418_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_1419_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_1420_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_1421_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_1422_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_1423_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_1424_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_1425_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_1426_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_1427_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_1428_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
     => ? [N2: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2) ) ).

% le_Suc_ex
tff(fact_1429_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_1430_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_1431_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_1432_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_1433_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_1434_add__nonpos__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y) = zero_zero_real )
        <=> ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1435_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1436_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),zero_zero_nat))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1437_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1438_add__nonneg__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y) = zero_zero_real )
        <=> ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1439_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1440_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1441_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1442_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_nonpos_nonpos
tff(fact_1443_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat)) ) ) ).

% add_nonpos_nonpos
tff(fact_1444_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),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,A),B)),zero_zero_nat)) ) ) ).

% add_nonpos_nonpos
tff(fact_1445_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),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,A),B)),zero_zero_int)) ) ) ).

% add_nonpos_nonpos
tff(fact_1446_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_1447_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_1448_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => 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,A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_1449_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => 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,A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_1450_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_increasing2
tff(fact_1451_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C))) ) ) ).

% add_increasing2
tff(fact_1452_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_increasing2
tff(fact_1453_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_increasing2
tff(fact_1454_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_1455_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_1456_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_1457_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_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_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_1458_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_increasing
tff(fact_1459_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C))) ) ) ).

% add_increasing
tff(fact_1460_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_increasing
tff(fact_1461_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_increasing
tff(fact_1462_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B)) ) ) ).

% add_decreasing
tff(fact_1463_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B)) ) ) ).

% add_decreasing
tff(fact_1464_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B)) ) ) ).

% add_decreasing
tff(fact_1465_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B)) ) ) ).

% add_decreasing
tff(fact_1466_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_less_le_mono
tff(fact_1467_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D))) ) ) ).

% add_less_le_mono
tff(fact_1468_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: 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_eq_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_less_le_mono
tff(fact_1469_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: 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_eq_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_less_le_mono
tff(fact_1470_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_le_less_mono
tff(fact_1471_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D))) ) ) ).

% add_le_less_mono
tff(fact_1472_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_le_less_mono
tff(fact_1473_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: 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,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_le_less_mono
tff(fact_1474_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_1475_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(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1476_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_1477_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_1478_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_1479_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(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1480_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_1481_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_1482_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_neg_neg
tff(fact_1483_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat)) ) ) ).

% add_neg_neg
tff(fact_1484_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),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,A),B)),zero_zero_nat)) ) ) ).

% add_neg_neg
tff(fact_1485_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),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,A),B)),zero_zero_int)) ) ) ).

% add_neg_neg
tff(fact_1486_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_pos_pos
tff(fact_1487_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))) ) ) ).

% add_pos_pos
tff(fact_1488_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => 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,A),B))) ) ) ).

% add_pos_pos
tff(fact_1489_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => 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,A),B))) ) ) ).

% add_pos_pos
tff(fact_1490_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ~ ! [C3: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1491_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% pos_add_strict
tff(fact_1492_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C))) ) ) ).

% pos_add_strict
tff(fact_1493_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% pos_add_strict
tff(fact_1494_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% pos_add_strict
tff(fact_1495_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_1496_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M5: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,M5)),aa_nat_nat(F,N2))) )
     => 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_1497_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_neg_nonpos
tff(fact_1498_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat)) ) ) ).

% add_neg_nonpos
tff(fact_1499_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),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,A),B)),zero_zero_nat)) ) ) ).

% add_neg_nonpos
tff(fact_1500_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),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,A),B)),zero_zero_int)) ) ) ).

% add_neg_nonpos
tff(fact_1501_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_1502_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_1503_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => 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,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_1504_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => 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,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_1505_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_nonpos_neg
tff(fact_1506_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat)) ) ) ).

% add_nonpos_neg
tff(fact_1507_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),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,A),B)),zero_zero_nat)) ) ) ).

% add_nonpos_neg
tff(fact_1508_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),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,A),B)),zero_zero_int)) ) ) ).

% add_nonpos_neg
tff(fact_1509_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_1510_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_1511_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => 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,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_1512_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => 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,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_1513_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_strict_increasing
tff(fact_1514_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C))) ) ) ).

% add_strict_increasing
tff(fact_1515_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_strict_increasing
tff(fact_1516_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_strict_increasing
tff(fact_1517_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_1518_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_1519_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_1520_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_1521_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1522_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1523_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1524_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1525_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1526_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
      | ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% verit_la_disequality
tff(fact_1527_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
      | ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ).

% verit_la_disequality
tff(fact_1528_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
      | ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).

% verit_la_disequality
tff(fact_1529_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ 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_eq_int,B),A)) ) ).

% verit_la_disequality
tff(fact_1530_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1531_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1532_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1533_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1534_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1535_ex__gt__or__lt,axiom,
    ! [A: real] :
    ? [B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B3))
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A)) ) ).

% ex_gt_or_lt
tff(fact_1536_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Y2: nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y2)),R2))
         => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y2)),S)) )
     => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(R2),S)) ) ).

% subrelI
tff(fact_1537_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ! [X4: int,Y2: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y2),R2)
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y2),S) )
     => ord_le2843351958646193337nt_int(R2,S) ) ).

% subrelI
tff(fact_1538_subrelI,axiom,
    ! [R2: set_Pr7592006421409863191nteger,S: set_Pr7592006421409863191nteger] :
      ( ! [X4: fun_Co6545587727236015215e_term,Y2: produc8923325533196201883nteger] :
          ( member4581088367259704696nteger(aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,X4),Y2),R2)
         => member4581088367259704696nteger(aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,X4),Y2),S) )
     => ord_le5347933966117430903nteger(R2,S) ) ).

% subrelI
tff(fact_1539_subrelI,axiom,
    ! [R2: set_Pr5663546166630115725nteger,S: set_Pr5663546166630115725nteger] :
      ( ! [X4: fun_Pr272337284026993017e_term,Y2: produc8923325533196201883nteger] :
          ( member6423679177046879214nteger(aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,X4),Y2),R2)
         => member6423679177046879214nteger(aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,X4),Y2),S) )
     => ord_le3337582088779512301nteger(R2,S) ) ).

% subrelI
tff(fact_1540_subrelI,axiom,
    ! [R2: set_Pr233528076387193594nt_int,S: set_Pr233528076387193594nt_int] :
      ( ! [X4: fun_Pr9164807099015279978e_term,Y2: product_prod_int_int] :
          ( member690604328116145115nt_int(aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,X4),Y2),R2)
         => member690604328116145115nt_int(aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,X4),Y2),S) )
     => ord_le5172437211506631514nt_int(R2,S) ) ).

% subrelI
tff(fact_1541_subrelI,axiom,
    ! [R2: set_Pr5033440245338114598nt_int,S: set_Pr5033440245338114598nt_int] :
      ( ! [X4: fun_in6429348457993719614e_term,Y2: product_prod_int_int] :
          ( member500491123954006279nt_int(aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,X4),Y2),R2)
         => member500491123954006279nt_int(aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,X4),Y2),S) )
     => ord_le172202941055982214nt_int(R2,S) ) ).

% subrelI
tff(fact_1542_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S: nat,R: set_Pr1261947904930325089at_nat,S3: nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(R2),S)),R))
     => ( ( S3 = S )
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(R2),S3)),R)) ) ) ).

% ssubst_Pair_rhs
tff(fact_1543_ssubst__Pair__rhs,axiom,
    ! [R2: int,S: int,R: set_Pr958786334691620121nt_int,S3: int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R2),S),R)
     => ( ( S3 = S )
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R2),S3),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1544_ssubst__Pair__rhs,axiom,
    ! [R2: fun_Co6545587727236015215e_term,S: produc8923325533196201883nteger,R: set_Pr7592006421409863191nteger,S3: produc8923325533196201883nteger] :
      ( member4581088367259704696nteger(aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,R2),S),R)
     => ( ( S3 = S )
       => member4581088367259704696nteger(aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,R2),S3),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1545_ssubst__Pair__rhs,axiom,
    ! [R2: fun_Pr272337284026993017e_term,S: produc8923325533196201883nteger,R: set_Pr5663546166630115725nteger,S3: produc8923325533196201883nteger] :
      ( member6423679177046879214nteger(aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,R2),S),R)
     => ( ( S3 = S )
       => member6423679177046879214nteger(aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,R2),S3),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1546_ssubst__Pair__rhs,axiom,
    ! [R2: fun_Pr9164807099015279978e_term,S: product_prod_int_int,R: set_Pr233528076387193594nt_int,S3: product_prod_int_int] :
      ( member690604328116145115nt_int(aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,R2),S),R)
     => ( ( S3 = S )
       => member690604328116145115nt_int(aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,R2),S3),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1547_ssubst__Pair__rhs,axiom,
    ! [R2: fun_in6429348457993719614e_term,S: product_prod_int_int,R: set_Pr5033440245338114598nt_int,S3: product_prod_int_int] :
      ( member500491123954006279nt_int(aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,R2),S),R)
     => ( ( S3 = S )
       => member500491123954006279nt_int(aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,R2),S3),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1548_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_1549_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_1550_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_1551_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_1552_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_1553_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_1554_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% double_eq_0_iff
tff(fact_1555_add__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,some_nat(X)),some_nat(Y)) = some_nat(Z) ) ) ).

% add_shift
tff(fact_1556_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% field_le_epsilon
tff(fact_1557_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),E))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% field_le_epsilon
tff(fact_1558_add__less__zeroD,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real)) ) ) ).

% add_less_zeroD
tff(fact_1559_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat)) ) ) ).

% add_less_zeroD
tff(fact_1560_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),zero_zero_int)) ) ) ).

% add_less_zeroD
tff(fact_1561_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_VEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),size_s6755466524823107622T_VEBT(Xs)))
     => ( nth_Pr744662078594809490T_VEBT(enumerate_VEBT_VEBT(N,Xs),M) = produc599794634098209291T_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M),nth_VEBT_VEBT(Xs,M)) ) ) ).

% nth_enumerate_eq
tff(fact_1562_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_bool,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),size_size_list_bool(Xs)))
     => ( nth_Pr396318242284072693t_bool(enumerate_bool(N,Xs),M) = produc3913204875133826030t_bool(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M),aa_nat_bool(nth_bool(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_1563_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),size_size_list_nat(Xs)))
     => ( nth_Pr7617993195940197384at_nat(enumerate_nat(N,Xs),M) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_1564_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),size_size_list_int(Xs)))
     => ( nth_Pr3440142176431000676at_int(enumerate_int(N,Xs),M) = product_Pair_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M),aa_nat_int(nth_int(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_1565_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B: 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,A),B)) ) ) ) ).

% Euclid_induct
tff(fact_1566_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B = aa_complex_complex(plus_plus_complex(B),A) )
    <=> ( A = zero_zero_complex ) ) ).

% add_0_iff
tff(fact_1567_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
    <=> ( A = zero_zero_real ) ) ).

% add_0_iff
tff(fact_1568_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_1569_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( A = zero_zero_nat ) ) ).

% add_0_iff
tff(fact_1570_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
    <=> ( A = zero_zero_int ) ) ).

% add_0_iff
tff(fact_1571_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z3))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X2)) ) ).

% minf(8)
tff(fact_1572_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z3))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X2)) ) ).

% minf(8)
tff(fact_1573_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z3))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X2)) ) ).

% minf(8)
tff(fact_1574_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( 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,T),X2)) ) ).

% minf(8)
tff(fact_1575_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z3))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X2)) ) ).

% minf(8)
tff(fact_1576_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),T)) ) ).

% minf(6)
tff(fact_1577_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),T)) ) ).

% minf(6)
tff(fact_1578_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z3))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),T)) ) ).

% minf(6)
tff(fact_1579_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( 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,X2),T)) ) ).

% minf(6)
tff(fact_1580_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),T)) ) ).

% minf(6)
tff(fact_1581_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X2)) ) ).

% pinf(8)
tff(fact_1582_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X2)) ) ).

% pinf(8)
tff(fact_1583_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X2)) ) ).

% pinf(8)
tff(fact_1584_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( 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,T),X2)) ) ).

% pinf(8)
tff(fact_1585_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X2)) ) ).

% pinf(8)
tff(fact_1586_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),T)) ) ).

% pinf(6)
tff(fact_1587_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X2))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),T)) ) ).

% pinf(6)
tff(fact_1588_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),T)) ) ).

% pinf(6)
tff(fact_1589_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( 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,X2),T)) ) ).

% pinf(6)
tff(fact_1590_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),T)) ) ).

% pinf(6)
tff(fact_1591_length__enumerate,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] : size_s4762443039079500285T_VEBT(enumerate_VEBT_VEBT(N,Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_enumerate
tff(fact_1592_length__enumerate,axiom,
    ! [N: nat,Xs: list_bool] : size_s783543119881238752t_bool(enumerate_bool(N,Xs)) = size_size_list_bool(Xs) ).

% length_enumerate
tff(fact_1593_length__enumerate,axiom,
    ! [N: nat,Xs: list_nat] : size_s5460976970255530739at_nat(enumerate_nat(N,Xs)) = size_size_list_nat(Xs) ).

% length_enumerate
tff(fact_1594_length__enumerate,axiom,
    ! [N: nat,Xs: list_int] : size_s2970893825323803983at_int(enumerate_int(N,Xs)) = size_size_list_int(Xs) ).

% length_enumerate
tff(fact_1595_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_1596_less__by__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( A2 = bot_bo2099793752762293965at_nat )
     => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B2)) ) ).

% less_by_empty
tff(fact_1597_linorder__neqE__linordered__idom,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1598_linorder__neqE__linordered__idom,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1599_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1600_linordered__field__no__ub,axiom,
    ! [X2: real] :
    ? [X_1: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),X_1)) ).

% linordered_field_no_ub
tff(fact_1601_linordered__field__no__ub,axiom,
    ! [X2: rat] :
    ? [X_1: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),X_1)) ).

% linordered_field_no_ub
tff(fact_1602_linordered__field__no__lb,axiom,
    ! [X2: real] :
    ? [Y2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),X2)) ).

% linordered_field_no_lb
tff(fact_1603_linordered__field__no__lb,axiom,
    ! [X2: rat] :
    ? [Y2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y2),X2)) ).

% linordered_field_no_lb
tff(fact_1604_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)) ) )
       => ? [Z3: real] :
          ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X2))
           => ( ( pp(aa_real_bool(P,X2))
                & pp(aa_real_bool(Q,X2)) )
            <=> ( pp(aa_real_bool(P3,X2))
                & pp(aa_real_bool(Q2,X2)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1605_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)) ) )
       => ? [Z3: rat] :
          ! [X2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X2))
           => ( ( pp(aa_rat_bool(P,X2))
                & pp(aa_rat_bool(Q,X2)) )
            <=> ( pp(aa_rat_bool(P3,X2))
                & pp(aa_rat_bool(Q2,X2)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1606_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)) ) )
       => ? [Z3: num] :
          ! [X2: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X2))
           => ( ( pp(aa_num_bool(P,X2))
                & pp(aa_num_bool(Q,X2)) )
            <=> ( pp(aa_num_bool(P3,X2))
                & pp(aa_num_bool(Q2,X2)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1607_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)) ) )
       => ? [Z3: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X2))
           => ( ( pp(aa_nat_bool(P,X2))
                & pp(aa_nat_bool(Q,X2)) )
            <=> ( pp(aa_nat_bool(P3,X2))
                & pp(aa_nat_bool(Q2,X2)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1608_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)) ) )
       => ? [Z3: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X2))
           => ( ( pp(aa_int_bool(P,X2))
                & pp(aa_int_bool(Q,X2)) )
            <=> ( pp(aa_int_bool(P3,X2))
                & pp(aa_int_bool(Q2,X2)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1609_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)) ) )
       => ? [Z3: real] :
          ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X2))
           => ( ( pp(aa_real_bool(P,X2))
                | pp(aa_real_bool(Q,X2)) )
            <=> ( pp(aa_real_bool(P3,X2))
                | pp(aa_real_bool(Q2,X2)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1610_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)) ) )
       => ? [Z3: rat] :
          ! [X2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X2))
           => ( ( pp(aa_rat_bool(P,X2))
                | pp(aa_rat_bool(Q,X2)) )
            <=> ( pp(aa_rat_bool(P3,X2))
                | pp(aa_rat_bool(Q2,X2)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1611_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)) ) )
       => ? [Z3: num] :
          ! [X2: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X2))
           => ( ( pp(aa_num_bool(P,X2))
                | pp(aa_num_bool(Q,X2)) )
            <=> ( pp(aa_num_bool(P3,X2))
                | pp(aa_num_bool(Q2,X2)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1612_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)) ) )
       => ? [Z3: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X2))
           => ( ( pp(aa_nat_bool(P,X2))
                | pp(aa_nat_bool(Q,X2)) )
            <=> ( pp(aa_nat_bool(P3,X2))
                | pp(aa_nat_bool(Q2,X2)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1613_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)) ) )
       => ? [Z3: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X2))
           => ( ( pp(aa_int_bool(P,X2))
                | pp(aa_int_bool(Q,X2)) )
            <=> ( pp(aa_int_bool(P3,X2))
                | pp(aa_int_bool(Q2,X2)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1614_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X2))
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_1615_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X2))
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_1616_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X2))
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_1617_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X2))
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_1618_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X2))
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_1619_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X2))
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_1620_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X2))
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_1621_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X2))
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_1622_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X2))
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_1623_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X2))
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_1624_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),T)) ) ).

% pinf(5)
tff(fact_1625_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X2))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),T)) ) ).

% pinf(5)
tff(fact_1626_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),T)) ) ).

% pinf(5)
tff(fact_1627_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( 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_nat,X2),T)) ) ).

% pinf(5)
tff(fact_1628_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),T)) ) ).

% pinf(5)
tff(fact_1629_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X2)) ) ).

% pinf(7)
tff(fact_1630_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X2)) ) ).

% pinf(7)
tff(fact_1631_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X2)) ) ).

% pinf(7)
tff(fact_1632_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( 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_nat,T),X2)) ) ).

% pinf(7)
tff(fact_1633_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X2)) ) ).

% pinf(7)
tff(fact_1634_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)) ) )
       => ? [Z3: real] :
          ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z3))
           => ( ( pp(aa_real_bool(P,X2))
                & pp(aa_real_bool(Q,X2)) )
            <=> ( pp(aa_real_bool(P3,X2))
                & pp(aa_real_bool(Q2,X2)) ) ) ) ) ) ).

% minf(1)
tff(fact_1635_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)) ) )
       => ? [Z3: rat] :
          ! [X2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z3))
           => ( ( pp(aa_rat_bool(P,X2))
                & pp(aa_rat_bool(Q,X2)) )
            <=> ( pp(aa_rat_bool(P3,X2))
                & pp(aa_rat_bool(Q2,X2)) ) ) ) ) ) ).

% minf(1)
tff(fact_1636_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)) ) )
       => ? [Z3: num] :
          ! [X2: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z3))
           => ( ( pp(aa_num_bool(P,X2))
                & pp(aa_num_bool(Q,X2)) )
            <=> ( pp(aa_num_bool(P3,X2))
                & pp(aa_num_bool(Q2,X2)) ) ) ) ) ) ).

% minf(1)
tff(fact_1637_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)) ) )
       => ? [Z3: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z3))
           => ( ( pp(aa_nat_bool(P,X2))
                & pp(aa_nat_bool(Q,X2)) )
            <=> ( pp(aa_nat_bool(P3,X2))
                & pp(aa_nat_bool(Q2,X2)) ) ) ) ) ) ).

% minf(1)
tff(fact_1638_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)) ) )
       => ? [Z3: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z3))
           => ( ( pp(aa_int_bool(P,X2))
                & pp(aa_int_bool(Q,X2)) )
            <=> ( pp(aa_int_bool(P3,X2))
                & pp(aa_int_bool(Q2,X2)) ) ) ) ) ) ).

% minf(1)
tff(fact_1639_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)) ) )
       => ? [Z3: real] :
          ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z3))
           => ( ( pp(aa_real_bool(P,X2))
                | pp(aa_real_bool(Q,X2)) )
            <=> ( pp(aa_real_bool(P3,X2))
                | pp(aa_real_bool(Q2,X2)) ) ) ) ) ) ).

% minf(2)
tff(fact_1640_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)) ) )
       => ? [Z3: rat] :
          ! [X2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z3))
           => ( ( pp(aa_rat_bool(P,X2))
                | pp(aa_rat_bool(Q,X2)) )
            <=> ( pp(aa_rat_bool(P3,X2))
                | pp(aa_rat_bool(Q2,X2)) ) ) ) ) ) ).

% minf(2)
tff(fact_1641_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)) ) )
       => ? [Z3: num] :
          ! [X2: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z3))
           => ( ( pp(aa_num_bool(P,X2))
                | pp(aa_num_bool(Q,X2)) )
            <=> ( pp(aa_num_bool(P3,X2))
                | pp(aa_num_bool(Q2,X2)) ) ) ) ) ) ).

% minf(2)
tff(fact_1642_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)) ) )
       => ? [Z3: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z3))
           => ( ( pp(aa_nat_bool(P,X2))
                | pp(aa_nat_bool(Q,X2)) )
            <=> ( pp(aa_nat_bool(P3,X2))
                | pp(aa_nat_bool(Q2,X2)) ) ) ) ) ) ).

% minf(2)
tff(fact_1643_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)) ) )
       => ? [Z3: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z3))
           => ( ( pp(aa_int_bool(P,X2))
                | pp(aa_int_bool(Q,X2)) )
            <=> ( pp(aa_int_bool(P3,X2))
                | pp(aa_int_bool(Q2,X2)) ) ) ) ) ) ).

% minf(2)
tff(fact_1644_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z3))
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_1645_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z3))
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_1646_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z3))
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_1647_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z3))
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_1648_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z3))
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_1649_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z3))
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_1650_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z3))
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_1651_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z3))
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_1652_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z3))
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_1653_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z3))
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_1654_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),T)) ) ).

% minf(5)
tff(fact_1655_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),T)) ) ).

% minf(5)
tff(fact_1656_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z3))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),T)) ) ).

% minf(5)
tff(fact_1657_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( 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_nat,X2),T)) ) ).

% minf(5)
tff(fact_1658_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),T)) ) ).

% minf(5)
tff(fact_1659_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z3))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X2)) ) ).

% minf(7)
tff(fact_1660_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z3))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X2)) ) ).

% minf(7)
tff(fact_1661_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z3))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X2)) ) ).

% minf(7)
tff(fact_1662_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( 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_nat,T),X2)) ) ).

% minf(7)
tff(fact_1663_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z3))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X2)) ) ).

% minf(7)
tff(fact_1664_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,N))))
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S4: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,N)),TreeList2,S4) ) ).

% deg_SUcn_Node
tff(fact_1665_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( 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,I),size_s6755466524823107622T_VEBT(Ys)))
       => ( nth_Pr4953567300277697838T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys),I) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,I),nth_VEBT_VEBT(Ys,I)) ) ) ) ).

% nth_zip
tff(fact_1666_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_bool] :
      ( 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,I),size_size_list_bool(Ys)))
       => ( nth_Pr1289831921885633937T_bool(zip_VEBT_VEBT_bool(Xs,Ys),I) = produc3035007008238225530T_bool(nth_VEBT_VEBT(Xs,I),aa_nat_bool(nth_bool(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_1667_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_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,I),size_size_list_nat(Ys)))
       => ( nth_Pr1791586995822124652BT_nat(zip_VEBT_VEBT_nat(Xs,Ys),I) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,I),aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,Ys),I)) ) ) ) ).

% nth_zip
tff(fact_1668_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( 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,I),size_size_list_int(Ys)))
       => ( nth_Pr6837108013167703752BT_int(zip_VEBT_VEBT_int(Xs,Ys),I) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,I),aa_nat_int(nth_int(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_1669_nth__zip,axiom,
    ! [I: nat,Xs: list_bool,Ys: list_VEBT_VEBT] :
      ( 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,I),size_s6755466524823107622T_VEBT(Ys)))
       => ( nth_Pr3250499561091541003T_VEBT(zip_bool_VEBT_VEBT(Xs,Ys),I) = produc9074556814250467700T_VEBT(aa_nat_bool(nth_bool(Xs),I),nth_VEBT_VEBT(Ys,I)) ) ) ) ).

% nth_zip
tff(fact_1670_nth__zip,axiom,
    ! [I: nat,Xs: list_bool,Ys: list_bool] :
      ( 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,I),size_size_list_bool(Ys)))
       => ( nth_Pr2856333732135533678l_bool(zip_bool_bool(Xs,Ys),I) = produc8297744024737662807l_bool(aa_nat_bool(nth_bool(Xs),I),aa_nat_bool(nth_bool(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_1671_nth__zip,axiom,
    ! [I: nat,Xs: list_bool,Ys: list_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,I),size_size_list_nat(Ys)))
       => ( nth_Pr2186422209488468047ol_nat(zip_bool_nat(Xs,Ys),I) = produc8874003862241262736ol_nat(aa_nat_bool(nth_bool(Xs),I),aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,Ys),I)) ) ) ) ).

% nth_zip
tff(fact_1672_nth__zip,axiom,
    ! [I: nat,Xs: list_bool,Ys: list_int] :
      ( 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,I),size_size_list_int(Ys)))
       => ( nth_Pr7231943226834047147ol_int(zip_bool_int(Xs,Ys),I) = produc8871513391732212460ol_int(aa_nat_bool(nth_bool(Xs),I),aa_nat_int(nth_int(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_1673_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( 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,I),size_s6755466524823107622T_VEBT(Ys)))
       => ( nth_Pr744662078594809490T_VEBT(zip_nat_VEBT_VEBT(Xs,Ys),I) = produc599794634098209291T_VEBT(aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,Xs),I),nth_VEBT_VEBT(Ys,I)) ) ) ) ).

% nth_zip
tff(fact_1674_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_bool] :
      ( 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,I),size_size_list_bool(Ys)))
       => ( nth_Pr396318242284072693t_bool(zip_nat_bool(Xs,Ys),I) = produc3913204875133826030t_bool(aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,Xs),I),aa_nat_bool(nth_bool(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_1675_find__Some__iff2,axiom,
    ! [X: product_prod_nat_nat,P: fun_Pr7757163693017731481t_bool,Xs: list_P6011104703257516679at_nat] :
      ( ( some_P7363390416028606310at_nat(X) = find_P8199882355184865565at_nat(P,Xs) )
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s5460976970255530739at_nat(Xs)))
          & pp(aa_Pro2232559814647311250t_bool(P,nth_Pr7617993195940197384at_nat(Xs,I4)))
          & ( X = nth_Pr7617993195940197384at_nat(Xs,I4) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_Pro2232559814647311250t_bool(P,nth_Pr7617993195940197384at_nat(Xs,J3))) ) ) ) ).

% find_Some_iff2
tff(fact_1676_find__Some__iff2,axiom,
    ! [X: num,P: fun_num_bool,Xs: list_num] :
      ( ( some_num(X) = find_num(P,Xs) )
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_num(Xs)))
          & pp(aa_num_bool(P,aa_nat_num(nth_num(Xs),I4)))
          & ( X = aa_nat_num(nth_num(Xs),I4) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_num_bool(P,aa_nat_num(nth_num(Xs),J3))) ) ) ) ).

% find_Some_iff2
tff(fact_1677_find__Some__iff2,axiom,
    ! [X: vEBT_VEBT,P: fun_VEBT_VEBT_bool,Xs: list_VEBT_VEBT] :
      ( ( some_VEBT_VEBT(X) = find_VEBT_VEBT(P,Xs) )
    <=> ? [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,nth_VEBT_VEBT(Xs,I4)))
          & ( X = nth_VEBT_VEBT(Xs,I4) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,J3))) ) ) ) ).

% find_Some_iff2
tff(fact_1678_find__Some__iff2,axiom,
    ! [X: bool,P: fun_bool_bool,Xs: list_bool] :
      ( ( some_bool(X) = find_bool(P,Xs) )
    <=> ? [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)))
          & ( pp(X)
          <=> pp(aa_nat_bool(nth_bool(Xs),I4)) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),J3))) ) ) ) ).

% find_Some_iff2
tff(fact_1679_find__Some__iff2,axiom,
    ! [X: nat,P: fun_nat_bool,Xs: list_nat] :
      ( ( some_nat(X) = find_nat(P,Xs) )
    <=> ? [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(aa_lis1299702591413342011at_nat(nth_nat,Xs),I4)))
          & ( X = aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,Xs),I4) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_nat_bool(P,aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,Xs),J3))) ) ) ) ).

% find_Some_iff2
tff(fact_1680_find__Some__iff2,axiom,
    ! [X: int,P: fun_int_bool,Xs: list_int] :
      ( ( some_int(X) = find_int(P,Xs) )
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
          & pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I4)))
          & ( X = aa_nat_int(nth_int(Xs),I4) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),J3))) ) ) ) ).

% find_Some_iff2
tff(fact_1681_find__Some__iff,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( ( find_P8199882355184865565at_nat(P,Xs) = some_P7363390416028606310at_nat(X) )
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s5460976970255530739at_nat(Xs)))
          & pp(aa_Pro2232559814647311250t_bool(P,nth_Pr7617993195940197384at_nat(Xs,I4)))
          & ( X = nth_Pr7617993195940197384at_nat(Xs,I4) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_Pro2232559814647311250t_bool(P,nth_Pr7617993195940197384at_nat(Xs,J3))) ) ) ) ).

% find_Some_iff
tff(fact_1682_find__Some__iff,axiom,
    ! [P: fun_num_bool,Xs: list_num,X: num] :
      ( ( find_num(P,Xs) = some_num(X) )
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_num(Xs)))
          & pp(aa_num_bool(P,aa_nat_num(nth_num(Xs),I4)))
          & ( X = aa_nat_num(nth_num(Xs),I4) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_num_bool(P,aa_nat_num(nth_num(Xs),J3))) ) ) ) ).

% find_Some_iff
tff(fact_1683_find__Some__iff,axiom,
    ! [P: fun_VEBT_VEBT_bool,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( find_VEBT_VEBT(P,Xs) = some_VEBT_VEBT(X) )
    <=> ? [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,nth_VEBT_VEBT(Xs,I4)))
          & ( X = nth_VEBT_VEBT(Xs,I4) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,J3))) ) ) ) ).

% find_Some_iff
tff(fact_1684_find__Some__iff,axiom,
    ! [P: fun_bool_bool,Xs: list_bool,X: bool] :
      ( ( find_bool(P,Xs) = some_bool(X) )
    <=> ? [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)))
          & ( pp(X)
          <=> pp(aa_nat_bool(nth_bool(Xs),I4)) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),J3))) ) ) ) ).

% find_Some_iff
tff(fact_1685_find__Some__iff,axiom,
    ! [P: fun_nat_bool,Xs: list_nat,X: nat] :
      ( ( find_nat(P,Xs) = some_nat(X) )
    <=> ? [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(aa_lis1299702591413342011at_nat(nth_nat,Xs),I4)))
          & ( X = aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,Xs),I4) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_nat_bool(P,aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,Xs),J3))) ) ) ) ).

% find_Some_iff
tff(fact_1686_find__Some__iff,axiom,
    ! [P: fun_int_bool,Xs: list_int,X: int] :
      ( ( find_int(P,Xs) = some_int(X) )
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
          & pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I4)))
          & ( X = aa_nat_int(nth_int(Xs),I4) )
          & ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),I4))
             => ~ pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),J3))) ) ) ) ).

% find_Some_iff
tff(fact_1687_count__notin,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
     => ( aa_Pro4243173770044397973at_nat(count_4203492906077236349at_nat(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1688_count__notin,axiom,
    ! [X: real,Xs: list_real] :
      ( ~ pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
     => ( aa_real_nat(count_list_real(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1689_count__notin,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(Xs)))
     => ( aa_set_nat_nat(count_list_set_nat(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1690_count__notin,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
     => ( aa_VEBT_VEBT_nat(count_list_VEBT_VEBT(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1691_count__notin,axiom,
    ! [X: int,Xs: list_int] :
      ( ~ pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
     => ( aa_int_nat(count_list_int(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1692_count__notin,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
     => ( aa_nat_nat(aa_lis1299702591413342011at_nat(count_list_nat,Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1693__C4_Ohyps_C_I9_J,axiom,
    ( ( mi != ma )
   => ! [I3: nat] :
        ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),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),na) = I3 )
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),na))) )
          & ! [X2: nat] :
              ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),na) = I3 )
                & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),na))) )
             => ( 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)) ) ) ) ) ) ).

% "4.hyps"(9)
tff(fact_1694_listrel1__iff__update,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
      ( member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(Xs,Ys),listrel1_VEBT_VEBT(R2))
    <=> ? [Y4: vEBT_VEBT,N3: nat] :
          ( member568628332442017744T_VEBT(produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,N3),Y4),R2)
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N3),size_s6755466524823107622T_VEBT(Xs)))
          & ( Ys = list_u1324408373059187874T_VEBT(Xs,N3,Y4) ) ) ) ).

% listrel1_iff_update
tff(fact_1695_listrel1__iff__update,axiom,
    ! [Xs: list_bool,Ys: list_bool,R2: set_Pr8621012493645324487l_bool] :
      ( member6300827366518646928t_bool(produc970142415094439895t_bool(Xs,Ys),listrel1_bool(R2))
    <=> ? [Y4: bool,N3: nat] :
          ( member4869097156692242192l_bool(produc8297744024737662807l_bool(aa_nat_bool(nth_bool(Xs),N3),Y4),R2)
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N3),size_size_list_bool(Xs)))
          & ( Ys = list_update_bool(Xs,N3,Y4) ) ) ) ).

% listrel1_iff_update
tff(fact_1696_listrel1__iff__update,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),listrel1_nat(R2))
    <=> ? [Y4: nat,N3: nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,Xs),N3)),Y4)),R2))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N3),size_size_list_nat(Xs)))
          & ( Ys = list_update_nat(Xs,N3,Y4) ) ) ) ).

% listrel1_iff_update
tff(fact_1697_listrel1__iff__update,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),listrel1_int(R2))
    <=> ? [Y4: int,N3: nat] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(nth_int(Xs),N3)),Y4),R2)
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N3),size_size_list_int(Xs)))
          & ( Ys = list_update_int(Xs,N3,Y4) ) ) ) ).

% listrel1_iff_update
tff(fact_1698_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),Vj) = none_nat ).

% vebt_pred.simps(6)
tff(fact_1699_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Vi) = none_nat ).

% vebt_succ.simps(5)
tff(fact_1700_pair__lessI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S),T))
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_1701_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N2: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N2)
     => ~ ! [N2: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(suc,N2)) ) ).

% even_odd_cases
tff(fact_1702_power__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,some_nat(X)),some_nat(Y)) = some_nat(Z) ) ) ).

% power_shift
tff(fact_1703__C4_Ohyps_C_I8_J,axiom,
    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))) ).

% "4.hyps"(8)
tff(fact_1704__C4_Ohyps_C_I2_J,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))),m) ).

% "4.hyps"(2)
tff(fact_1705_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_1706_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( aa_nat_nat(suc,X22) = aa_nat_nat(suc,Y22) )
    <=> ( X22 = Y22 ) ) ).

% nat.inject
tff(fact_1707_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_1708_dp,axiom,
    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)) ).

% dp
tff(fact_1709__C4_Ohyps_C_I5_J,axiom,
    ! [I3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m)))
     => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I3)),X_13))
      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,summary),I3)) ) ) ).

% "4.hyps"(5)
tff(fact_1710_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_1711_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(Tree),X))
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),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_1712_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_1713_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_1714_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_1715_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_1716_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_1717_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_1718_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_1719_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_1720_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_1721_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_1722_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_1723_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_1724_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_1725_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_1726_misiz,axiom,
    ! [T: vEBT_VEBT,N: nat,M: nat] :
      ( pp(aa_nat_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_1727_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_1728_helpyd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_succ(T,X) = some_nat(Y) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),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_1729_helpypredd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_pred(T,X) = some_nat(Y) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),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_1730__092_060open_062x_A_060_A2_A_094_Adeg_092_060close_062,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg))) ).

% \<open>x < 2 ^ deg\<close>
tff(fact_1731__092_060open_062ma_A_092_060le_062_A2_A_094_Adeg_092_060close_062,axiom,
    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))),deg))) ).

% \<open>ma \<le> 2 ^ deg\<close>
tff(fact_1732_delt__out__of__range,axiom,
    ! [X: 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,X),Mi))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X)) )
     => ( 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))
       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) ) ) ) ).

% delt_out_of_range
tff(fact_1733_del__single__cont,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( X = 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))
       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) ) ) ) ).

% del_single_cont
tff(fact_1734_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_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_1735_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X: 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),X))
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = some_nat(Ma) ) ) ) ).

% pred_max
tff(fact_1736_succ__min,axiom,
    ! [Deg: nat,X: 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,X),Mi))
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = some_nat(Mi) ) ) ) ).

% succ_min
tff(fact_1737_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_1738_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_1739_nothprolist,axiom,
    ! [I: nat] :
      ( ( ( I != aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))) )
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na))),I) = nth_VEBT_VEBT(treeList,I) ) ) ).

% nothprolist
tff(fact_1740_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z)),fun_pair_less)
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z)) ) ).

% pair_less_iff1
tff(fact_1741_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa_nat_nat(suc,N) ).

% n_not_Suc_n
tff(fact_1742_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(suc,X) = aa_nat_nat(suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_1743_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_1744_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_1745_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_1746_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_1747_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),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,X),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_1748_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),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,X),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_1749_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_1750_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_complex != numera6690914467698888265omplex(N) ).

% zero_neq_numeral
tff(fact_1751_zero__neq__numeral,axiom,
    ! [N: num] : zero_z5237406670263579293d_enat != numera1916890842035813515d_enat(N) ).

% zero_neq_numeral
tff(fact_1752_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_real != aa_num_real(numeral_numeral_real,N) ).

% zero_neq_numeral
tff(fact_1753_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,N) ).

% zero_neq_numeral
tff(fact_1754_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,N) ).

% zero_neq_numeral
tff(fact_1755_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_num_int(numeral_numeral_int,N) ).

% zero_neq_numeral
tff(fact_1756_vebt__buildup_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va2: nat] : X != aa_nat_nat(suc,aa_nat_nat(suc,Va2)) ) ) ).

% vebt_buildup.cases
tff(fact_1757_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero_nat != aa_nat_nat(suc,X22) ).

% nat.distinct(1)
tff(fact_1758_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_1759_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_1760_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa_nat_nat(suc,X22) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_1761_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_1762_nat__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,N2))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% nat_induct
tff(fact_1763_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))
     => ( ! [Y2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,zero_zero_nat),aa_nat_nat(suc,Y2)))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(suc,X4)),aa_nat_nat(suc,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ) ).

% diff_induct
tff(fact_1764_zero__induct,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(P,N2)) )
       => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ).

% zero_induct
tff(fact_1765_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_1766_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_1767_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_1768_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M5: nat] : N = aa_nat_nat(suc,M5) ) ).

% not0_implies_Suc
tff(fact_1769_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_1770_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_1771_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_1772_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_1773_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_1774_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_1775_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_1776_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_1777_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_1778_All__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)) ) ) ) ).

% All_less_Suc
tff(fact_1779_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_1780_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_1781_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_1782_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_1783_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_1784_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_1785_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_1786_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_1787_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_1788_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_1789_Suc__le__D,axiom,
    ! [N: nat,M7: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M7))
     => ? [M5: nat] : M7 = aa_nat_nat(suc,M5) ) ).

% Suc_le_D
tff(fact_1790_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_1791_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_1792_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_1793_full__nat__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M2)),N2))
             => pp(aa_nat_bool(P,M2)) )
         => pp(aa_nat_bool(P,N2)) )
     => pp(aa_nat_bool(P,N)) ) ).

% full_nat_induct
tff(fact_1794_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))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
             => ( pp(aa_nat_bool(P,N2))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct_at_least
tff(fact_1795_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,Y2: nat,Z3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Y2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y2),Z3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Z3)) ) )
         => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R,N2),aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,M),N)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_1796_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_1797_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_1798_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(suc,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_1799_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_1800_listrel1__eq__len,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
      ( member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(Xs,Ys),listrel1_VEBT_VEBT(R2))
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% listrel1_eq_len
tff(fact_1801_listrel1__eq__len,axiom,
    ! [Xs: list_bool,Ys: list_bool,R2: set_Pr8621012493645324487l_bool] :
      ( member6300827366518646928t_bool(produc970142415094439895t_bool(Xs,Ys),listrel1_bool(R2))
     => ( size_size_list_bool(Xs) = size_size_list_bool(Ys) ) ) ).

% listrel1_eq_len
tff(fact_1802_listrel1__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),listrel1_nat(R2))
     => ( size_size_list_nat(Xs) = size_size_list_nat(Ys) ) ) ).

% listrel1_eq_len
tff(fact_1803_listrel1__eq__len,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),listrel1_int(R2))
     => ( size_size_list_int(Xs) = size_size_list_int(Ys) ) ) ).

% listrel1_eq_len
tff(fact_1804_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(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N)) )
     => ( pp(aa_nat_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_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_1))
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                     => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) )
                 => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_1805_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(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N)) )
     => ( pp(aa_nat_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_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_1))
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                     => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) )
                 => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_1806_find__cong,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,P: fun_Pr7757163693017731481t_bool,Q: fun_Pr7757163693017731481t_bool] :
      ( ( Xs = Ys )
     => ( ! [X4: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),set_Pr5648618587558075414at_nat(Ys)))
           => ( pp(aa_Pro2232559814647311250t_bool(P,X4))
            <=> pp(aa_Pro2232559814647311250t_bool(Q,X4)) ) )
       => ( find_P8199882355184865565at_nat(P,Xs) = find_P8199882355184865565at_nat(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1807_find__cong,axiom,
    ! [Xs: list_real,Ys: list_real,P: fun_real_bool,Q: fun_real_bool] :
      ( ( Xs = Ys )
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),set_real2(Ys)))
           => ( pp(aa_real_bool(P,X4))
            <=> pp(aa_real_bool(Q,X4)) ) )
       => ( find_real(P,Xs) = find_real(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1808_find__cong,axiom,
    ! [Xs: list_set_nat,Ys: list_set_nat,P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ( Xs = Ys )
     => ( ! [X4: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X4),set_set_nat2(Ys)))
           => ( pp(aa_set_nat_bool(P,X4))
            <=> pp(aa_set_nat_bool(Q,X4)) ) )
       => ( find_set_nat(P,Xs) = find_set_nat(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1809_find__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,Q: fun_VEBT_VEBT_bool] :
      ( ( Xs = Ys )
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Ys)))
           => ( pp(aa_VEBT_VEBT_bool(P,X4))
            <=> pp(aa_VEBT_VEBT_bool(Q,X4)) ) )
       => ( find_VEBT_VEBT(P,Xs) = find_VEBT_VEBT(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1810_find__cong,axiom,
    ! [Xs: list_int,Ys: list_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ( Xs = Ys )
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),set_int2(Ys)))
           => ( pp(aa_int_bool(P,X4))
            <=> pp(aa_int_bool(Q,X4)) ) )
       => ( find_int(P,Xs) = find_int(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1811_find__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,P: fun_nat_bool,Q: fun_nat_bool] :
      ( ( Xs = Ys )
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Ys)))
           => ( pp(aa_nat_bool(P,X4))
            <=> pp(aa_nat_bool(Q,X4)) ) )
       => ( find_nat(P,Xs) = find_nat(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1812_zip__same,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(A,B),set_Pr5518436109238095868at_nat(zip_Pr4664179122662387191at_nat(Xs,Xs)))
    <=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),set_Pr5648618587558075414at_nat(Xs)))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1813_zip__same,axiom,
    ! [A: real,B: real,Xs: list_real] :
      ( member7849222048561428706l_real(produc4511245868158468465l_real(A,B),set_Pr5999470521830281550l_real(zip_real_real(Xs,Xs)))
    <=> ( pp(aa_set_real_bool(member_real(A),set_real2(Xs)))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1814_zip__same,axiom,
    ! [A: set_nat,B: set_nat,Xs: list_set_nat] :
      ( member8277197624267554838et_nat(produc4532415448927165861et_nat(A,B),set_Pr9040384385603167362et_nat(zip_set_nat_set_nat(Xs,Xs)))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(A),set_set_nat2(Xs)))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1815_zip__same,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member568628332442017744T_VEBT(produc537772716801021591T_VEBT(A,B),set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Xs)))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),set_VEBT_VEBT2(Xs)))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1816_zip__same,axiom,
    ! [A: nat,B: nat,Xs: list_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Xs))))
    <=> ( pp(aa_set_nat_bool(member_nat(A),set_nat2(Xs)))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1817_zip__same,axiom,
    ! [A: int,B: int,Xs: list_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Xs)))
    <=> ( pp(aa_set_int_bool(member_int(A),set_int2(Xs)))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1818_in__set__zipE,axiom,
    ! [X: real,Y: real,Xs: list_real,Ys: list_real] :
      ( member7849222048561428706l_real(produc4511245868158468465l_real(X,Y),set_Pr5999470521830281550l_real(zip_real_real(Xs,Ys)))
     => ~ ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
         => ~ pp(aa_set_real_bool(member_real(Y),set_real2(Ys))) ) ) ).

% in_set_zipE
tff(fact_1819_in__set__zipE,axiom,
    ! [X: real,Y: vEBT_VEBT,Xs: list_real,Ys: list_VEBT_VEBT] :
      ( member7262085504369356948T_VEBT(produc6931449550656315951T_VEBT(X,Y),set_Pr8897343066327330088T_VEBT(zip_real_VEBT_VEBT(Xs,Ys)))
     => ~ ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
         => ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y),set_VEBT_VEBT2(Ys))) ) ) ).

% in_set_zipE
tff(fact_1820_in__set__zipE,axiom,
    ! [X: real,Y: int,Xs: list_real,Ys: list_int] :
      ( member1627681773268152802al_int(produc3179012173361985393al_int(X,Y),set_Pr8219819362198175822al_int(zip_real_int(Xs,Ys)))
     => ~ ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
         => ~ pp(aa_set_int_bool(member_int(Y),set_int2(Ys))) ) ) ).

% in_set_zipE
tff(fact_1821_in__set__zipE,axiom,
    ! [X: real,Y: nat,Xs: list_real,Ys: list_nat] :
      ( member5805532792777349510al_nat(produc3181502643871035669al_nat(X,Y),set_Pr3174298344852596722al_nat(zip_real_nat(Xs,Ys)))
     => ~ ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
         => ~ pp(aa_set_nat_bool(member_nat(Y),set_nat2(Ys))) ) ) ).

% in_set_zipE
tff(fact_1822_in__set__zipE,axiom,
    ! [X: vEBT_VEBT,Y: real,Xs: list_VEBT_VEBT,Ys: list_real] :
      ( member8675245146396747942T_real(produc8117437818029410057T_real(X,Y),set_Pr1087130671499945274T_real(zip_VEBT_VEBT_real(Xs,Ys)))
     => ~ ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
         => ~ pp(aa_set_real_bool(member_real(Y),set_real2(Ys))) ) ) ).

% in_set_zipE
tff(fact_1823_in__set__zipE,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( member568628332442017744T_VEBT(produc537772716801021591T_VEBT(X,Y),set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys)))
     => ~ ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
         => ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y),set_VEBT_VEBT2(Ys))) ) ) ).

% in_set_zipE
tff(fact_1824_in__set__zipE,axiom,
    ! [X: vEBT_VEBT,Y: int,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( member5419026705395827622BT_int(produc736041933913180425BT_int(X,Y),set_Pr2853735649769556538BT_int(zip_VEBT_VEBT_int(Xs,Ys)))
     => ~ ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
         => ~ pp(aa_set_int_bool(member_int(Y),set_int2(Ys))) ) ) ).

% in_set_zipE
tff(fact_1825_in__set__zipE,axiom,
    ! [X: vEBT_VEBT,Y: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( member373505688050248522BT_nat(produc738532404422230701BT_nat(X,Y),set_Pr7031586669278753246BT_nat(zip_VEBT_VEBT_nat(Xs,Ys)))
     => ~ ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
         => ~ pp(aa_set_nat_bool(member_nat(Y),set_nat2(Ys))) ) ) ).

% in_set_zipE
tff(fact_1826_in__set__zipE,axiom,
    ! [X: int,Y: real,Xs: list_int,Ys: list_real] :
      ( member2744130022092475746t_real(produc801115645435158769t_real(X,Y),set_Pr112895574167722958t_real(zip_int_real(Xs,Ys)))
     => ~ ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
         => ~ pp(aa_set_real_bool(member_real(Y),set_real2(Ys))) ) ) ).

% in_set_zipE
tff(fact_1827_in__set__zipE,axiom,
    ! [X: int,Y: vEBT_VEBT,Xs: list_int,Ys: list_VEBT_VEBT] :
      ( member2056185340421749780T_VEBT(produc3329399203697025711T_VEBT(X,Y),set_Pr8714266321650254504T_VEBT(zip_int_VEBT_VEBT(Xs,Ys)))
     => ~ ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
         => ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y),set_VEBT_VEBT2(Ys))) ) ) ).

% in_set_zipE
tff(fact_1828_set__zip__leftD,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Ys))))
     => pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs))) ) ).

% set_zip_leftD
tff(fact_1829_set__zip__leftD,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Ys)))
     => pp(aa_set_int_bool(member_int(X),set_int2(Xs))) ) ).

% set_zip_leftD
tff(fact_1830_set__zip__leftD,axiom,
    ! [X: fun_Co6545587727236015215e_term,Y: produc8923325533196201883nteger,Xs: list_f4599320226441621759e_term,Ys: list_P5578671422887162913nteger] :
      ( member4581088367259704696nteger(aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,X),Y),set_Pr4317899816602322444nteger(zip_fu734256677615756147nteger(Xs,Ys)))
     => member4312719604404396038e_term(X,set_fu7445659897732421274e_term(Xs)) ) ).

% set_zip_leftD
tff(fact_1831_set__zip__leftD,axiom,
    ! [X: fun_Pr272337284026993017e_term,Y: produc8923325533196201883nteger,Xs: list_f4295387047331035401e_term,Ys: list_P5578671422887162913nteger] :
      ( member6423679177046879214nteger(aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,X),Y),set_Pr8538088337767396482nteger(zip_fu197485079333405161nteger(Xs,Ys)))
     => member4565687819040644368e_term(X,set_fu53807654081409956e_term(Xs)) ) ).

% set_zip_leftD
tff(fact_1832_set__zip__leftD,axiom,
    ! [X: fun_Pr9164807099015279978e_term,Y: product_prod_int_int,Xs: list_f8504123576994025594e_term,Ys: list_P5707943133018811711nt_int] :
      ( member690604328116145115nt_int(aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,X),Y),set_Pr7767296851990987887nt_int(zip_fu2408773928738089174nt_int(Xs,Ys)))
     => member2948136515907314945e_term(X,set_fu4765906003373919125e_term(Xs)) ) ).

% set_zip_leftD
tff(fact_1833_set__zip__leftD,axiom,
    ! [X: fun_in6429348457993719614e_term,Y: product_prod_int_int,Xs: list_f6829383385585334350e_term,Ys: list_P5707943133018811711nt_int] :
      ( member500491123954006279nt_int(aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,X),Y),set_Pr2267922651014402459nt_int(zip_fu9102111156773635586nt_int(Xs,Ys)))
     => member7723273712176309973e_term(X,set_fu1494195493796392297e_term(Xs)) ) ).

% set_zip_leftD
tff(fact_1834_set__zip__rightD,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Ys))))
     => pp(aa_set_nat_bool(member_nat(Y),set_nat2(Ys))) ) ).

% set_zip_rightD
tff(fact_1835_set__zip__rightD,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Ys)))
     => pp(aa_set_int_bool(member_int(Y),set_int2(Ys))) ) ).

% set_zip_rightD
tff(fact_1836_set__zip__rightD,axiom,
    ! [X: fun_Co6545587727236015215e_term,Y: produc8923325533196201883nteger,Xs: list_f4599320226441621759e_term,Ys: list_P5578671422887162913nteger] :
      ( member4581088367259704696nteger(aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,X),Y),set_Pr4317899816602322444nteger(zip_fu734256677615756147nteger(Xs,Ys)))
     => member157494554546826820nteger(Y,set_Pr920681315882439344nteger(Ys)) ) ).

% set_zip_rightD
tff(fact_1837_set__zip__rightD,axiom,
    ! [X: fun_Pr272337284026993017e_term,Y: produc8923325533196201883nteger,Xs: list_f4295387047331035401e_term,Ys: list_P5578671422887162913nteger] :
      ( member6423679177046879214nteger(aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,X),Y),set_Pr8538088337767396482nteger(zip_fu197485079333405161nteger(Xs,Ys)))
     => member157494554546826820nteger(Y,set_Pr920681315882439344nteger(Ys)) ) ).

% set_zip_rightD
tff(fact_1838_set__zip__rightD,axiom,
    ! [X: fun_Pr9164807099015279978e_term,Y: product_prod_int_int,Xs: list_f8504123576994025594e_term,Ys: list_P5707943133018811711nt_int] :
      ( member690604328116145115nt_int(aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,X),Y),set_Pr7767296851990987887nt_int(zip_fu2408773928738089174nt_int(Xs,Ys)))
     => member5262025264175285858nt_int(Y,set_Pr2470121279949933262nt_int(Ys)) ) ).

% set_zip_rightD
tff(fact_1839_set__zip__rightD,axiom,
    ! [X: fun_in6429348457993719614e_term,Y: product_prod_int_int,Xs: list_f6829383385585334350e_term,Ys: list_P5707943133018811711nt_int] :
      ( member500491123954006279nt_int(aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,X),Y),set_Pr2267922651014402459nt_int(zip_fu9102111156773635586nt_int(Xs,Ys)))
     => member5262025264175285858nt_int(Y,set_Pr2470121279949933262nt_int(Ys)) ) ).

% set_zip_rightD
tff(fact_1840_zip__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] : zip_VE537291747668921783T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),list_u1324408373059187874T_VEBT(Ys,I,Y)) = list_u6961636818849549845T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys),I,produc537772716801021591T_VEBT(X,Y)) ).

% zip_update
tff(fact_1841_zip__update,axiom,
    ! [Xs: list_int,I: nat,X: int,Ys: list_int,Y: int] : zip_int_int(list_update_int(Xs,I,X),list_update_int(Ys,I,Y)) = list_u3002344382305578791nt_int(zip_int_int(Xs,Ys),I,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)) ).

% zip_update
tff(fact_1842_zip__update,axiom,
    ! [Xs: list_f4599320226441621759e_term,I: nat,X: fun_Co6545587727236015215e_term,Ys: list_P5578671422887162913nteger,Y: produc8923325533196201883nteger] : zip_fu734256677615756147nteger(list_u339823752026759297e_term(Xs,I,X),list_u2254550707601501961nteger(Ys,I,Y)) = list_u3060434466275295603nteger(zip_fu734256677615756147nteger(Xs,Ys),I,aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,X),Y)) ).

% zip_update
tff(fact_1843_zip__update,axiom,
    ! [Xs: list_f4295387047331035401e_term,I: nat,X: fun_Pr272337284026993017e_term,Ys: list_P5578671422887162913nteger,Y: produc8923325533196201883nteger] : zip_fu197485079333405161nteger(list_u6547189781710427531e_term(Xs,I,X),list_u2254550707601501961nteger(Ys,I,Y)) = list_u5284324012848482281nteger(zip_fu197485079333405161nteger(Xs,Ys),I,aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,X),Y)) ).

% zip_update
tff(fact_1844_zip__update,axiom,
    ! [Xs: list_f8504123576994025594e_term,I: nat,X: fun_Pr9164807099015279978e_term,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] : zip_fu2408773928738089174nt_int(list_u3754337643414342012e_term(Xs,I,X),list_u3002344382305578791nt_int(Ys,I,Y)) = list_u2433715936953936854nt_int(zip_fu2408773928738089174nt_int(Xs,Ys),I,aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,X),Y)) ).

% zip_update
tff(fact_1845_zip__update,axiom,
    ! [Xs: list_f6829383385585334350e_term,I: nat,X: fun_in6429348457993719614e_term,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] : zip_fu9102111156773635586nt_int(list_u8051947491296383824e_term(Xs,I,X),list_u3002344382305578791nt_int(Ys,I,Y)) = list_u6288444477985464578nt_int(zip_fu9102111156773635586nt_int(Xs,Ys),I,aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,X),Y)) ).

% zip_update
tff(fact_1846_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(N)),zero_z5237406670263579293d_enat)) ).

% not_numeral_le_zero
tff(fact_1847_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_1848_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_1849_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_1850_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_1851_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(N))) ).

% zero_le_numeral
tff(fact_1852_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_1853_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_1854_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_1855_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_1856_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(N))) ).

% zero_less_numeral
tff(fact_1857_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_1858_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_1859_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_1860_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_1861_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(N)),zero_z5237406670263579293d_enat)) ).

% not_numeral_less_zero
tff(fact_1862_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_1863_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_1864_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_1865_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_1866_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,N6))) ) ) ).

% lift_Suc_mono_less
tff(fact_1867_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
     => ( 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,aa_nat_rat(F,N)),aa_nat_rat(F,N6))) ) ) ).

% lift_Suc_mono_less
tff(fact_1868_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,N6))) ) ) ).

% lift_Suc_mono_less
tff(fact_1869_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( 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,aa_nat_nat(F,N)),aa_nat_nat(F,N6))) ) ) ).

% lift_Suc_mono_less
tff(fact_1870_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( 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,aa_nat_int(F,N)),aa_nat_int(F,N6))) ) ) ).

% lift_Suc_mono_less
tff(fact_1871_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( 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_1872_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
     => ( 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_1873_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
     => ( 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_1874_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( 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_1875_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( 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_1876_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_int,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,N2)),aa_nat_set_int(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,N)),aa_nat_set_int(F,N6))) ) ) ).

% lift_Suc_mono_le
tff(fact_1877_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
     => ( 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,aa_nat_rat(F,N)),aa_nat_rat(F,N6))) ) ) ).

% lift_Suc_mono_le
tff(fact_1878_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N)),aa_nat_num(F,N6))) ) ) ).

% lift_Suc_mono_le
tff(fact_1879_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( 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,aa_nat_nat(F,N)),aa_nat_nat(F,N6))) ) ) ).

% lift_Suc_mono_le
tff(fact_1880_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( 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,aa_nat_int(F,N)),aa_nat_int(F,N6))) ) ) ).

% lift_Suc_mono_le
tff(fact_1881_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_int,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,aa_nat_nat(suc,N2))),aa_nat_set_int(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,N6)),aa_nat_set_int(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1882_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N2))),aa_nat_rat(F,N2)))
     => ( 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,aa_nat_rat(F,N6)),aa_nat_rat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1883_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N2))),aa_nat_num(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N6)),aa_nat_num(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1884_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N2))),aa_nat_nat(F,N2)))
     => ( 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,aa_nat_nat(F,N6)),aa_nat_nat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1885_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N2))),aa_nat_int(F,N2)))
     => ( 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,aa_nat_int(F,N6)),aa_nat_int(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1886_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_1887_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ? [M3: nat] : N = aa_nat_nat(suc,M3) ) ).

% gr0_conv_Suc
tff(fact_1888_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_1889_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_1890_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_1891_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_1892_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_1893_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_1894_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_1895_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_1896_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))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J))
               => ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
                 => pp(aa_nat_bool(P,N2)) ) ) )
         => pp(aa_nat_bool(P,I)) ) ) ) ).

% inc_induct
tff(fact_1897_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))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J))
               => ( pp(aa_nat_bool(P,N2))
                 => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) )
         => pp(aa_nat_bool(P,J)) ) ) ) ).

% dec_induct
tff(fact_1898_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_1899_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_1900_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_1901_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_1902_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_1903_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_1904_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_1905_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_1906_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ~ ! [Q3: nat] : N != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)) ) ).

% less_natE
tff(fact_1907_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(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N)) )
     => ( pp(aa_nat_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_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I2)),X_13))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I2)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                       => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) ) )
                 => ( 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,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,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(vEBT_invar_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_1908_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_real,Y: real] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Ys) )
     => ( pp(aa_set_real_bool(member_real(Y),set_real2(Ys)))
       => ~ ! [X4: vEBT_VEBT] : ~ member8675245146396747942T_real(produc8117437818029410057T_real(X4,Y),set_Pr1087130671499945274T_real(zip_VEBT_VEBT_real(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1909_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y),set_VEBT_VEBT2(Ys)))
       => ~ ! [X4: vEBT_VEBT] : ~ member568628332442017744T_VEBT(produc537772716801021591T_VEBT(X4,Y),set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1910_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_bool,Y: bool] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_bool(Ys) )
     => ( member_bool(Y,set_bool2(Ys))
       => ~ ! [X4: vEBT_VEBT] : ~ member3504487111852181555T_bool(produc3035007008238225530T_bool(X4,Y),set_Pr3399341499479959199T_bool(zip_VEBT_VEBT_bool(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1911_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,Y: nat] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) )
     => ( pp(aa_set_nat_bool(member_nat(Y),set_nat2(Ys)))
       => ~ ! [X4: vEBT_VEBT] : ~ member373505688050248522BT_nat(produc738532404422230701BT_nat(X4,Y),set_Pr7031586669278753246BT_nat(zip_VEBT_VEBT_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1912_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int,Y: int] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Ys) )
     => ( pp(aa_set_int_bool(member_int(Y),set_int2(Ys)))
       => ~ ! [X4: vEBT_VEBT] : ~ member5419026705395827622BT_int(produc736041933913180425BT_int(X4,Y),set_Pr2853735649769556538BT_int(zip_VEBT_VEBT_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1913_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_bool,Ys: list_real,Y: real] :
      ( ( size_size_list_bool(Xs) = size_size_list_real(Ys) )
     => ( pp(aa_set_real_bool(member_real(Y),set_real2(Ys)))
       => ~ ! [X4: bool] : ~ member3680586897080482057l_real(produc7587643690356850540l_real(X4,Y),set_Pr3516335181496812957l_real(zip_bool_real(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1914_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_bool,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( size_size_list_bool(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y),set_VEBT_VEBT2(Ys)))
       => ~ ! [X4: bool] : ~ member5465154751058088621T_VEBT(produc9074556814250467700T_VEBT(X4,Y),set_Pr5360009138685866265T_VEBT(zip_bool_VEBT_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1915_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_bool,Ys: list_bool,Y: bool] :
      ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys) )
     => ( member_bool(Y,set_bool2(Ys))
       => ~ ! [X4: bool] : ~ member4869097156692242192l_bool(produc8297744024737662807l_bool(X4,Y),set_Pr2237862708767489404l_bool(zip_bool_bool(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1916_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_bool,Ys: list_nat,Y: nat] :
      ( ( size_size_list_bool(Xs) = size_size_list_nat(Ys) )
     => ( pp(aa_set_nat_bool(member_nat(Y),set_nat2(Ys)))
       => ~ ! [X4: bool] : ~ member2564532549135826221ol_nat(produc8874003862241262736ol_nat(X4,Y),set_Pr4953607175514100673ol_nat(zip_bool_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1917_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_bool,Ys: list_int,Y: int] :
      ( ( size_size_list_bool(Xs) = size_size_list_int(Ys) )
     => ( pp(aa_set_int_bool(member_int(Y),set_int2(Ys)))
       => ~ ! [X4: bool] : ~ member7610053566481405321ol_int(produc8871513391732212460ol_int(X4,Y),set_Pr775756156004903965ol_int(zip_bool_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1918_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_VEBT_VEBT,X: real] :
      ( ( size_size_list_real(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
       => ~ ! [Y2: vEBT_VEBT] : ~ member7262085504369356948T_VEBT(produc6931449550656315951T_VEBT(X,Y2),set_Pr8897343066327330088T_VEBT(zip_real_VEBT_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1919_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_bool,X: real] :
      ( ( size_size_list_real(Xs) = size_size_list_bool(Ys) )
     => ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
       => ~ ! [Y2: bool] : ~ member3295258257541327095l_bool(produc3350604258665657362l_bool(X,Y2),set_Pr3131006541957657995l_bool(zip_real_bool(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1920_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_nat,X: real] :
      ( ( size_size_list_real(Xs) = size_size_list_nat(Ys) )
     => ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
       => ~ ! [Y2: nat] : ~ member5805532792777349510al_nat(produc3181502643871035669al_nat(X,Y2),set_Pr3174298344852596722al_nat(zip_real_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1921_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_int,X: real] :
      ( ( size_size_list_real(Xs) = size_size_list_int(Ys) )
     => ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
       => ~ ! [Y2: int] : ~ member1627681773268152802al_int(produc3179012173361985393al_int(X,Y2),set_Pr8219819362198175822al_int(zip_real_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1922_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
       => ~ ! [Y2: vEBT_VEBT] : ~ member568628332442017744T_VEBT(produc537772716801021591T_VEBT(X,Y2),set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1923_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_bool,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_bool(Ys) )
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
       => ~ ! [Y2: bool] : ~ member3504487111852181555T_bool(produc3035007008238225530T_bool(X,Y2),set_Pr3399341499479959199T_bool(zip_VEBT_VEBT_bool(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1924_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) )
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
       => ~ ! [Y2: nat] : ~ member373505688050248522BT_nat(produc738532404422230701BT_nat(X,Y2),set_Pr7031586669278753246BT_nat(zip_VEBT_VEBT_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1925_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Ys) )
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
       => ~ ! [Y2: int] : ~ member5419026705395827622BT_int(produc736041933913180425BT_int(X,Y2),set_Pr2853735649769556538BT_int(zip_VEBT_VEBT_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1926_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_bool,Ys: list_VEBT_VEBT,X: bool] :
      ( ( size_size_list_bool(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( member_bool(X,set_bool2(Xs))
       => ~ ! [Y2: vEBT_VEBT] : ~ member5465154751058088621T_VEBT(produc9074556814250467700T_VEBT(X,Y2),set_Pr5360009138685866265T_VEBT(zip_bool_VEBT_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1927_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_bool,Ys: list_bool,X: bool] :
      ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys) )
     => ( member_bool(X,set_bool2(Xs))
       => ~ ! [Y2: bool] : ~ member4869097156692242192l_bool(produc8297744024737662807l_bool(X,Y2),set_Pr2237862708767489404l_bool(zip_bool_bool(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1928_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_Pr2477239776880934807at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(F,some_P7363390416028606310at_nat(A),some_P7363390416028606310at_nat(B)) = some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1929_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_num_fun_num_num,A: num,B: num] : vEBT_V819420779217536731ft_num(F,some_num(A),some_num(B)) = some_num(aa_num_num(aa_num_fun_num_num(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1930_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(F),some_nat(A)),some_nat(B)) = some_nat(aa_nat_nat(aa_nat_fun_nat_nat(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1931_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_1932_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_num_fun_num_num,Uv: option_num] : vEBT_V819420779217536731ft_num(Uu,none_num,Uv) = none_num ).

% VEBT_internal.option_shift.simps(1)
tff(fact_1933_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_1934_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))
            & ! [I3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),K2))
               => ~ pp(aa_nat_bool(P,I3)) )
            & pp(aa_nat_bool(P,aa_nat_nat(suc,K2))) ) ) ) ).

% ex_least_nat_less
tff(fact_1935_find__None__iff,axiom,
    ! [P: fun_real_bool,Xs: list_real] :
      ( ( find_real(P,Xs) = none_real )
    <=> ~ ? [X3: real] :
            ( pp(aa_set_real_bool(member_real(X3),set_real2(Xs)))
            & pp(aa_real_bool(P,X3)) ) ) ).

% find_None_iff
tff(fact_1936_find__None__iff,axiom,
    ! [P: fun_set_nat_bool,Xs: list_set_nat] :
      ( ( find_set_nat(P,Xs) = none_set_nat )
    <=> ~ ? [X3: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(Xs)))
            & pp(aa_set_nat_bool(P,X3)) ) ) ).

% find_None_iff
tff(fact_1937_find__None__iff,axiom,
    ! [P: fun_VEBT_VEBT_bool,Xs: list_VEBT_VEBT] :
      ( ( find_VEBT_VEBT(P,Xs) = none_VEBT_VEBT )
    <=> ~ ? [X3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
            & pp(aa_VEBT_VEBT_bool(P,X3)) ) ) ).

% find_None_iff
tff(fact_1938_find__None__iff,axiom,
    ! [P: fun_int_bool,Xs: list_int] :
      ( ( find_int(P,Xs) = none_int )
    <=> ~ ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),set_int2(Xs)))
            & pp(aa_int_bool(P,X3)) ) ) ).

% find_None_iff
tff(fact_1939_find__None__iff,axiom,
    ! [P: fun_nat_bool,Xs: list_nat] :
      ( ( find_nat(P,Xs) = none_nat )
    <=> ~ ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
            & pp(aa_nat_bool(P,X3)) ) ) ).

% find_None_iff
tff(fact_1940_find__None__iff,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,Xs: list_P6011104703257516679at_nat] :
      ( ( find_P8199882355184865565at_nat(P,Xs) = none_P5556105721700978146at_nat )
    <=> ~ ? [X3: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),set_Pr5648618587558075414at_nat(Xs)))
            & pp(aa_Pro2232559814647311250t_bool(P,X3)) ) ) ).

% find_None_iff
tff(fact_1941_find__None__iff,axiom,
    ! [P: fun_num_bool,Xs: list_num] :
      ( ( find_num(P,Xs) = none_num )
    <=> ~ ? [X3: num] :
            ( pp(aa_set_num_bool(member_num(X3),set_num2(Xs)))
            & pp(aa_num_bool(P,X3)) ) ) ).

% find_None_iff
tff(fact_1942_find__None__iff2,axiom,
    ! [P: fun_real_bool,Xs: list_real] :
      ( ( none_real = find_real(P,Xs) )
    <=> ~ ? [X3: real] :
            ( pp(aa_set_real_bool(member_real(X3),set_real2(Xs)))
            & pp(aa_real_bool(P,X3)) ) ) ).

% find_None_iff2
tff(fact_1943_find__None__iff2,axiom,
    ! [P: fun_set_nat_bool,Xs: list_set_nat] :
      ( ( none_set_nat = find_set_nat(P,Xs) )
    <=> ~ ? [X3: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(Xs)))
            & pp(aa_set_nat_bool(P,X3)) ) ) ).

% find_None_iff2
tff(fact_1944_find__None__iff2,axiom,
    ! [P: fun_VEBT_VEBT_bool,Xs: list_VEBT_VEBT] :
      ( ( none_VEBT_VEBT = find_VEBT_VEBT(P,Xs) )
    <=> ~ ? [X3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
            & pp(aa_VEBT_VEBT_bool(P,X3)) ) ) ).

% find_None_iff2
tff(fact_1945_find__None__iff2,axiom,
    ! [P: fun_int_bool,Xs: list_int] :
      ( ( none_int = find_int(P,Xs) )
    <=> ~ ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),set_int2(Xs)))
            & pp(aa_int_bool(P,X3)) ) ) ).

% find_None_iff2
tff(fact_1946_find__None__iff2,axiom,
    ! [P: fun_nat_bool,Xs: list_nat] :
      ( ( none_nat = find_nat(P,Xs) )
    <=> ~ ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
            & pp(aa_nat_bool(P,X3)) ) ) ).

% find_None_iff2
tff(fact_1947_find__None__iff2,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,Xs: list_P6011104703257516679at_nat] :
      ( ( none_P5556105721700978146at_nat = find_P8199882355184865565at_nat(P,Xs) )
    <=> ~ ? [X3: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),set_Pr5648618587558075414at_nat(Xs)))
            & pp(aa_Pro2232559814647311250t_bool(P,X3)) ) ) ).

% find_None_iff2
tff(fact_1948_find__None__iff2,axiom,
    ! [P: fun_num_bool,Xs: list_num] :
      ( ( none_num = find_num(P,Xs) )
    <=> ~ ? [X3: num] :
            ( pp(aa_set_num_bool(member_num(X3),set_num2(Xs)))
            & pp(aa_num_bool(P,X3)) ) ) ).

% find_None_iff2
tff(fact_1949_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(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N)) )
     => ( pp(aa_nat_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_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I2)),X_13))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I2)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                       => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) ) )
                 => ( 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,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,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(vEBT_invar_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_1950_pair__lessI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_1951_count__le__length,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(count_list_VEBT_VEBT(Xs),X)),size_s6755466524823107622T_VEBT(Xs))) ).

% count_le_length
tff(fact_1952_count__le__length,axiom,
    ! [Xs: list_bool,X: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(count_list_bool(Xs),X)),size_size_list_bool(Xs))) ).

% count_le_length
tff(fact_1953_count__le__length,axiom,
    ! [Xs: list_nat,X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_lis1299702591413342011at_nat(count_list_nat,Xs),X)),size_size_list_nat(Xs))) ).

% count_le_length
tff(fact_1954_count__le__length,axiom,
    ! [Xs: list_int,X: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(count_list_int(Xs),X)),size_size_list_int(Xs))) ).

% count_le_length
tff(fact_1955_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_1956_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_1957_option_Osize_I4_J,axiom,
    ! [X22: num] : size_size_option_num(some_num(X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_1958_option_Osize_I3_J,axiom,
    size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1959_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1960_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1961_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),X)) ).

% vebt_member.simps(4)
tff(fact_1962_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: fun_Pr2477239776880934807at_nat,Xa2: option4927543243414619207at_nat,Xb2: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(X,Xa2,Xb2) = Y )
     => ( ( ( Xa2 = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] : Xa2 = some_P7363390416028606310at_nat(V2)
           => ( ( Xb2 = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A3: product_prod_nat_nat] :
                ( ( Xa2 = some_P7363390416028606310at_nat(A3) )
               => ! [B3: product_prod_nat_nat] :
                    ( ( Xb2 = some_P7363390416028606310at_nat(B3) )
                   => ( Y != some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(X,A3),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1963_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: fun_num_fun_num_num,Xa2: option_num,Xb2: option_num,Y: option_num] :
      ( ( vEBT_V819420779217536731ft_num(X,Xa2,Xb2) = Y )
     => ( ( ( Xa2 = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V2: num] : Xa2 = some_num(V2)
           => ( ( Xb2 = none_num )
             => ( Y != none_num ) ) )
         => ~ ! [A3: num] :
                ( ( Xa2 = some_num(A3) )
               => ! [B3: num] :
                    ( ( Xb2 = some_num(B3) )
                   => ( Y != some_num(aa_num_num(aa_num_fun_num_num(X,A3),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1964_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa2: option_nat,Xb2: option_nat,Y: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(X),Xa2),Xb2) = Y )
     => ( ( ( Xa2 = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V2: nat] : Xa2 = some_nat(V2)
           => ( ( Xb2 = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A3: nat] :
                ( ( Xa2 = some_nat(A3) )
               => ! [B3: nat] :
                    ( ( Xb2 = some_nat(B3) )
                   => ( Y != some_nat(aa_nat_nat(aa_nat_fun_nat_nat(X,A3),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1965_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_1966_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_num_fun_num_num,V: num] : vEBT_V819420779217536731ft_num(Uw,some_num(V),none_num) = none_num ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1967_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_1968_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X: nat] : vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm),X) = 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_1969_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1970_sum__power2__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1971_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1972_zero__less__power2,axiom,
    ! [A: 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,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_1973_zero__less__power2,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_1974_zero__less__power2,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_1975_power2__eq__iff__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1976_power2__eq__iff__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1977_power2__eq__iff__nonneg,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1978_power2__eq__iff__nonneg,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1979_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real))
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_1980_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat))
    <=> ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_1981_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int))
    <=> ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_1982_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),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,Y),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(vEBT_vebt_member(vEBT_vebt_insert(T,X)),Y))
           => ( pp(aa_nat_bool(vEBT_vebt_member(T),Y))
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_1983_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),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,vEBT_vebt_insert(T,X)),X)) ) ) ).

% valid_insert_both_member_options_add
tff(fact_1984_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),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,Y),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),X))
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,Y)),X)) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_1985_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        | ( X = 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))
       => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) ) ) ) ).

% insert_simp_mima
tff(fact_1986_power__mono__iff,axiom,
    ! [A: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => ( 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,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_1987_power__mono__iff,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => ( 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(A),N)),aa_nat_rat(power_power_rat(B),N)))
          <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_1988_power__mono__iff,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),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(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
          <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_1989_power__mono__iff,axiom,
    ! [A: int,B: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => ( 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(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
          <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_1990_zero__eq__power2,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_power2
tff(fact_1991_zero__eq__power2,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat )
    <=> ( A = zero_zero_nat ) ) ).

% zero_eq_power2
tff(fact_1992_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_1993_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% zero_eq_power2
tff(fact_1994_zero__eq__power2,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% zero_eq_power2
tff(fact_1995_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_1996_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_1997_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_1998_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,N)) = zero_zero_int ).

% power_0_Suc
tff(fact_1999_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,N)) = zero_zero_complex ).

% power_0_Suc
tff(fact_2000_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_2001_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_2002_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_2003_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_int ).

% power_zero_numeral
tff(fact_2004_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_2005_power__Suc0__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_2006_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_2007_power__Suc0__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_2008_power__Suc0__right,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_2009_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = zero_zero_nat )
        | ( X = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_2010_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_2011_nat__zero__less__power__iff,axiom,
    ! [X: 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,X),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_2012_power__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( aa_nat_rat(power_power_rat(A),N) = zero_zero_rat )
    <=> ( ( A = 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_2013_power__eq__0__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = zero_zero_nat )
    <=> ( ( A = 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_2014_power__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) = zero_zero_real )
    <=> ( ( A = 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_2015_power__eq__0__iff,axiom,
    ! [A: int,N: nat] :
      ( ( aa_nat_int(power_power_int(A),N) = zero_zero_int )
    <=> ( ( A = 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_2016_power__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( aa_nat_complex(power_power_complex(A),N) = zero_zero_complex )
    <=> ( ( A = 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_2017_power__not__zero,axiom,
    ! [A: rat,N: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),N) != zero_zero_rat ) ) ).

% power_not_zero
tff(fact_2018_power__not__zero,axiom,
    ! [A: nat,N: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) != zero_zero_nat ) ) ).

% power_not_zero
tff(fact_2019_power__not__zero,axiom,
    ! [A: real,N: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_2020_power__not__zero,axiom,
    ! [A: int,N: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),N) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_2021_power__not__zero,axiom,
    ! [A: complex,N: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),N) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_2022_power__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N))) ) ) ).

% power_mono
tff(fact_2023_power__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ).

% power_mono
tff(fact_2024_power__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))) ) ) ).

% power_mono
tff(fact_2025_power__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( 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_eq_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ).

% power_mono
tff(fact_2026_zero__le__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => 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,A),N))) ) ).

% zero_le_power
tff(fact_2027_zero__le__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))) ) ).

% zero_le_power
tff(fact_2028_zero__le__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => 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,A),N))) ) ).

% zero_le_power
tff(fact_2029_zero__le__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).

% zero_le_power
tff(fact_2030_zero__less__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ).

% zero_less_power
tff(fact_2031_zero__less__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))) ) ).

% zero_less_power
tff(fact_2032_zero__less__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => 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,A),N))) ) ).

% zero_less_power
tff(fact_2033_zero__less__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).

% zero_less_power
tff(fact_2034_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_2035_power__less__imp__less__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_2036_power__less__imp__less__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_2037_power__less__imp__less__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_2038_power__less__imp__less__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_2039_power__inject__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,N)) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2040_power__inject__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N)) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2041_power__inject__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2042_power__inject__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,N)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2043_power__le__imp__le__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,N))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_2044_power__le__imp__le__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_2045_power__le__imp__le__base,axiom,
    ! [A: nat,N: nat,B: 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,A),aa_nat_nat(suc,N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_2046_power__le__imp__le__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,N))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_2047_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_2048_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_2049_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_2050_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ).

% zero_power
tff(fact_2051_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = zero_zero_complex ) ) ).

% zero_power
tff(fact_2052_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_2053_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_2054_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_2055_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_2056_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_2057_zero__power2,axiom,
    aa_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_2058_zero__power2,axiom,
    aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex ).

% zero_power2
tff(fact_2059_power__eq__imp__eq__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2060_power__eq__imp__eq__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),N) = aa_nat_rat(power_power_rat(B),N) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2061_power__eq__imp__eq__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2062_power__eq__imp__eq__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),N) = aa_nat_int(power_power_int(B),N) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2063_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: real,B: 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),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2064_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: rat,B: 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),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( ( aa_nat_rat(power_power_rat(A),N) = aa_nat_rat(power_power_rat(B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2065_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: nat,B: 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),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2066_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: int,B: 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),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( ( aa_nat_int(power_power_int(A),N) = aa_nat_int(power_power_int(B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2067_less__exp,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))) ).

% less_exp
tff(fact_2068_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_2069_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_2070_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_2071_zero__le__power2,axiom,
    ! [A: 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,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_2072_zero__le__power2,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_2073_zero__le__power2,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_2074_power2__eq__imp__eq,axiom,
    ! [X: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),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),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2075_power2__eq__imp__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),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),X))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2076_power2__eq__imp__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),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),X))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2077_power2__eq__imp__eq,axiom,
    ! [X: int,Y: int] :
      ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),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),X))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2078_power2__le__imp__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),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),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_2079_power2__le__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),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),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_2080_power2__le__imp__le,axiom,
    ! [X: nat,Y: 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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),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),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_2081_power2__le__imp__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),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),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_2082_power2__less__0,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)) ).

% power2_less_0
tff(fact_2083_power2__less__0,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat)) ).

% power2_less_0
tff(fact_2084_power2__less__0,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int)) ).

% power2_less_0
tff(fact_2085_power__strict__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( 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,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N))) ) ) ) ).

% power_strict_mono
tff(fact_2086_power__strict__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( 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(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ) ).

% power_strict_mono
tff(fact_2087_power__strict__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_eq_nat,zero_zero_nat),A))
       => ( 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,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))) ) ) ) ).

% power_strict_mono
tff(fact_2088_power__strict__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( 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_eq_int,zero_zero_int),A))
       => ( 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(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ) ).

% power_strict_mono
tff(fact_2089_power2__less__imp__less,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),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),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_2090_power2__less__imp__less,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),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),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_2091_power2__less__imp__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),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),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_2092_power2__less__imp__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),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),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_2093_sum__power2__le__zero__iff,axiom,
    ! [X: real,Y: 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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real))
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2094_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat))
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2095_sum__power2__le__zero__iff,axiom,
    ! [X: int,Y: 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(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int))
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2096_sum__power2__ge__zero,axiom,
    ! [X: real,Y: 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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_2097_sum__power2__ge__zero,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_2098_sum__power2__ge__zero,axiom,
    ! [X: int,Y: 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(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_2099_not__sum__power2__lt__zero,axiom,
    ! [X: real,Y: 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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)) ).

% not_sum_power2_lt_zero
tff(fact_2100_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)) ).

% not_sum_power2_lt_zero
tff(fact_2101_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y: 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(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)) ).

% not_sum_power2_lt_zero
tff(fact_2102_sum__power2__gt__zero__iff,axiom,
    ! [X: real,Y: 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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2103_sum__power2__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2104_sum__power2__gt__zero__iff,axiom,
    ! [X: int,Y: 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(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2105_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),X))
     => ( 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))
        & ( ( X = Mi )
          | ( X = Ma )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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_2106_pred__list__to__short,axiom,
    ! [Deg: nat,X: 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,X),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,X),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(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = none_nat ) ) ) ) ).

% pred_list_to_short
tff(fact_2107_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X: 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),X))
       => ( 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,X),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(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = none_nat ) ) ) ) ).

% succ_list_to_short
tff(fact_2108_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),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,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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,vEBT_Node(Info,Deg,TreeList,Summary)),X)) ) ) ) ).

% both_member_options_ding
tff(fact_2109_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_2110_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_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_2111_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),X)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary) ).

% vebt_insert.simps(4)
tff(fact_2112_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_2113_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_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(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_2114_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_2115_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_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(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_2116_bit__concat__def,axiom,
    ! [H: nat,L: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(H),L),D) = 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))),D))),L) ).

% bit_concat_def
tff(fact_2117_pow__sum,axiom,
    ! [A: nat,B: 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,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ).

% pow_sum
tff(fact_2118_high__def,axiom,
    ! [X: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),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_2119__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = na ).

% \<open>deg div 2 = n\<close>
tff(fact_2120_mult__zero__left,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A) = zero_zero_complex ).

% mult_zero_left
tff(fact_2121_mult__zero__left,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_zero_real),A) = zero_zero_real ).

% mult_zero_left
tff(fact_2122_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_2123_mult__zero__left,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A) = zero_zero_nat ).

% mult_zero_left
tff(fact_2124_mult__zero__left,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),A) = zero_zero_int ).

% mult_zero_left
tff(fact_2125_mult__zero__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_2126_mult__zero__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),zero_zero_real) = zero_zero_real ).

% mult_zero_right
tff(fact_2127_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_2128_mult__zero__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),zero_zero_nat) = zero_zero_nat ).

% mult_zero_right
tff(fact_2129_mult__zero__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),zero_zero_int) = zero_zero_int ).

% mult_zero_right
tff(fact_2130_mult__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
tff(fact_2131_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_2132_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_2133_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_2134_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_2135_mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2136_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2137_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2138_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2139_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2140_mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2141_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2142_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2143_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2144_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2145_high__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),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,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),X)),N) = Y ) ) ).

% high_inv
tff(fact_2146_low__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),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,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),X)),N) = X ) ) ).

% low_inv
tff(fact_2147_bits__div__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).

% bits_div_0
tff(fact_2148_bits__div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_2149_bits__div__0,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% bits_div_0
tff(fact_2150_bits__div__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).

% bits_div_by_0
tff(fact_2151_bits__div__by__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_2152_bits__div__by__0,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% bits_div_by_0
tff(fact_2153_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_2154_div__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).

% div_0
tff(fact_2155_div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_2156_div__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,zero_zero_real),A) = zero_zero_real ).

% div_0
tff(fact_2157_div__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(zero_zero_rat),A) = zero_zero_rat ).

% div_0
tff(fact_2158_div__0,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% div_0
tff(fact_2159_divide__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
tff(fact_2160_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_2161_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_2162_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_2163_div__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).

% div_by_0
tff(fact_2164_div__by__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_2165_div__by__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_2166_div__by__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_2167_div__by__0,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% div_by_0
tff(fact_2168_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(C),A) = aa_complex_complex(divide1717551699836669952omplex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_2169_divide__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_2170_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(C),A) = aa_rat_rat(divide_divide_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_2171_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_2172_divide__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_2173_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_2174_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_2175_division__ring__divide__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_2176_division__ring__divide__zero,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_2177_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_2178_mult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% mult_1
tff(fact_2179_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ).

% mult_1
tff(fact_2180_mult__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ).

% mult_1
tff(fact_2181_mult__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).

% mult_1
tff(fact_2182_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_2183_mult_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),one_one_real) = A ).

% mult.right_neutral
tff(fact_2184_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_2185_mult_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ).

% mult.right_neutral
tff(fact_2186_mult_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).

% mult.right_neutral
tff(fact_2187_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_2188_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_2189_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_2190_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_2191_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_2192_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_2193_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_2194_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_2195_sum__squares__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2196_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2197_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2198_mult__cancel__left1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(times_times_complex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_2199_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_2200_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_2201_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_2202_mult__cancel__left2,axiom,
    ! [C: complex,A: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
tff(fact_2203_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_2204_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_2205_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_2206_mult__cancel__right1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(times_times_complex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_2207_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_2208_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_2209_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_2210_mult__cancel__right2,axiom,
    ! [A: complex,C: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
tff(fact_2211_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_2212_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_2213_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_2214_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( ( C = zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = zero_zero_complex ) )
      & ( ( C != zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_2215_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( C = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_2216_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( C = zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_2217_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2218_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2219_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2220_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2221_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != 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,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2222_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2223_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2224_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2225_nonzero__mult__div__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2226_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2227_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2228_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2229_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2230_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2231_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2232_nonzero__mult__div__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2233_nonzero__mult__div__cancel__right,axiom,
    ! [B: nat,A: nat] :
      ( ( B != 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,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2234_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2235_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2236_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2237_nonzero__mult__div__cancel__right,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2238_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2239_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2240_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2241_divide__eq__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = one_one_complex )
    <=> ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2242_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = one_one_real )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2243_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2244_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_2245_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),A) = one_one_nat ) ) ).

% div_self
tff(fact_2246_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(A),A) = one_one_int ) ) ).

% div_self
tff(fact_2247_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ).

% div_self
tff(fact_2248_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% div_self
tff(fact_2249_div__self,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(A,A) = one_one_Code_integer ) ) ).

% div_self
tff(fact_2250_one__eq__divide__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( one_one_complex = aa_complex_complex(divide1717551699836669952omplex(A),B) )
    <=> ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2251_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2252_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(A),B) )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2253_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_2254_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ).

% divide_self
tff(fact_2255_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% divide_self
tff(fact_2256_divide__self__if,axiom,
    ! [A: complex] :
      ( ( ( A = zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = zero_zero_complex ) )
      & ( ( A != zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ) ).

% divide_self_if
tff(fact_2257_divide__self__if,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ) ).

% divide_self_if
tff(fact_2258_divide__self__if,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(A),A) = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ) ).

% divide_self_if
tff(fact_2259_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) = one_one_real )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_2260_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_2261_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_2262_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(B),A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_2263_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_2264_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(one_one_rat),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_2265_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_2266_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(divide_divide_rat(one_one_rat),A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_2267_power__inject__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_2268_power__inject__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( ( aa_nat_rat(power_power_rat(A),M) = aa_nat_rat(power_power_rat(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_2269_power__inject__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_2270_power__inject__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( ( aa_nat_int(power_power_int(A),M) = aa_nat_int(power_power_int(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_2271_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_2272_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_2273_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_2274_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_2275_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_2276_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_2277_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_2278_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_2279_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: 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,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),X))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_2280_zero__le__divide__1__iff,axiom,
    ! [A: 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),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% zero_le_divide_1_iff
tff(fact_2281_zero__le__divide__1__iff,axiom,
    ! [A: 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),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% zero_le_divide_1_iff
tff(fact_2282_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% divide_le_0_1_iff
tff(fact_2283_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% divide_le_0_1_iff
tff(fact_2284_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% divide_less_0_1_iff
tff(fact_2285_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% divide_less_0_1_iff
tff(fact_2286_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% divide_less_eq_1_neg
tff(fact_2287_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% divide_less_eq_1_neg
tff(fact_2288_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% divide_less_eq_1_pos
tff(fact_2289_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% divide_less_eq_1_pos
tff(fact_2290_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),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,B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% less_divide_eq_1_neg
tff(fact_2291_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% less_divide_eq_1_neg
tff(fact_2292_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% less_divide_eq_1_pos
tff(fact_2293_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% less_divide_eq_1_pos
tff(fact_2294_zero__less__divide__1__iff,axiom,
    ! [A: 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),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% zero_less_divide_1_iff
tff(fact_2295_zero__less__divide__1__iff,axiom,
    ! [A: 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),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% zero_less_divide_1_iff
tff(fact_2296_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2))),B)) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2297_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2))),B)) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2298_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2)))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2299_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2300_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W2: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W2)) )
    <=> ( ( ( numera6690914467698888265omplex(W2) != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W2)) = B ) )
        & ( ( numera6690914467698888265omplex(W2) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2301_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2)) )
    <=> ( ( ( aa_num_real(numeral_numeral_real,W2) != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2)) = B ) )
        & ( ( aa_num_real(numeral_numeral_real,W2) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2302_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2)) )
    <=> ( ( ( aa_num_rat(numeral_numeral_rat,W2) != zero_zero_rat )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2)) = B ) )
        & ( ( aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2303_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W2: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W2)) = A )
    <=> ( ( ( numera6690914467698888265omplex(W2) != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W2)) ) )
        & ( ( numera6690914467698888265omplex(W2) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2304_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2)) = A )
    <=> ( ( ( aa_num_real(numeral_numeral_real,W2) != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2)) ) )
        & ( ( aa_num_real(numeral_numeral_real,W2) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2305_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2)) = A )
    <=> ( ( ( aa_num_rat(numeral_numeral_rat,W2) != zero_zero_rat )
         => ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2)) ) )
        & ( ( aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2306_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2)))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_2307_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_2308_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2))),B)) ) ).

% less_divide_eq_numeral1(1)
tff(fact_2309_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2))),B)) ) ).

% less_divide_eq_numeral1(1)
tff(fact_2310_nonzero__divide__mult__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2311_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2312_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2313_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2314_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2315_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2316_power__strict__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2317_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2318_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2319_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2320_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: 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,X),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,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),X)) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_2321_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_2322_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_2323_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_2324_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% divide_le_eq_1_neg
tff(fact_2325_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% divide_le_eq_1_neg
tff(fact_2326_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% divide_le_eq_1_pos
tff(fact_2327_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% divide_le_eq_1_pos
tff(fact_2328_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),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,B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% le_divide_eq_1_neg
tff(fact_2329_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),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(B),A)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% le_divide_eq_1_neg
tff(fact_2330_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% le_divide_eq_1_pos
tff(fact_2331_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% le_divide_eq_1_pos
tff(fact_2332_power__strict__decreasing__iff,axiom,
    ! [B: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2333_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2334_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),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,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2335_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2336_power__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2337_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2338_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2339_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2340_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(N)),one_on7984719198319812577d_enat))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_2341_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_2342_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_2343_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_2344_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_2345_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_2346_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_2347_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_2348_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_2349_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_2350_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_2351_bits__1__div__2,axiom,
    aa_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_2352_bits__1__div__2,axiom,
    divide6298287555418463151nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% bits_1_div_2
tff(fact_2353_power__decreasing__iff,axiom,
    ! [B: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),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,B),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_2354_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_2355_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),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,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_2356_power__decreasing__iff,axiom,
    ! [B: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_2357__092_060open_062y_A_061_A_Iif_Ax_A_061_Ama_Athen_Ahigh_Ax_An_A_K_A2_A_094_A_Ideg_Adiv_A2_J_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Ahigh_Ax_An_J_J_Aelse_Ama_J_092_060close_062,axiom,
    ( ( ( xa = ma )
     => ( ya = 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,xa),na)),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_nat2(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na))))) ) )
    & ( ( xa != ma )
     => ( ya = ma ) ) ) ).

% \<open>y = (if x = ma then high x n * 2 ^ (deg div 2) + the (vebt_maxt (treeList [high x n := vebt_delete (treeList ! high x n) (low x n)] ! high x n)) else ma)\<close>
tff(fact_2358_ssms,axiom,
    pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,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,xa),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,xa),na)),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_nat2(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)))))),ma))),deg,list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na))),summary)),ya)) ).

% ssms
tff(fact_2359_right__inverse__eq,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = one_one_complex )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2360_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = one_one_real )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2361_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2362_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_2363_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(aa_rat_fun_rat_rat(times_times_rat,M),N))) ) ) ).

% less_1_mult
tff(fact_2364_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_2365_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_2366_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_2367_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
      <=> ( aa_complex_complex(times_times_complex(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2368_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2369_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A = aa_rat_rat(divide_divide_rat(B),C) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2370_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
      <=> ( B = aa_complex_complex(times_times_complex(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2371_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
      <=> ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2372_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(B),C) = A )
      <=> ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2373_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = B )
       => ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2374_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B )
       => ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2375_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B )
       => ( A = aa_rat_rat(divide_divide_rat(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2376_divide__eq__imp,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B = aa_complex_complex(times_times_complex(A),C) )
       => ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2377_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2378_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) )
       => ( aa_rat_rat(divide_divide_rat(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2379_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),C) = B ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
tff(fact_2380_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq
tff(fact_2381_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
tff(fact_2382_divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
    <=> ( ( ( C != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(A),C) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
tff(fact_2383_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
    <=> ( ( ( C != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq
tff(fact_2384_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = A )
    <=> ( ( ( C != zero_zero_rat )
         => ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
tff(fact_2385_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( aa_complex_complex(divide1717551699836669952omplex(X),Y) = aa_complex_complex(divide1717551699836669952omplex(W2),Z) )
        <=> ( aa_complex_complex(times_times_complex(X),Z) = aa_complex_complex(times_times_complex(W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2386_frac__eq__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z) )
        <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2387_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( aa_rat_rat(divide_divide_rat(X),Y) = aa_rat_rat(divide_divide_rat(W2),Z) )
        <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2388_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_2389_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
    <=> ( X = one_one_complex ) ) ).

% one_reorient
tff(fact_2390_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
    <=> ( X = one_one_real ) ) ).

% one_reorient
tff(fact_2391_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
    <=> ( X = one_one_rat ) ) ).

% one_reorient
tff(fact_2392_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
    <=> ( X = one_one_nat ) ) ).

% one_reorient
tff(fact_2393_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
    <=> ( X = one_one_int ) ) ).

% one_reorient
tff(fact_2394_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_2395_mult_Oleft__commute,axiom,
    ! [B: complex,A: complex,C: complex] : aa_complex_complex(times_times_complex(B),aa_complex_complex(times_times_complex(A),C)) = aa_complex_complex(times_times_complex(A),aa_complex_complex(times_times_complex(B),C)) ).

% mult.left_commute
tff(fact_2396_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% mult.left_commute
tff(fact_2397_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% mult.left_commute
tff(fact_2398_mult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.left_commute
tff(fact_2399_mult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.left_commute
tff(fact_2400_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_2401_mult_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),one_one_real) = A ).

% mult.comm_neutral
tff(fact_2402_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_2403_mult_Ocomm__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ).

% mult.comm_neutral
tff(fact_2404_mult_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).

% mult.comm_neutral
tff(fact_2405_mult_Ocommute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(A),B) = aa_complex_complex(times_times_complex(B),A) ).

% mult.commute
tff(fact_2406_mult_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A) ).

% mult.commute
tff(fact_2407_mult_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A) ).

% mult.commute
tff(fact_2408_mult_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) ).

% mult.commute
tff(fact_2409_mult_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,B),A) ).

% mult.commute
tff(fact_2410_mult_Oassoc,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(A),B)),C) = aa_complex_complex(times_times_complex(A),aa_complex_complex(times_times_complex(B),C)) ).

% mult.assoc
tff(fact_2411_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% mult.assoc
tff(fact_2412_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% mult.assoc
tff(fact_2413_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.assoc
tff(fact_2414_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.assoc
tff(fact_2415_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2416_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2417_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2418_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2419_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2420_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(A),B)),C) = aa_complex_complex(times_times_complex(A),aa_complex_complex(times_times_complex(B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2421_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2422_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2423_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2424_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2425_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_2426_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_2427_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ) ) ).

% divide_less_eq
tff(fact_2428_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% divide_less_eq
tff(fact_2429_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq
tff(fact_2430_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq
tff(fact_2431_neg__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).

% neg_divide_less_eq
tff(fact_2432_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B)) ) ) ).

% neg_divide_less_eq
tff(fact_2433_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% neg_less_divide_eq
tff(fact_2434_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) ) ) ).

% neg_less_divide_eq
tff(fact_2435_pos__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% pos_divide_less_eq
tff(fact_2436_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) ) ) ).

% pos_divide_less_eq
tff(fact_2437_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).

% pos_less_divide_eq
tff(fact_2438_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B)) ) ) ).

% pos_less_divide_eq
tff(fact_2439_mult__imp__div__pos__less,axiom,
    ! [Y: real,X: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z)) ) ) ).

% mult_imp_div_pos_less
tff(fact_2440_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),Z)) ) ) ).

% mult_imp_div_pos_less
tff(fact_2441_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% mult_imp_less_div_pos
tff(fact_2442_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),X))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% mult_imp_less_div_pos
tff(fact_2443_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B))) ) ) ) ).

% divide_strict_left_mono
tff(fact_2444_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B))) ) ) ) ).

% divide_strict_left_mono
tff(fact_2445_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B))) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2446_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B))) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2447_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: complex,C: complex] :
      ( ( numera6690914467698888265omplex(W2) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),C) = B ) )
        & ( ( C = zero_zero_complex )
         => ( numera6690914467698888265omplex(W2) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2448_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( aa_num_real(numeral_numeral_real,W2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C) = B ) )
        & ( ( C = zero_zero_real )
         => ( aa_num_real(numeral_numeral_real,W2) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2449_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W2) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C) = B ) )
        & ( ( C = zero_zero_rat )
         => ( aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2450_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W2: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = numera6690914467698888265omplex(W2) )
    <=> ( ( ( C != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),C) ) )
        & ( ( C = zero_zero_complex )
         => ( numera6690914467698888265omplex(W2) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2451_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = aa_num_real(numeral_numeral_real,W2) )
    <=> ( ( ( C != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C) ) )
        & ( ( C = zero_zero_real )
         => ( aa_num_real(numeral_numeral_real,W2) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2452_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_num_rat(numeral_numeral_rat,W2) )
    <=> ( ( ( C != zero_zero_rat )
         => ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C) ) )
        & ( ( C = zero_zero_rat )
         => ( aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2453_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = B ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_2454_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = B ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_2455_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(A),Z)),B) = B ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(A),Z)),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_2456_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = A ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_2457_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = A ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),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,A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_2458_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(divide_divide_rat(B),Z)) = A ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(divide_divide_rat(B),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_2459_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(W2),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),Z)),aa_complex_complex(times_times_complex(W2),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2460_add__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != 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,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),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,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2461_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W2),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2462_add__frac__num,axiom,
    ! [Y: complex,X: complex,Z: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2463_add__frac__num,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2464_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(X),Y)),Z) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2465_add__num__frac,axiom,
    ! [Y: complex,Z: complex,X: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2466_add__num__frac,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2467_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_rat_rat(divide_divide_rat(X),Y)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2468_add__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(X),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2469_add__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),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,X),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2470_add__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(divide_divide_rat(Y),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2471_divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2472_divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2473_divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(X),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2474_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_2475_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_2476_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% less_divide_eq_1
tff(fact_2477_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% less_divide_eq_1
tff(fact_2478_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real)))) ) ).

% less_half_sum
tff(fact_2479_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)))) ) ).

% less_half_sum
tff(fact_2480_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => 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,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))),B)) ) ).

% gt_half_sum
tff(fact_2481_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))),B)) ) ).

% gt_half_sum
tff(fact_2482_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),A)) ) ) ).

% mult_left_le
tff(fact_2483_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),one_one_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),A)) ) ) ).

% mult_left_le
tff(fact_2484_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A)) ) ) ).

% mult_left_le
tff(fact_2485_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),A)) ) ) ).

% mult_left_le
tff(fact_2486_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),one_one_real)) ) ) ) ).

% mult_le_one
tff(fact_2487_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),one_one_rat)) ) ) ) ).

% mult_le_one
tff(fact_2488_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),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,A),B)),one_one_nat)) ) ) ) ).

% mult_le_one
tff(fact_2489_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),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,A),B)),one_one_int)) ) ) ) ).

% mult_le_one
tff(fact_2490_mult__right__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_2491_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_2492_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),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,X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_2493_mult__left__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_2494_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_2495_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),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,Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_2496_power__less__power__Suc,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))) ) ).

% power_less_power_Suc
tff(fact_2497_power__less__power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N)))) ) ).

% power_less_power_Suc
tff(fact_2498_power__less__power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))) ) ).

% power_less_power_Suc
tff(fact_2499_power__less__power__Suc,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)))) ) ).

% power_less_power_Suc
tff(fact_2500_power__gt1__lemma,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))) ) ).

% power_gt1_lemma
tff(fact_2501_power__gt1__lemma,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N)))) ) ).

% power_gt1_lemma
tff(fact_2502_power__gt1__lemma,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => 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,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))) ) ).

% power_gt1_lemma
tff(fact_2503_power__gt1__lemma,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => 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,A),aa_nat_int(power_power_int(A),N)))) ) ).

% power_gt1_lemma
tff(fact_2504_mult__not__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex )
     => ( ( A != zero_zero_complex )
        & ( B != zero_zero_complex ) ) ) ).

% mult_not_zero
tff(fact_2505_mult__not__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_2506_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_2507_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_2508_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_2509_divisors__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
     => ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divisors_zero
tff(fact_2510_divisors__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_2511_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_2512_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_2513_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_2514_no__zero__divisors,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex ) ) ) ).

% no_zero_divisors
tff(fact_2515_no__zero__divisors,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_2516_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_2517_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_2518_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_2519_mult__left__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2520_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2521_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2522_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2523_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2524_mult__right__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2525_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2526_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2527_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2528_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2529_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_2530_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_2531_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_2532_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_2533_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_2534_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_2535_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_2536_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_2537_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_2538_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_2539_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_2540_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_2541_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_2542_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_2543_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_2544_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_2545_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_2546_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_2547_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_2548_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_2549_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_2550_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_2551_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_2552_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_2553_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ) ) ).

% divide_le_eq
tff(fact_2554_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% divide_le_eq
tff(fact_2555_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq
tff(fact_2556_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq
tff(fact_2557_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B))) ) ) ) ).

% divide_left_mono
tff(fact_2558_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B))) ) ) ) ).

% divide_left_mono
tff(fact_2559_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).

% neg_divide_le_eq
tff(fact_2560_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B)) ) ) ).

% neg_divide_le_eq
tff(fact_2561_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% neg_le_divide_eq
tff(fact_2562_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) ) ) ).

% neg_le_divide_eq
tff(fact_2563_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% pos_divide_le_eq
tff(fact_2564_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) ) ) ).

% pos_divide_le_eq
tff(fact_2565_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).

% pos_le_divide_eq
tff(fact_2566_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B)) ) ) ).

% pos_le_divide_eq
tff(fact_2567_mult__imp__div__pos__le,axiom,
    ! [Y: real,X: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z)) ) ) ).

% mult_imp_div_pos_le
tff(fact_2568_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),Z)) ) ) ).

% mult_imp_div_pos_le
tff(fact_2569_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% mult_imp_le_div_pos
tff(fact_2570_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),X))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% mult_imp_le_div_pos
tff(fact_2571_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),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,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B))) ) ) ) ).

% divide_left_mono_neg
tff(fact_2572_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B))) ) ) ) ).

% divide_left_mono_neg
tff(fact_2573_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_num_real(numeral_numeral_real,W2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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,W2)),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,W2))) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2574_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W2))) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2575_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,W2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => 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,W2)),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,W2)),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2576_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2577_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_2578_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_2579_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% le_divide_eq_1
tff(fact_2580_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% le_divide_eq_1
tff(fact_2581_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2582_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2583_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2584_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2585_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2586_mult__le__cancel__left2,axiom,
    ! [C: int,A: 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,C),A)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2587_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2588_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2589_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2590_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2591_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2592_mult__le__cancel__right2,axiom,
    ! [A: int,C: 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,A),C)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2593_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2594_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2595_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2596_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2597_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2598_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2599_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2600_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2601_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2602_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2603_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2604_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2605_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z3),X)),Y)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% field_le_mult_one_interval
tff(fact_2606_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z3))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),one_one_rat))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z3),X)),Y)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% field_le_mult_one_interval
tff(fact_2607_convex__bound__le,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),A))
       => ( 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),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2608_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),A))
       => ( 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(aa_rat_fun_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(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2609_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),A))
       => ( 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),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2610_power__Suc__less,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ).

% power_Suc_less
tff(fact_2611_power__Suc__less,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N))),aa_nat_rat(power_power_rat(A),N))) ) ) ).

% power_Suc_less
tff(fact_2612_power__Suc__less,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),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,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).

% power_Suc_less
tff(fact_2613_power__Suc__less,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),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,A),aa_nat_int(power_power_int(A),N))),aa_nat_int(power_power_int(A),N))) ) ) ).

% power_Suc_less
tff(fact_2614_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,W2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => 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,W2)),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,W2)),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2615_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2616_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_num_real(numeral_numeral_real,W2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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,W2)),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,W2))) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2617_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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,W2))) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2618_convex__bound__lt,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),A))
       => ( 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),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2619_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),A))
       => ( 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(aa_rat_fun_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(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2620_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),A))
       => ( 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),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2621_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C))) ) ) ).

% divide_right_mono_neg
tff(fact_2622_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(divide_divide_rat(A),C))) ) ) ).

% divide_right_mono_neg
tff(fact_2623_divide__nonpos__nonpos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),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,X),Y))) ) ) ).

% divide_nonpos_nonpos
tff(fact_2624_divide__nonpos__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),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(X),Y))) ) ) ).

% divide_nonpos_nonpos
tff(fact_2625_divide__nonpos__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_nonpos_nonneg
tff(fact_2626_divide__nonpos__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_nonpos_nonneg
tff(fact_2627_divide__nonneg__nonpos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_nonneg_nonpos
tff(fact_2628_divide__nonneg__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_nonneg_nonpos
tff(fact_2629_divide__nonneg__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_nonneg_nonneg
tff(fact_2630_divide__nonneg__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_nonneg_nonneg
tff(fact_2631_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) ) ) ).

% zero_le_divide_iff
tff(fact_2632_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) ) ) ).

% zero_le_divide_iff
tff(fact_2633_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))) ) ) ).

% divide_right_mono
tff(fact_2634_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))) ) ) ).

% divide_right_mono
tff(fact_2635_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ) ) ).

% divide_le_0_iff
tff(fact_2636_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ) ) ).

% divide_le_0_iff
tff(fact_2637_divide__neg__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),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,X),Y))) ) ) ).

% divide_neg_neg
tff(fact_2638_divide__neg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_neg_neg
tff(fact_2639_divide__neg__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_neg_pos
tff(fact_2640_divide__neg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_neg_pos
tff(fact_2641_divide__pos__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_pos_neg
tff(fact_2642_divide__pos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_pos_neg
tff(fact_2643_divide__pos__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_pos_pos
tff(fact_2644_divide__pos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_pos_pos
tff(fact_2645_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ) ).

% divide_less_0_iff
tff(fact_2646_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ) ).

% divide_less_0_iff
tff(fact_2647_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_2648_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_2649_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) ) ) ) ).

% zero_less_divide_iff
tff(fact_2650_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) ) ) ) ).

% zero_less_divide_iff
tff(fact_2651_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))) ) ) ).

% divide_strict_right_mono
tff(fact_2652_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))) ) ) ).

% divide_strict_right_mono
tff(fact_2653_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2654_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2655_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2656_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2657_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2658_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: 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_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2659_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) ) ) ).

% zero_le_mult_iff
tff(fact_2660_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) ) ) ).

% zero_le_mult_iff
tff(fact_2661_zero__le__mult__iff,axiom,
    ! [A: int,B: 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,A),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) ) ) ).

% zero_le_mult_iff
tff(fact_2662_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2663_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2664_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),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,B),A)),zero_zero_nat)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2665_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),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,B),A)),zero_zero_int)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2666_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_nonpos_nonneg
tff(fact_2667_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)) ) ) ).

% mult_nonpos_nonneg
tff(fact_2668_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).

% mult_nonpos_nonneg
tff(fact_2669_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).

% mult_nonpos_nonneg
tff(fact_2670_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_nonneg_nonpos
tff(fact_2671_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos
tff(fact_2672_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),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,A),B)),zero_zero_nat)) ) ) ).

% mult_nonneg_nonpos
tff(fact_2673_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),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,A),B)),zero_zero_int)) ) ) ).

% mult_nonneg_nonpos
tff(fact_2674_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_2675_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_2676_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => 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,A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_2677_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => 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,A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_2678_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ).

% split_mult_neg_le
tff(fact_2679_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)) ) ).

% split_mult_neg_le
tff(fact_2680_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat)) )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ).

% split_mult_neg_le
tff(fact_2681_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ).

% split_mult_neg_le
tff(fact_2682_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ) ) ).

% mult_le_0_iff
tff(fact_2683_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ) ) ).

% mult_le_0_iff
tff(fact_2684_mult__le__0__iff,axiom,
    ! [A: int,B: 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,A),B)),zero_zero_int))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ) ) ).

% mult_le_0_iff
tff(fact_2685_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_right_mono
tff(fact_2686_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))) ) ) ).

% mult_right_mono
tff(fact_2687_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ) ).

% mult_right_mono
tff(fact_2688_mult__right__mono,axiom,
    ! [A: int,B: int,C: 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_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).

% mult_right_mono
tff(fact_2689_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_2690_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_2691_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),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,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_2692_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_left_mono
tff(fact_2693_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))) ) ) ).

% mult_left_mono
tff(fact_2694_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).

% mult_left_mono
tff(fact_2695_mult__left__mono,axiom,
    ! [A: int,B: int,C: 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_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% mult_left_mono
tff(fact_2696_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),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,A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_2697_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_2698_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),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,A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_2699_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_2700_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_2701_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),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,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_2702_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),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,A),B))) ) ).

% split_mult_pos_le
tff(fact_2703_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ).

% split_mult_pos_le
tff(fact_2704_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),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,A),B))) ) ).

% split_mult_pos_le
tff(fact_2705_zero__le__square,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A))) ).

% zero_le_square
tff(fact_2706_zero__le__square,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A))) ).

% zero_le_square
tff(fact_2707_zero__le__square,axiom,
    ! [A: 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,A),A))) ).

% zero_le_square
tff(fact_2708_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2709_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2710_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2711_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: 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_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2712_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_2713_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_2714_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_2715_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: 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_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_2716_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),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,A),B))) ) ) ).

% mult_neg_neg
tff(fact_2717_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).

% mult_neg_neg
tff(fact_2718_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),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,A),B))) ) ) ).

% mult_neg_neg
tff(fact_2719_not__square__less__zero,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),zero_zero_real)) ).

% not_square_less_zero
tff(fact_2720_not__square__less__zero,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)),zero_zero_rat)) ).

% not_square_less_zero
tff(fact_2721_not__square__less__zero,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),zero_zero_int)) ).

% not_square_less_zero
tff(fact_2722_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ) ).

% mult_less_0_iff
tff(fact_2723_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ) ).

% mult_less_0_iff
tff(fact_2724_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ) ).

% mult_less_0_iff
tff(fact_2725_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_neg_pos
tff(fact_2726_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)) ) ) ).

% mult_neg_pos
tff(fact_2727_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).

% mult_neg_pos
tff(fact_2728_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).

% mult_neg_pos
tff(fact_2729_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_pos_neg
tff(fact_2730_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)) ) ) ).

% mult_pos_neg
tff(fact_2731_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),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,A),B)),zero_zero_nat)) ) ) ).

% mult_pos_neg
tff(fact_2732_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),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,A),B)),zero_zero_int)) ) ) ).

% mult_pos_neg
tff(fact_2733_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_pos_pos
tff(fact_2734_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).

% mult_pos_pos
tff(fact_2735_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => 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,A),B))) ) ) ).

% mult_pos_pos
tff(fact_2736_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => 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,A),B))) ) ) ).

% mult_pos_pos
tff(fact_2737_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real)) ) ) ).

% mult_pos_neg2
tff(fact_2738_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat)) ) ) ).

% mult_pos_neg2
tff(fact_2739_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),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,B),A)),zero_zero_nat)) ) ) ).

% mult_pos_neg2
tff(fact_2740_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),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,B),A)),zero_zero_int)) ) ) ).

% mult_pos_neg2
tff(fact_2741_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) ) ) ) ).

% zero_less_mult_iff
tff(fact_2742_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) ) ) ) ).

% zero_less_mult_iff
tff(fact_2743_zero__less__mult__iff,axiom,
    ! [A: int,B: 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,A),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int)) ) ) ) ).

% zero_less_mult_iff
tff(fact_2744_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ).

% zero_less_mult_pos
tff(fact_2745_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ).

% zero_less_mult_pos
tff(fact_2746_zero__less__mult__pos,axiom,
    ! [A: nat,B: 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,A),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ) ).

% zero_less_mult_pos
tff(fact_2747_zero__less__mult__pos,axiom,
    ! [A: int,B: 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,A),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ).

% zero_less_mult_pos
tff(fact_2748_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_2749_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_2750_zero__less__mult__pos2,axiom,
    ! [B: nat,A: 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,B),A)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_2751_zero__less__mult__pos2,axiom,
    ! [B: int,A: 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,B),A)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_2752_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2753_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2754_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),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,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2755_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2756_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2757_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2758_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2759_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2760_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),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,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2761_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_2762_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_2763_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_2764_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: 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),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_2765_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2766_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2767_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
          & 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,C),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2768_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2769_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2770_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),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,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2771_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_2772_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_2773_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_2774_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: 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),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_2775_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2776_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2777_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
          & 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,C),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2778_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2779_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2780_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2781_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: 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),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2782_add__scale__eq__noteq,axiom,
    ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(R2),C)) != aa_complex_complex(plus_plus_complex(B),aa_complex_complex(times_times_complex(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2783_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),C)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2784_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),C)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2785_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2786_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R2),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2787_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_2788_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_2789_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_2790_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_2791_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_2792_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_2793_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_2794_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_2795_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_2796_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_2797_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_2798_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_2799_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_2800_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_2801_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_2802_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_2803_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_2804_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_2805_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_2806_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_2807_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_2808_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_2809_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_2810_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_2811_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_2812_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_2813_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_2814_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_2815_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_2816_one__le__numeral,axiom,
    ! [N: num] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(N))) ).

% one_le_numeral
tff(fact_2817_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_2818_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_2819_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_2820_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_2821_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(N)),one_on7984719198319812577d_enat)) ).

% not_numeral_less_one
tff(fact_2822_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_2823_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_2824_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_2825_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_2826_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_2827_add__mono1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),one_one_real))) ) ).

% add_mono1
tff(fact_2828_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),one_one_rat))) ) ).

% add_mono1
tff(fact_2829_add__mono1,axiom,
    ! [A: nat,B: 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,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),one_one_nat))) ) ).

% add_mono1
tff(fact_2830_add__mono1,axiom,
    ! [A: int,B: 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,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int))) ) ).

% add_mono1
tff(fact_2831_less__add__one,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real))) ).

% less_add_one
tff(fact_2832_less__add__one,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat))) ).

% less_add_one
tff(fact_2833_less__add__one,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat))) ).

% less_add_one
tff(fact_2834_less__add__one,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int))) ).

% less_add_one
tff(fact_2835_one__le__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
     => 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,A),N))) ) ).

% one_le_power
tff(fact_2836_one__le__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N))) ) ).

% one_le_power
tff(fact_2837_one__le__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
     => 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,A),N))) ) ).

% one_le_power
tff(fact_2838_one__le__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),N))) ) ).

% one_le_power
tff(fact_2839_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_2840_power__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),zero_zero_nat) = one_one_nat ).

% power_0
tff(fact_2841_power__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_2842_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_2843_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_2844_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_2845_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_2846_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_2847_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_2848_frac__le,axiom,
    ! [Y: real,X: real,W2: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W2),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2))) ) ) ) ) ).

% frac_le
tff(fact_2849_frac__le,axiom,
    ! [Y: rat,X: rat,W2: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W2))
         => ( 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,aa_rat_rat(divide_divide_rat(X),Z)),aa_rat_rat(divide_divide_rat(Y),W2))) ) ) ) ) ).

% frac_le
tff(fact_2850_frac__less,axiom,
    ! [X: real,Y: real,W2: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W2),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2))) ) ) ) ) ).

% frac_less
tff(fact_2851_frac__less,axiom,
    ! [X: rat,Y: rat,W2: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W2))
         => ( 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_rat,aa_rat_rat(divide_divide_rat(X),Z)),aa_rat_rat(divide_divide_rat(Y),W2))) ) ) ) ) ).

% frac_less
tff(fact_2852_frac__less2,axiom,
    ! [X: real,Y: real,W2: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W2),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2))) ) ) ) ) ).

% frac_less2
tff(fact_2853_frac__less2,axiom,
    ! [X: rat,Y: rat,W2: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W2),Z))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Z)),aa_rat_rat(divide_divide_rat(Y),W2))) ) ) ) ) ).

% frac_less2
tff(fact_2854_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% divide_le_cancel
tff(fact_2855_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% divide_le_cancel
tff(fact_2856_divide__nonneg__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_nonneg_neg
tff(fact_2857_divide__nonneg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_nonneg_neg
tff(fact_2858_divide__nonneg__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_nonneg_pos
tff(fact_2859_divide__nonneg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_nonneg_pos
tff(fact_2860_divide__nonpos__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),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,X),Y))) ) ) ).

% divide_nonpos_neg
tff(fact_2861_divide__nonpos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),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(X),Y))) ) ) ).

% divide_nonpos_neg
tff(fact_2862_divide__nonpos__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_nonpos_pos
tff(fact_2863_divide__nonpos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_nonpos_pos
tff(fact_2864_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_2865_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_2866_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: 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,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => 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,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_2867_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_2868_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_2869_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: 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,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => 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,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_2870_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_2871_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_2872_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_2873_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_2874_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2875_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2876_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: 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,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2877_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: 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,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2878_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_2879_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_2880_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => 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_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_2881_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_2882_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_2883_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_2884_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_2885_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2886_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2887_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: 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,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2888_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: 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,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2889_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_2890_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_2891_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => 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_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_2892_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2893_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2894_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),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,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2895_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2896_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2897_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2898_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_2899_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_2900_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: 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,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_2901_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: 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,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_2902_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_2903_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_2904_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: 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,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_2905_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: 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,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_2906_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2907_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2908_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2909_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: 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,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2910_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2911_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2912_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: 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_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2913_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: 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_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2914_sum__squares__ge__zero,axiom,
    ! [X: real,Y: 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,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_2915_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_2916_sum__squares__ge__zero,axiom,
    ! [X: int,Y: 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,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_2917_sum__squares__le__zero__iff,axiom,
    ! [X: real,Y: 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,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real))
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2918_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat))
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2919_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: 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,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int))
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2920_not__sum__squares__lt__zero,axiom,
    ! [X: real,Y: 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,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real)) ).

% not_sum_squares_lt_zero
tff(fact_2921_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat)) ).

% not_sum_squares_lt_zero
tff(fact_2922_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: 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,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)) ).

% not_sum_squares_lt_zero
tff(fact_2923_sum__squares__gt__zero__iff,axiom,
    ! [X: real,Y: 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,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))))
    <=> ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2924_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2925_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: 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,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2926_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_2927_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_2928_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_2929_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_2930_zero__less__two,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))) ).

% zero_less_two
tff(fact_2931_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_2932_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_2933_power__le__one,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),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,A),N)),one_one_real)) ) ) ).

% power_le_one
tff(fact_2934_power__le__one,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),one_one_rat)) ) ) ).

% power_le_one
tff(fact_2935_power__le__one,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),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,A),N)),one_one_nat)) ) ) ).

% power_le_one
tff(fact_2936_power__le__one,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),one_one_int)) ) ) ).

% power_le_one
tff(fact_2937_power__gt1,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_2938_power__gt1,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_2939_power__gt1,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => 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,A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_2940_power__gt1,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_2941_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_2942_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_2943_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_2944_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(power_power_int(zero_zero_int),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ) ).

% power_0_left
tff(fact_2945_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = zero_zero_complex ) ) ) ).

% power_0_left
tff(fact_2946_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_2947_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_2948_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_2949_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_2950_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4))) ) ) ).

% power_strict_increasing
tff(fact_2951_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(A),N4))) ) ) ).

% power_strict_increasing
tff(fact_2952_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4))) ) ) ).

% power_strict_increasing
tff(fact_2953_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N4))) ) ) ).

% power_strict_increasing
tff(fact_2954_power__increasing,axiom,
    ! [N: nat,N4: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4))) ) ) ).

% power_increasing
tff(fact_2955_power__increasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(A),N4))) ) ) ).

% power_increasing
tff(fact_2956_power__increasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4))) ) ) ).

% power_increasing
tff(fact_2957_power__increasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N4))) ) ) ).

% power_increasing
tff(fact_2958_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))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => ( pp(aa_nat_bool(P,N2))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct_non_zero
tff(fact_2959_power__Suc__le__self,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),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,A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2960_power__Suc__le__self,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2961_power__Suc__le__self,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),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,A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2962_power__Suc__le__self,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2963_power__Suc__less__one,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N))),one_one_real)) ) ) ).

% power_Suc_less_one
tff(fact_2964_power__Suc__less__one,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),one_one_rat)) ) ) ).

% power_Suc_less_one
tff(fact_2965_power__Suc__less__one,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),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,A),aa_nat_nat(suc,N))),one_one_nat)) ) ) ).

% power_Suc_less_one
tff(fact_2966_power__Suc__less__one,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),one_one_int)) ) ) ).

% power_Suc_less_one
tff(fact_2967_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2968_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N4)),aa_nat_rat(power_power_rat(A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2969_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),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,A),N4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2970_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N4)),aa_nat_int(power_power_int(A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2971_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),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,A),N4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ) ).

% power_decreasing
tff(fact_2972_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N4)),aa_nat_rat(power_power_rat(A),N))) ) ) ) ).

% power_decreasing
tff(fact_2973_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),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,A),N4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ) ).

% power_decreasing
tff(fact_2974_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N4)),aa_nat_int(power_power_int(A),N))) ) ) ) ).

% power_decreasing
tff(fact_2975_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2976_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2977_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2978_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2979_self__le__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
     => ( 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,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ).

% self_le_power
tff(fact_2980_self__le__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
     => ( 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,A),aa_nat_rat(power_power_rat(A),N))) ) ) ).

% self_le_power
tff(fact_2981_self__le__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
     => ( 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,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).

% self_le_power
tff(fact_2982_self__le__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
     => ( 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,A),aa_nat_int(power_power_int(A),N))) ) ) ).

% self_le_power
tff(fact_2983_one__less__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( 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,A),N))) ) ) ).

% one_less_power
tff(fact_2984_one__less__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( 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(A),N))) ) ) ).

% one_less_power
tff(fact_2985_one__less__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( 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,A),N))) ) ) ).

% one_less_power
tff(fact_2986_one__less__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( 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(power_power_int(A),N))) ) ) ).

% one_less_power
tff(fact_2987_nat__bit__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,N2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => 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))),N2))) ) )
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(P,N2))
             => 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))),N2)))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_bit_induct
tff(fact_2988_half__gt__zero,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% half_gt_zero
tff(fact_2989_half__gt__zero,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ).

% half_gt_zero
tff(fact_2990_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% half_gt_zero_iff
tff(fact_2991_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),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),A)) ) ).

% half_gt_zero_iff
tff(fact_2992_field__less__half__sum,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% field_less_half_sum
tff(fact_2993_field__less__half__sum,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ).

% field_less_half_sum
tff(fact_2994_zero__le__even__power_H,axiom,
    ! [A: 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,A),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_2995_zero__le__even__power_H,axiom,
    ! [A: 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(A),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_2996_zero__le__even__power_H,axiom,
    ! [A: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),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_2997_odd__0__le__power__imp__0__le,axiom,
    ! [A: 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,A),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),A)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2998_odd__0__le__power__imp__0__le,axiom,
    ! [A: 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(A),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),A)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2999_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),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),A)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3000_odd__power__less__zero,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),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_3001_odd__power__less__zero,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),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_3002_odd__power__less__zero,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),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_3003_ex__power__ivl2,axiom,
    ! [B: 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))),B))
     => ( 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))
       => ? [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),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,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_3004_ex__power__ivl1,axiom,
    ! [B: 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))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),K))
       => ? [N2: 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,B),N2)),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,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_3005_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero_nat,Ts,S),X) = vEBT_Node(Info,zero_zero_nat,Ts,S) ).

% vebt_insert.simps(2)
tff(fact_3006_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S),X) = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S) ).

% vebt_insert.simps(3)
tff(fact_3007_aaaa,axiom,
    ( ( ya = mi )
    | ( ( ( xa = ma )
       => ( ya = 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,xa),na)),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_nat2(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na))))) ) )
      & ( ( xa != ma )
       => ( ya = ma ) ) )
    | ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ya),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ya),na)))
      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ya),na)),size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na)))))) ) ) ).

% aaaa
tff(fact_3008__092_060open_062vebt__delete_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_061_ANode_A_ISome_A_Imi_M_Aif_Ax_A_061_Ama_Athen_Ahigh_Ax_An_A_K_A2_A_094_A_Ideg_Adiv_A2_J_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Ahigh_Ax_An_J_J_Aelse_Ama_J_J_Adeg_A_ItreeList_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_J_Asummary_092_060close_062,axiom,
    vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary),xa) = 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,xa),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,xa),na)),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_nat2(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)))))),ma))),deg,list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na))),summary) ).

% \<open>vebt_delete (Node (Some (mi, ma)) deg treeList summary) x = Node (Some (mi, if x = ma then high x n * 2 ^ (deg div 2) + the (vebt_maxt (treeList [high x n := vebt_delete (treeList ! high x n) (low x n)] ! high x n)) else ma)) deg (treeList[high x n := vebt_delete (treeList ! high x n) (low x n)]) summary\<close>
tff(fact_3009_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_3010_one__div__two__eq__zero,axiom,
    aa_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_3011_one__div__two__eq__zero,axiom,
    divide6298287555418463151nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% one_div_two_eq_zero
tff(fact_3012__092_060open_062_Iif_Ax_A_061_Ama_Athen_Ahigh_Ax_An_A_K_A2_A_094_A_Ideg_Adiv_A2_J_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Ahigh_Ax_An_J_J_Aelse_Ama_J_A_061_Ahigh_Ax_An_A_K_A2_A_094_A_Ideg_Adiv_A2_J_A_L_Athe_A_Ivebt__maxt_A_ItreeList_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Ahigh_Ax_An_J_J_092_060close_062,axiom,
    ( ( xa != ma )
   => ( 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,xa),na)),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_nat2(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na))))) ) ) ).

% \<open>(if x = ma then high x n * 2 ^ (deg div 2) + the (vebt_maxt (treeList [high x n := vebt_delete (treeList ! high x n) (low x n)] ! high x n)) else ma) = high x n * 2 ^ (deg div 2) + the (vebt_maxt (treeList[high x n := vebt_delete (treeList ! high x n) (low x n)] ! high x n))\<close>
tff(fact_3013_del__x__mi__lets__in__not__minNull,axiom,
    ! [X: 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] :
      ( ( ( X = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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 = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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_nat2(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H))))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_3014_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X: 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),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),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,X),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,X),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 = vEBT_vebt_delete(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,X),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_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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,X),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_nat2(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H))))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_3015_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X: 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,X),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))
         => ( ( X != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),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)))),vEBT_vebt_insert(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)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,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)))))),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_3016_insert__simp__norm,axiom,
    ! [X: 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,X),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),X))
       => ( 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))
         => ( ( X != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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_3017_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
      ( pp(aa_nat_bool(vEBT_invar_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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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_3018_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_3019_half__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_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_3020_max__bot2,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ord_ma7524802468073614006at_nat(X,bot_bo2099793752762293965at_nat) = X ).

% max_bot2
tff(fact_3021_max__bot2,axiom,
    ! [X: set_real] : ord_max_set_real(X,bot_bot_set_real) = X ).

% max_bot2
tff(fact_3022_max__bot2,axiom,
    ! [X: set_nat] : ord_max_set_nat(X,bot_bot_set_nat) = X ).

% max_bot2
tff(fact_3023_max__bot2,axiom,
    ! [X: set_int] : ord_max_set_int(X,bot_bot_set_int) = X ).

% max_bot2
tff(fact_3024_max__bot2,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),bot_bot_nat) = X ).

% max_bot2
tff(fact_3025_max__bot,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ord_ma7524802468073614006at_nat(bot_bo2099793752762293965at_nat,X) = X ).

% max_bot
tff(fact_3026_max__bot,axiom,
    ! [X: set_real] : ord_max_set_real(bot_bot_set_real,X) = X ).

% max_bot
tff(fact_3027_max__bot,axiom,
    ! [X: set_nat] : ord_max_set_nat(bot_bot_set_nat,X) = X ).

% max_bot
tff(fact_3028_max__bot,axiom,
    ! [X: set_int] : ord_max_set_int(bot_bot_set_int,X) = X ).

% max_bot
tff(fact_3029_max__bot,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),X) = X ).

% max_bot
tff(fact_3030_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_3031_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_3032_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_3033_max__nat_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),zero_zero_nat) = A ).

% max_nat.right_neutral
tff(fact_3034_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_3035_max__nat_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),A) = A ).

% max_nat.left_neutral
tff(fact_3036_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_3037_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Deg))
     => ( ( Mi != Ma )
       => ( the_nat2(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_3038_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( C = 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,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = zero_zero_nat ) )
      & ( ( C != 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,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% div_mult_mult1_if
tff(fact_3039_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( C = zero_zero_int )
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(divide_divide_int(A),B) ) ) ) ).

% div_mult_mult1_if
tff(fact_3040_div__mult__mult1__if,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ( C = zero_z3403309356797280102nteger )
       => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) = zero_z3403309356797280102nteger ) )
      & ( ( C != zero_z3403309356797280102nteger )
       => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) = divide6298287555418463151nteger(A,B) ) ) ) ).

% div_mult_mult1_if
tff(fact_3041_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != 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,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult2
tff(fact_3042_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(A),B) ) ) ).

% div_mult_mult2
tff(fact_3043_div__mult__mult2,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(A,B) ) ) ).

% div_mult_mult2
tff(fact_3044_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != 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,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult1
tff(fact_3045_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(divide_divide_int(A),B) ) ) ).

% div_mult_mult1
tff(fact_3046_div__mult__mult1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) = divide6298287555418463151nteger(A,B) ) ) ).

% div_mult_mult1
tff(fact_3047_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_3048_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_3049_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_3050_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_3051_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(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(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_3052_max__0__1_I3_J,axiom,
    ! [X: num] : ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,numera1916890842035813515d_enat(X)) = numera1916890842035813515d_enat(X) ).

% max_0_1(3)
tff(fact_3053_max__0__1_I3_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),aa_num_real(numeral_numeral_real,X)) = aa_num_real(numeral_numeral_real,X) ).

% max_0_1(3)
tff(fact_3054_max__0__1_I3_J,axiom,
    ! [X: num] : aa_rat_rat(ord_max_rat(zero_zero_rat),aa_num_rat(numeral_numeral_rat,X)) = aa_num_rat(numeral_numeral_rat,X) ).

% max_0_1(3)
tff(fact_3055_max__0__1_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_num_nat(numeral_numeral_nat,X) ).

% max_0_1(3)
tff(fact_3056_max__0__1_I3_J,axiom,
    ! [X: num] : aa_int_int(ord_max_int(zero_zero_int),aa_num_int(numeral_numeral_int,X)) = aa_num_int(numeral_numeral_int,X) ).

% max_0_1(3)
tff(fact_3057_max__0__1_I4_J,axiom,
    ! [X: num] : ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(X),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(X) ).

% max_0_1(4)
tff(fact_3058_max__0__1_I4_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,X)),zero_zero_real) = aa_num_real(numeral_numeral_real,X) ).

% max_0_1(4)
tff(fact_3059_max__0__1_I4_J,axiom,
    ! [X: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,X)),zero_zero_rat) = aa_num_rat(numeral_numeral_rat,X) ).

% max_0_1(4)
tff(fact_3060_max__0__1_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,X)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,X) ).

% max_0_1(4)
tff(fact_3061_max__0__1_I4_J,axiom,
    ! [X: num] : aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,X)),zero_zero_int) = aa_num_int(numeral_numeral_int,X) ).

% max_0_1(4)
tff(fact_3062_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_3063_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_3064_max__0__1_I2_J,axiom,
    aa_int_int(ord_max_int(one_one_int),zero_zero_int) = one_one_int ).

% max_0_1(2)
tff(fact_3065_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_3066_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_3067_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_3068_max__0__1_I1_J,axiom,
    aa_int_int(ord_max_int(zero_zero_int),one_one_int) = one_one_int ).

% max_0_1(1)
tff(fact_3069_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_3070_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_3071_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_3072_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( some_nat(the_nat2(Option)) = Option ) ) ).

% option.collapse
tff(fact_3073_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_3074_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( some_num(the_num(Option)) = Option ) ) ).

% option.collapse
tff(fact_3075_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != 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,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self4
tff(fact_3076_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_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,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self4
tff(fact_3077_div__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C),A),B) = plus_p5714425477246183910nteger(C,divide6298287555418463151nteger(A,B)) ) ) ).

% div_mult_self4
tff(fact_3078_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != 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,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self3
tff(fact_3079_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_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,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self3
tff(fact_3080_div__mult__self3,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B),A),B) = plus_p5714425477246183910nteger(C,divide6298287555418463151nteger(A,B)) ) ) ).

% div_mult_self3
tff(fact_3081_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != 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,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self2
tff(fact_3082_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self2
tff(fact_3083_div__mult__self2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),B) = plus_p5714425477246183910nteger(C,divide6298287555418463151nteger(A,B)) ) ) ).

% div_mult_self2
tff(fact_3084_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != 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,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self1
tff(fact_3085_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self1
tff(fact_3086_div__mult__self1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),B) = plus_p5714425477246183910nteger(C,divide6298287555418463151nteger(A,B)) ) ) ).

% div_mult_self1
tff(fact_3087_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_3088_max__absorb2,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( ord_max_set_int(X,Y) = Y ) ) ).

% max_absorb2
tff(fact_3089_max__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( aa_rat_rat(ord_max_rat(X),Y) = Y ) ) ).

% max_absorb2
tff(fact_3090_max__absorb2,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_3091_max__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_3092_max__absorb2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( aa_int_int(ord_max_int(X),Y) = Y ) ) ).

% max_absorb2
tff(fact_3093_max__absorb1,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X))
     => ( ord_max_set_int(X,Y) = X ) ) ).

% max_absorb1
tff(fact_3094_max__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ( aa_rat_rat(ord_max_rat(X),Y) = X ) ) ).

% max_absorb1
tff(fact_3095_max__absorb1,axiom,
    ! [Y: num,X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y) = X ) ) ).

% max_absorb1
tff(fact_3096_max__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y) = X ) ) ).

% max_absorb1
tff(fact_3097_max__absorb1,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ( aa_int_int(ord_max_int(X),Y) = X ) ) ).

% max_absorb1
tff(fact_3098_max__def,axiom,
    ! [A: set_int,B: set_int] :
      ( ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
       => ( ord_max_set_int(A,B) = B ) )
      & ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
       => ( ord_max_set_int(A,B) = A ) ) ) ).

% max_def
tff(fact_3099_max__def,axiom,
    ! [A: rat,B: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => ( aa_rat_rat(ord_max_rat(A),B) = B ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ) ).

% max_def
tff(fact_3100_max__def,axiom,
    ! [A: num,B: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
       => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
       => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ) ).

% max_def
tff(fact_3101_max__def,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ) ).

% max_def
tff(fact_3102_max__def,axiom,
    ! [A: int,B: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => ( aa_int_int(ord_max_int(A),B) = B ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => ( aa_int_int(ord_max_int(A),B) = A ) ) ) ).

% max_def
tff(fact_3103_max__add__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),Z)) ).

% max_add_distrib_left
tff(fact_3104_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(ord_max_rat(X),Y)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),Z)) ).

% max_add_distrib_left
tff(fact_3105_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(ord_max_int(X),Y)),Z) = aa_int_int(ord_max_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z)) ).

% max_add_distrib_left
tff(fact_3106_max__add__distrib__left,axiom,
    ! [X: nat,Y: 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,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z)) ).

% max_add_distrib_left
tff(fact_3107_max__add__distrib__right,axiom,
    ! [X: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Z)) ).

% max_add_distrib_right
tff(fact_3108_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(ord_max_rat(Y),Z)) = aa_rat_rat(ord_max_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Z)) ).

% max_add_distrib_right
tff(fact_3109_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(ord_max_int(Y),Z)) = aa_int_int(ord_max_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z)) ).

% max_add_distrib_right
tff(fact_3110_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z)) ).

% max_add_distrib_right
tff(fact_3111_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q4: 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)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),Q4)) ).

% nat_add_max_left
tff(fact_3112_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q4: 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),Q4)) = 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),Q4)) ).

% nat_add_max_right
tff(fact_3113_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q4: 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)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4)) ).

% nat_mult_max_left
tff(fact_3114_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q4: 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),Q4)) = 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),Q4)) ).

% nat_mult_max_right
tff(fact_3115_option_Osel,axiom,
    ! [X22: nat] : the_nat2(some_nat(X22)) = X22 ).

% option.sel
tff(fact_3116_option_Osel,axiom,
    ! [X22: product_prod_nat_nat] : the_Pr8591224930841456533at_nat(some_P7363390416028606310at_nat(X22)) = X22 ).

% option.sel
tff(fact_3117_option_Osel,axiom,
    ! [X22: num] : the_num(some_num(X22)) = X22 ).

% option.sel
tff(fact_3118_option_Oexpand,axiom,
    ! [Option: option_nat,Option2: option_nat] :
      ( ( ( Option = none_nat )
      <=> ( Option2 = none_nat ) )
     => ( ( ( Option != none_nat )
         => ( ( Option2 != none_nat )
           => ( the_nat2(Option) = the_nat2(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_3119_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_3120_option_Oexpand,axiom,
    ! [Option: option_num,Option2: option_num] :
      ( ( ( Option = none_num )
      <=> ( Option2 = none_num ) )
     => ( ( ( Option != none_num )
         => ( ( Option2 != none_num )
           => ( the_num(Option) = the_num(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_3121_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option = some_nat(the_nat2(Option)) ) ) ).

% option.exhaust_sel
tff(fact_3122_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_3123_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option = some_num(the_num(Option)) ) ) ).

% option.exhaust_sel
tff(fact_3124_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_3125_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_3126_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_3127_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_3128_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_3129_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_3130_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_3131_div__add__self2,axiom,
    ! [B: nat,A: nat] :
      ( ( B != 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,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self2
tff(fact_3132_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),B)),one_one_int) ) ) ).

% div_add_self2
tff(fact_3133_div__add__self2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),B) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,B),one_one_Code_integer) ) ) ).

% div_add_self2
tff(fact_3134_div__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( B != 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,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self1
tff(fact_3135_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),B)),one_one_int) ) ) ).

% div_add_self1
tff(fact_3136_div__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(B,A),B) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,B),one_one_Code_integer) ) ) ).

% div_add_self1
tff(fact_3137_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_3138_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_3139_div__less__iff__less__mult,axiom,
    ! [Q4: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q4)),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),Q4))) ) ) ).

% div_less_iff_less_mult
tff(fact_3140_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_3141_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_3142_div__nat__eqI,axiom,
    ! [N: nat,Q4: 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),Q4)),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,Q4))))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = Q4 ) ) ) ).

% div_nat_eqI
tff(fact_3143_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q4: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q4))
     => ( 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),Q4)))
      <=> 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),Q4)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_3144_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_3145_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_3146_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_3147_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_3148_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_3149_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_3150_arith__geo__mean,axiom,
    ! [U: real,X: real,Y: 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,X),Y) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => 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,X),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ).

% arith_geo_mean
tff(fact_3151_arith__geo__mean,axiom,
    ! [U: rat,X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,U),aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ) ) ).

% arith_geo_mean
tff(fact_3152_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_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_3153_sum__squares__bound,axiom,
    ! [X: real,Y: 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))),X)),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_squares_bound
tff(fact_3154_sum__squares__bound,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_squares_bound
tff(fact_3155_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_3156_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))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(P,N2))
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct2
tff(fact_3157_set__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),A) = 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,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_3158_set__bit__0,axiom,
    ! [A: code_integer] : bit_se2793503036327961859nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_3159_set__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se7879613467334960850it_int(zero_zero_nat),A) = 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(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_3160_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(A1),A22))
    <=> ( ( ? [A6: bool,B6: bool] : A1 = vEBT_Leaf(A6,B6)
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X3),N3)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary2),N3))
            & ( 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))),N3) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3) )
            & ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_13))
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_13)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X3),N3)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N3)))
            & ( 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,N3)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) )
            & ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_13))
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_13)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),A22,TreeList3,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X3),N3)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary2),N3))
            & ( 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))),N3) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3) )
            & ! [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))),N3)))
               => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),X_13))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
                 => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),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))),A22)))
            & ( ( Mi2 != Ma2 )
             => ! [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))),N3)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N3) = I4 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N3))) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N3) = I4 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N3))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X3))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma2)) ) ) ) ) ) )
        | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),A22,TreeList3,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X3),N3)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N3)))
            & ( 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,N3)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) )
            & ! [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,N3))))
               => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),X_13))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
                 => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),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))),A22)))
            & ( ( Mi2 != Ma2 )
             => ! [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,N3))))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N3) = I4 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N3))) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N3) = I4 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N3))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X3))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma2)) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_3161_Leaf__0__not,axiom,
    ! [A: bool,B: bool] : ~ pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Leaf(A,B)),zero_zero_nat)) ).

% Leaf_0_not
tff(fact_3162_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( N = one_one_nat )
       => ? [A3: bool,B3: bool] : T = vEBT_Leaf(A3,B3) ) ) ).

% deg_1_Leafy
tff(fact_3163_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),one_one_nat))
     => ? [A3: bool,B3: bool] : T = vEBT_Leaf(A3,B3) ) ).

% deg_1_Leaf
tff(fact_3164_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),one_one_nat))
    <=> ? [A6: bool,B6: bool] : T = vEBT_Leaf(A6,B6) ) ).

% deg1Leaf
tff(fact_3165_mul__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(X)),some_nat(Y)) = some_nat(Z) ) ) ).

% mul_shift
tff(fact_3166_zle__add1__eq__le,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),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,W2),Z)) ) ).

% zle_add1_eq_le
tff(fact_3167_VEBT_Oinject_I2_J,axiom,
    ! [X21: bool,X222: bool,Y21: bool,Y222: bool] :
      ( ( vEBT_Leaf(X21,X222) = vEBT_Leaf(Y21,Y222) )
    <=> ( ( pp(X21)
        <=> pp(Y21) )
        & ( pp(X222)
        <=> pp(Y222) ) ) ) ).

% VEBT.inject(2)
tff(fact_3168_power__minus__is__div,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( 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,A),B)) = 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))),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% power_minus_is_div
tff(fact_3169_diff__self,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% diff_self
tff(fact_3170_diff__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% diff_self
tff(fact_3171_diff__self,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% diff_self
tff(fact_3172_diff__self,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).

% diff_self
tff(fact_3173_diff__0__right,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_0_right
tff(fact_3174_diff__0__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_0_right
tff(fact_3175_diff__0__right,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_0_right
tff(fact_3176_diff__0__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).

% diff_0_right
tff(fact_3177_zero__diff,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),A) = zero_zero_nat ).

% zero_diff
tff(fact_3178_diff__zero,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_zero
tff(fact_3179_diff__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_zero
tff(fact_3180_diff__zero,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_zero
tff(fact_3181_diff__zero,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),zero_zero_nat) = A ).

% diff_zero
tff(fact_3182_diff__zero,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).

% diff_zero
tff(fact_3183_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3184_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3185_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3186_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),A) = zero_zero_nat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3187_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3188_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3189_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3190_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3191_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3192_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).

% add_diff_cancel_right
tff(fact_3193_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(minus_minus_rat(A),B) ).

% add_diff_cancel_right
tff(fact_3194_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_right
tff(fact_3195_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_right
tff(fact_3196_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3197_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3198_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3199_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3200_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).

% add_diff_cancel_left
tff(fact_3201_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) = aa_rat_rat(minus_minus_rat(A),B) ).

% add_diff_cancel_left
tff(fact_3202_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_left
tff(fact_3203_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_left
tff(fact_3204_diff__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ).

% diff_add_cancel
tff(fact_3205_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(A),B)),B) = A ).

% diff_add_cancel
tff(fact_3206_diff__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ).

% diff_add_cancel
tff(fact_3207_add__diff__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).

% add_diff_cancel
tff(fact_3208_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B) = A ).

% add_diff_cancel
tff(fact_3209_add__diff__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).

% add_diff_cancel
tff(fact_3210_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_3211_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_3212_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_3213_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_3214_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_3215_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_3216_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se7879613467334960850it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% set_bit_negative_int_iff
tff(fact_3217_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_3218_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_3219_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: 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,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_3220_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_3221_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_3222_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: 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,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_3223_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3224_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3225_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3226_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3227_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3228_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3229_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3230_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3231_diff__add__zero,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = zero_zero_nat ).

% diff_add_zero
tff(fact_3232_diff__numeral__special_I9_J,axiom,
    aa_complex_complex(minus_minus_complex(one_one_complex),one_one_complex) = zero_zero_complex ).

% diff_numeral_special(9)
tff(fact_3233_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_3234_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_3235_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_3236_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_3237_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_3238_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_3239_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_3240_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_3241_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_3242_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_3243_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(divide_divide_int(K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_3244_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(divide_divide_int(K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_3245_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_3246_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_3247_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_3248_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_3249_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_3250_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: 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,zero_zero_int),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J))) ) ) ).

% zmult_zless_mono2
tff(fact_3251_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_3252_add1__zle__eq,axiom,
    ! [W2: 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,W2),one_one_int)),Z))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ).

% add1_zle_eq
tff(fact_3253_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_3254_zless__imp__add1__zle,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),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,W2),one_one_int)),Z)) ) ).

% zless_imp_add1_zle
tff(fact_3255_incr__mult__lemma,axiom,
    ! [D: int,P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [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),D))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X2: int] :
              ( pp(aa_int_bool(P,X2))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_3256_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_3257_zless__add1__eq,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),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,W2),Z))
        | ( W2 = Z ) ) ) ).

% zless_add1_eq
tff(fact_3258_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_3259_less__int__code_I1_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_zero_int)) ).

% less_int_code(1)
tff(fact_3260_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_3261_diff__right__commute,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) ).

% diff_right_commute
tff(fact_3262_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) ).

% diff_right_commute
tff(fact_3263_diff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),B) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) ).

% diff_right_commute
tff(fact_3264_diff__right__commute,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% diff_right_commute
tff(fact_3265_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_3266_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_3267_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_3268_VEBT_Osize_I4_J,axiom,
    ! [X21: bool,X222: bool] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf(X21,X222)) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_3269_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,D),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D))) ) ) ).

% diff_mono
tff(fact_3270_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,D),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D))) ) ) ).

% diff_mono
tff(fact_3271_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: 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_eq_int,D),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))) ) ) ).

% diff_mono
tff(fact_3272_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))) ) ).

% diff_left_mono
tff(fact_3273_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B))) ) ).

% diff_left_mono
tff(fact_3274_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))) ) ).

% diff_left_mono
tff(fact_3275_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C))) ) ).

% diff_right_mono
tff(fact_3276_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C))) ) ).

% diff_right_mono
tff(fact_3277_diff__right__mono,axiom,
    ! [A: int,B: int,C: 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_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C))) ) ).

% diff_right_mono
tff(fact_3278_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3279_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3280_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( 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_eq_int,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3281_eq__iff__diff__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = B )
    <=> ( aa_complex_complex(minus_minus_complex(A),B) = zero_zero_complex ) ) ).

% eq_iff_diff_eq_0
tff(fact_3282_eq__iff__diff__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_3283_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_rat(minus_minus_rat(A),B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_3284_eq__iff__diff__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_3285_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: bool,X222: bool] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf(X21,X222) ).

% VEBT.distinct(1)
tff(fact_3286_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : Y != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: bool,X223: bool] : Y != vEBT_Leaf(X212,X223) ) ).

% VEBT.exhaust
tff(fact_3287_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool,D4: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),D4)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,Deg3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary3),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_3288_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D))) ) ) ).

% diff_strict_mono
tff(fact_3289_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,D),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D))) ) ) ).

% diff_strict_mono
tff(fact_3290_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: 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,D),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))) ) ) ).

% diff_strict_mono
tff(fact_3291_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_3292_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_3293_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( 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,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_3294_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))) ) ).

% diff_strict_left_mono
tff(fact_3295_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B))) ) ).

% diff_strict_left_mono
tff(fact_3296_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))) ) ).

% diff_strict_left_mono
tff(fact_3297_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C))) ) ).

% diff_strict_right_mono
tff(fact_3298_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C))) ) ).

% diff_strict_right_mono
tff(fact_3299_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: 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,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C))) ) ).

% diff_strict_right_mono
tff(fact_3300_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% diff_diff_eq
tff(fact_3301_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% diff_diff_eq
tff(fact_3302_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% diff_diff_eq
tff(fact_3303_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% diff_diff_eq
tff(fact_3304_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) = A )
     => ( C = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ) ) ).

% add_implies_diff
tff(fact_3305_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) = A )
     => ( C = aa_rat_rat(minus_minus_rat(A),B) ) ) ).

% add_implies_diff
tff(fact_3306_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B) = A )
     => ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).

% add_implies_diff
tff(fact_3307_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) = A )
     => ( C = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ) ).

% add_implies_diff
tff(fact_3308_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_3309_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_3310_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_3311_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).

% diff_add_eq
tff(fact_3312_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ).

% diff_add_eq
tff(fact_3313_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_add_eq
tff(fact_3314_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).

% diff_diff_eq2
tff(fact_3315_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ).

% diff_diff_eq2
tff(fact_3316_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_diff_eq2
tff(fact_3317_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ).

% add_diff_eq
tff(fact_3318_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ).

% add_diff_eq
tff(fact_3319_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% add_diff_eq
tff(fact_3320_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_3321_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A = aa_rat_rat(minus_minus_rat(C),B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_3322_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_3323_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = C )
    <=> ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) ) ) ).

% diff_eq_eq
tff(fact_3324_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = C )
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) ) ) ).

% diff_eq_eq
tff(fact_3325_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = C )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) ) ) ).

% diff_eq_eq
tff(fact_3326_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_3327_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(minus_minus_rat(A2),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_3328_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_3329_zero__induct__lemma,axiom,
    ! [P: fun_nat_bool,K: nat,I: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(P,N2)) )
       => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I))) ) ) ).

% zero_induct_lemma
tff(fact_3330_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_3331_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_3332_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_3333_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_3334_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_3335_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_3336_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_3337_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_3338_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_3339_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),B)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ) ).

% le_diff_iff'
tff(fact_3340_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_3341_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_3342_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_3343_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_3344_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_3345_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_3346_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_3347_max__diff__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z)) ).

% max_diff_distrib_left
tff(fact_3348_max__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(ord_max_rat(X),Y)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(minus_minus_rat(X),Z)),aa_rat_rat(minus_minus_rat(Y),Z)) ).

% max_diff_distrib_left
tff(fact_3349_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(ord_max_int(X),Y)),Z) = aa_int_int(ord_max_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ).

% max_diff_distrib_left
tff(fact_3350_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: bool] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(Uu,fTrue))) ).

% VEBT_internal.minNull.simps(3)
tff(fact_3351_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: bool] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(fTrue,Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_3352_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(fFalse,fFalse))) ).

% VEBT_internal.minNull.simps(1)
tff(fact_3353_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,Uw: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Leaf(Uu,Uv)),Uw)) ).

% VEBT_internal.membermima.simps(1)
tff(fact_3354_le__iff__diff__le__0,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real)) ) ).

% le_iff_diff_le_0
tff(fact_3355_le__iff__diff__le__0,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat)) ) ).

% le_iff_diff_le_0
tff(fact_3356_le__iff__diff__le__0,axiom,
    ! [A: int,B: 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_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int)) ) ).

% le_iff_diff_le_0
tff(fact_3357_less__iff__diff__less__0,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real)) ) ).

% less_iff_diff_less_0
tff(fact_3358_less__iff__diff__less__0,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat)) ) ).

% less_iff_diff_less_0
tff(fact_3359_less__iff__diff__less__0,axiom,
    ! [A: int,B: 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,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int)) ) ).

% less_iff_diff_less_0
tff(fact_3360_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% diff_le_eq
tff(fact_3361_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),C))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))) ) ).

% diff_le_eq
tff(fact_3362_diff__le__eq,axiom,
    ! [A: int,B: int,C: 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,A),B)),C))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% diff_le_eq
tff(fact_3363_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C)) ) ).

% le_diff_eq
tff(fact_3364_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(minus_minus_rat(C),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C)) ) ).

% le_diff_eq
tff(fact_3365_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C)) ) ).

% le_diff_eq
tff(fact_3366_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),A) = B ) ) ).

% diff_add
tff(fact_3367_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A))) ) ).

% le_add_diff
tff(fact_3368_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_3369_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_3370_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_3371_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_3372_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_3373_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_3374_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_3375_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_3376_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_3377_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(aa_rat_fun_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(aa_rat_fun_rat_rat(plus_plus_rat,J),K)))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_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(aa_rat_fun_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_3378_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_3379_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_3380_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_3381_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(aa_rat_fun_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_3382_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_3383_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_3384_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C)) ) ).

% less_diff_eq
tff(fact_3385_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(minus_minus_rat(C),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C)) ) ).

% less_diff_eq
tff(fact_3386_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C)) ) ).

% less_diff_eq
tff(fact_3387_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% diff_less_eq
tff(fact_3388_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),C))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))) ) ).

% diff_less_eq
tff(fact_3389_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% diff_less_eq
tff(fact_3390_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3391_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3392_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3393_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3394_vebt__delete_Osimps_I3_J,axiom,
    ! [A: bool,B: bool,N: nat] : vEBT_vebt_delete(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,N))) = vEBT_Leaf(A,B) ).

% vebt_delete.simps(3)
tff(fact_3395_vebt__delete_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] : vEBT_vebt_delete(vEBT_Leaf(A,B),zero_zero_nat) = vEBT_Leaf(fFalse,B) ).

% vebt_delete.simps(1)
tff(fact_3396_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_3397_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_3398_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_3399_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_3400_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: 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_eq_nat,C),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C))) ) ) ).

% diff_less_mono
tff(fact_3401_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_3402_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_3403_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_3404_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_3405_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_3406_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_3407_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_3408_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_3409_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_3410_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_3411_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(1)
tff(fact_3412_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,D: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(vEBT_Leaf(Uu,Uv)),D))
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_3413_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_3414_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X4)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S4: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S4),X4) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_3415_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: 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,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),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,B),A)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3416_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(B),A)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3417_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: 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,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),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,B),A)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3418_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: 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,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D)))
    <=> 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,A),B)),E2)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3419_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(A),B)),E2)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3420_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: 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,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D)))
    <=> 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,A),B)),E2)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3421_less__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: 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,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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,B),A)),E2)),D))) ) ).

% less_add_iff2
tff(fact_3422_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(B),A)),E2)),D))) ) ).

% less_add_iff2
tff(fact_3423_less__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: 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,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),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,B),A)),E2)),D))) ) ).

% less_add_iff2
tff(fact_3424_less__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: 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,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D)))
    <=> 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,A),B)),E2)),C)),D)) ) ).

% less_add_iff1
tff(fact_3425_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(A),B)),E2)),C)),D)) ) ).

% less_add_iff1
tff(fact_3426_less__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: 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,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D)))
    <=> 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,A),B)),E2)),C)),D)) ) ).

% less_add_iff1
tff(fact_3427_invar__vebt_Ointros_I1_J,axiom,
    ! [A: bool,B: bool] : pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Leaf(A,B)),aa_nat_nat(suc,zero_zero_nat))) ).

% invar_vebt.intros(1)
tff(fact_3428_divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(X),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3429_divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3430_divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(X),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3431_diff__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(X),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3432_diff__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),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,X),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3433_diff__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(Y),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3434_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(W2),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Z)),aa_complex_complex(times_times_complex(W2),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3435_diff__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != 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,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),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,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3436_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W2),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3437_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = A ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_3438_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = A ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),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,A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_3439_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = A ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_3440_vebt__delete_Osimps_I2_J,axiom,
    ! [A: bool,B: bool] : vEBT_vebt_delete(vEBT_Leaf(A,B),aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf(A,fFalse) ).

% vebt_delete.simps(2)
tff(fact_3441_VEBT__internal_OminNull_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X != vEBT_Leaf(fFalse,fFalse) )
     => ( ! [Uv2: bool] : X != vEBT_Leaf(fTrue,Uv2)
       => ( ! [Uu2: bool] : X != vEBT_Leaf(Uu2,fTrue)
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_3442_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_3443_vebt__member_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,X: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Leaf(A,B)),X))
    <=> ( ( ( X = zero_zero_nat )
         => pp(A) )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => pp(B) )
            & ( X = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
tff(fact_3444_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(2)
tff(fact_3445_nat__diff__split,axiom,
    ! [P: fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
         => pp(aa_nat_bool(P,zero_zero_nat)) )
        & ! [D5: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D5) )
           => pp(aa_nat_bool(P,D5)) ) ) ) ).

% nat_diff_split
tff(fact_3446_nat__diff__split__asm,axiom,
    ! [P: fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)))
    <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
            & ~ pp(aa_nat_bool(P,zero_zero_nat)) )
          | ? [D5: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D5) )
              & ~ pp(aa_nat_bool(P,D5)) ) ) ) ).

% nat_diff_split_asm
tff(fact_3447_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_3448_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( ! [Uv2: bool] : X != vEBT_Leaf(fTrue,Uv2)
       => ( ! [Uu2: bool] : X != vEBT_Leaf(Uu2,fTrue)
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_3449_vebt__insert_Osimps_I1_J,axiom,
    ! [X: nat,A: bool,B: bool] :
      ( ( ( X = zero_zero_nat )
       => ( vEBT_vebt_insert(vEBT_Leaf(A,B),X) = vEBT_Leaf(fTrue,B) ) )
      & ( ( X != zero_zero_nat )
       => ( ( ( X = one_one_nat )
           => ( vEBT_vebt_insert(vEBT_Leaf(A,B),X) = vEBT_Leaf(A,fTrue) ) )
          & ( ( X != one_one_nat )
           => ( vEBT_vebt_insert(vEBT_Leaf(A,B),X) = vEBT_Leaf(A,B) ) ) ) ) ) ).

% vebt_insert.simps(1)
tff(fact_3450_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_3451_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_3452_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_3453_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_3454_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_3455_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_3456_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Leaf(A,B)),X))
    <=> ( ( ( X = zero_zero_nat )
         => pp(A) )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => pp(B) )
            & ( X = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_3457_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: bool,Uw: bool,N: nat] : vEBT_vebt_succ(vEBT_Leaf(Uv,Uw),aa_nat_nat(suc,N)) = none_nat ).

% vebt_succ.simps(2)
tff(fact_3458_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool] : vEBT_vebt_pred(vEBT_Leaf(Uu,Uv),zero_zero_nat) = none_nat ).

% vebt_pred.simps(1)
tff(fact_3459_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( ( X != vEBT_Leaf(fFalse,fFalse) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_3460_realpow__pos__nth2,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ? [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)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_3461_frac__le__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != 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,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),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,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real)) ) ) ) ).

% frac_le_eq
tff(fact_3462_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != 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(X),Y)),aa_rat_rat(divide_divide_rat(W2),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(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),zero_zero_rat)) ) ) ) ).

% frac_le_eq
tff(fact_3463_frac__less__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != 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,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),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,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real)) ) ) ) ).

% frac_less_eq
tff(fact_3464_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != 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(X),Y)),aa_rat_rat(divide_divide_rat(W2),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(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),zero_zero_rat)) ) ) ) ).

% frac_less_eq
tff(fact_3465_power__diff,axiom,
    ! [A: complex,N: nat,M: nat] :
      ( ( A != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N)) ) ) ) ).

% power_diff
tff(fact_3466_power__diff,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != 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,A),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,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ) ).

% power_diff
tff(fact_3467_power__diff,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) ) ) ) ).

% power_diff
tff(fact_3468_power__diff,axiom,
    ! [A: real,N: nat,M: nat] :
      ( ( A != 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,A),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,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ) ) ) ).

% power_diff
tff(fact_3469_power__diff,axiom,
    ! [A: rat,N: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_rat(power_power_rat(A),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(A),M)),aa_nat_rat(power_power_rat(A),N)) ) ) ) ).

% power_diff
tff(fact_3470_power__diff,axiom,
    ! [A: code_integer,N: nat,M: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ) ).

% power_diff
tff(fact_3471_vebt__mint_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A3: bool,B3: bool] : X != vEBT_Leaf(A3,B3)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
       => ~ ! [Mi3: nat,Ma3: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),Ux2,Uy2,Uz2) ) ) ).

% vebt_mint.cases
tff(fact_3472_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_3473_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_3474_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_3475_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_3476_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_3477_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_3478_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_3479_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: bool] :
      ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
      <=> pp(Y) )
     => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
         => ~ pp(Y) )
       => ( ( ? [Uv2: bool] : X = vEBT_Leaf(fTrue,Uv2)
           => pp(Y) )
         => ( ( ? [Uu2: bool] : X = vEBT_Leaf(Uu2,fTrue)
             => pp(Y) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
               => ~ pp(Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)
                 => pp(Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_3480_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S: 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),S))
         => 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))),S))),V)) ) ) ) ).

% scaling_mono
tff(fact_3481_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S: 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),S))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),aa_rat_rat(minus_minus_rat(V),U))),S))),V)) ) ) ) ).

% scaling_mono
tff(fact_3482_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_3483_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) != zero_zero_int )
     => ( aa_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_3484_power__diff__power__eq,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != 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,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),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,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),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,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_3485_power__diff__power__eq,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) = aa_nat_int(power_power_int(A),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(divide_divide_int(aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) = aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_3486_power__diff__power__eq,axiom,
    ! [A: code_integer,N: nat,M: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),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))
         => ( divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) = divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_3487_power__eq__if,axiom,
    ! [M: nat,P2: complex] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(P2),M) = one_one_complex ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(P2),M) = aa_complex_complex(times_times_complex(P2),aa_nat_complex(power_power_complex(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_3488_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_3489_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(aa_rat_fun_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_3490_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_3491_power__eq__if,axiom,
    ! [M: nat,P2: int] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_int(power_power_int(P2),M) = one_one_int ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_int(power_power_int(P2),M) = aa_int_int(aa_int_fun_int_int(times_times_int,P2),aa_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_3492_vebt__maxt_Osimps_I1_J,axiom,
    ! [B: bool,A: bool] :
      ( ( pp(B)
       => ( vEBT_vebt_maxt(vEBT_Leaf(A,B)) = some_nat(one_one_nat) ) )
      & ( ~ pp(B)
       => ( ( pp(A)
           => ( vEBT_vebt_maxt(vEBT_Leaf(A,B)) = some_nat(zero_zero_nat) ) )
          & ( ~ pp(A)
           => ( vEBT_vebt_maxt(vEBT_Leaf(A,B)) = none_nat ) ) ) ) ) ).

% vebt_maxt.simps(1)
tff(fact_3493_power__minus__mult,axiom,
    ! [N: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),N) ) ) ).

% power_minus_mult
tff(fact_3494_power__minus__mult,axiom,
    ! [N: nat,A: 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,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) ) ) ).

% power_minus_mult
tff(fact_3495_power__minus__mult,axiom,
    ! [N: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),N) ) ) ).

% power_minus_mult
tff(fact_3496_power__minus__mult,axiom,
    ! [N: nat,A: 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,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) ) ) ).

% power_minus_mult
tff(fact_3497_power__minus__mult,axiom,
    ! [N: nat,A: 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(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_int(power_power_int(A),N) ) ) ).

% power_minus_mult
tff(fact_3498_vebt__mint_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] :
      ( ( pp(A)
       => ( vEBT_vebt_mint(vEBT_Leaf(A,B)) = some_nat(zero_zero_nat) ) )
      & ( ~ pp(A)
       => ( ( pp(B)
           => ( vEBT_vebt_mint(vEBT_Leaf(A,B)) = some_nat(one_one_nat) ) )
          & ( ~ pp(B)
           => ( vEBT_vebt_mint(vEBT_Leaf(A,B)) = none_nat ) ) ) ) ) ).

% vebt_mint.simps(1)
tff(fact_3499_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_3500_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_3501_vebt__pred_Osimps_I2_J,axiom,
    ! [A: bool,Uw: bool] :
      ( ( pp(A)
       => ( vEBT_vebt_pred(vEBT_Leaf(A,Uw),aa_nat_nat(suc,zero_zero_nat)) = some_nat(zero_zero_nat) ) )
      & ( ~ pp(A)
       => ( vEBT_vebt_pred(vEBT_Leaf(A,Uw),aa_nat_nat(suc,zero_zero_nat)) = none_nat ) ) ) ).

% vebt_pred.simps(2)
tff(fact_3502_vebt__succ_Osimps_I1_J,axiom,
    ! [B: bool,Uu: bool] :
      ( ( pp(B)
       => ( vEBT_vebt_succ(vEBT_Leaf(Uu,B),zero_zero_nat) = some_nat(one_one_nat) ) )
      & ( ~ pp(B)
       => ( vEBT_vebt_succ(vEBT_Leaf(Uu,B),zero_zero_nat) = none_nat ) ) ) ).

% vebt_succ.simps(1)
tff(fact_3503_vebt__insert_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X4)
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S4),X4)
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S4),X4)
         => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary3),X4)
           => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3),X4) ) ) ) ) ).

% vebt_insert.cases
tff(fact_3504_vebt__pred_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),zero_zero_nat)
     => ( ! [A3: bool,Uw2: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,Uw2),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A3: bool,B3: bool,Va2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Vb2)
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2),Vf2)
             => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Vj2)
               => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3),X4) ) ) ) ) ) ) ).

% vebt_pred.cases
tff(fact_3505_vebt__succ_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,B3: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,B3),zero_zero_nat)
     => ( ! [Uv2: bool,Uw2: bool,N2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uv2,Uw2),aa_nat_nat(suc,N2))
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Va3)
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2),Ve2)
           => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Vi2)
             => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3),X4) ) ) ) ) ) ).

% vebt_succ.cases
tff(fact_3506_vebt__delete_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),zero_zero_nat)
     => ( ! [A3: bool,B3: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A3: bool,B3: bool,N2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),aa_nat_nat(suc,aa_nat_nat(suc,N2)))
         => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,Uu2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary3),Uu2)
           => ( ! [Mi3: nat,Ma3: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,TrLst2,Smry2),X4)
             => ( ! [Mi3: nat,Ma3: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2),X4)
               => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3),X4) ) ) ) ) ) ) ).

% vebt_delete.cases
tff(fact_3507_vebt__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X4)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),X4)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(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] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X4)
           => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3),X4) ) ) ) ) ).

% vebt_member.cases
tff(fact_3508_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool,Uw2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Uw2)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Uz2)
       => ( ! [Mi3: nat,Ma3: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,Va3,Vb2),X4)
         => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList2,Vc2),X4)
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2),X4) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_3509_vebt__pred_Osimps_I3_J,axiom,
    ! [B: bool,A: bool,Va: nat] :
      ( ( pp(B)
       => ( vEBT_vebt_pred(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = some_nat(one_one_nat) ) )
      & ( ~ pp(B)
       => ( ( pp(A)
           => ( vEBT_vebt_pred(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = some_nat(zero_zero_nat) ) )
          & ( ~ pp(A)
           => ( vEBT_vebt_pred(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
tff(fact_3510_realpow__pos__nth,axiom,
    ! [N: nat,A: 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),A))
       => ? [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) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_3511_realpow__pos__nth__unique,axiom,
    ! [N: nat,A: 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),A))
       => ? [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) = A )
            & ! [Y3: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),N) = A ) )
               => ( Y3 = X4 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_3512_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(X) = Y )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ~ ( ( pp(B3)
                 => ( Y = some_nat(one_one_nat) ) )
                & ( ~ pp(B3)
                 => ( ( pp(A3)
                     => ( Y = some_nat(zero_zero_nat) ) )
                    & ( ~ pp(A3)
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi3: nat,Ma3: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),Ux2,Uy2,Uz2)
               => ( Y != some_nat(Ma3) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_3513_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(X) = Y )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ~ ( ( pp(A3)
                 => ( Y = some_nat(zero_zero_nat) ) )
                & ( ~ pp(A3)
                 => ( ( pp(B3)
                     => ( Y = some_nat(one_one_nat) ) )
                    & ( ~ pp(B3)
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi3: nat] :
                ( ? [Ma3: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),Ux2,Uy2,Uz2)
               => ( Y != some_nat(Mi3) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_3514_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_3515_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(A1),A22))
     => ( ( ? [A3: bool,B3: bool] : A1 = vEBT_Leaf(A3,B3)
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat] :
              ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary3) )
             => ( ( A22 = Deg2 )
               => ( ! [X2: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                     => pp(aa_nat_bool(vEBT_invar_vebt(X2),N2)) )
                 => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary3),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 = N2 )
                       => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M5) )
                         => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_12))
                           => ~ ! [X2: vEBT_VEBT] :
                                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                                 => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_12)) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat] :
                ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary3) )
               => ( ( A22 = Deg2 )
                 => ( ! [X2: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                       => pp(aa_nat_bool(vEBT_invar_vebt(X2),N2)) )
                   => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary3),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,N2) )
                         => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M5) )
                           => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_12))
                             => ~ ! [X2: vEBT_VEBT] :
                                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                                   => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_12)) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat,Mi3: nat,Ma3: nat] :
                  ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),Deg2,TreeList2,Summary3) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X2: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                         => pp(aa_nat_bool(vEBT_invar_vebt(X2),N2)) )
                     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary3),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 = N2 )
                           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M5) )
                             => ( ! [I3: nat] :
                                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M5)))
                                   => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),X_13))
                                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I3)) ) )
                               => ( ( ( Mi3 = Ma3 )
                                   => ! [X2: vEBT_VEBT] :
                                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                                       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_12)) ) )
                                 => ( 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))),Deg2)))
                                     => ~ ( ( Mi3 != Ma3 )
                                         => ! [I3: nat] :
                                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),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,Ma3),N2) = I3 )
                                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N2))) )
                                                & ! [X2: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N2) = I3 )
                                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N2))) )
                                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X2))
                                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Ma3)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat,Mi3: nat,Ma3: nat] :
                    ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),Deg2,TreeList2,Summary3) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X2: vEBT_VEBT] :
                            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                           => pp(aa_nat_bool(vEBT_invar_vebt(X2),N2)) )
                       => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary3),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,N2) )
                             => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M5) )
                               => ( ! [I3: nat] :
                                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M5)))
                                     => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),X_13))
                                      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I3)) ) )
                                 => ( ( ( Mi3 = Ma3 )
                                     => ! [X2: vEBT_VEBT] :
                                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                                         => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_12)) ) )
                                   => ( 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))),Deg2)))
                                       => ~ ( ( Mi3 != Ma3 )
                                           => ! [I3: nat] :
                                                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),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,Ma3),N2) = I3 )
                                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N2))) )
                                                  & ! [X2: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N2) = I3 )
                                                        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N2))) )
                                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X2))
                                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Ma3)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_3516_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(divide_divide_int(aa_nat_int(power_power_int(K),M)),K) = aa_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_3517_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_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_3518_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = A ) ) ).

% max.absorb3
tff(fact_3519_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb3
tff(fact_3520_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb3
tff(fact_3521_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb3
tff(fact_3522_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( aa_int_int(ord_max_int(A),B) = A ) ) ).

% max.absorb3
tff(fact_3523_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = B ) ) ).

% max.absorb4
tff(fact_3524_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb4
tff(fact_3525_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb4
tff(fact_3526_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb4
tff(fact_3527_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( aa_int_int(ord_max_int(A),B) = B ) ) ).

% max.absorb4
tff(fact_3528_max__less__iff__conj,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)),Z))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_3529_max__less__iff__conj,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(ord_max_rat(X),Y)),Z))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_3530_max__less__iff__conj,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y)),Z))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_3531_max__less__iff__conj,axiom,
    ! [X: nat,Y: 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,X),Y)),Z))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_3532_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ord_max_int(X),Y)),Z))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_3533_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb1
tff(fact_3534_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb1
tff(fact_3535_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb1
tff(fact_3536_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( aa_int_int(ord_max_int(A),B) = A ) ) ).

% max.absorb1
tff(fact_3537_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb2
tff(fact_3538_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb2
tff(fact_3539_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb2
tff(fact_3540_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( aa_int_int(ord_max_int(A),B) = B ) ) ).

% max.absorb2
tff(fact_3541_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),A))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A)) ) ) ).

% max.bounded_iff
tff(fact_3542_max_Obounded__iff,axiom,
    ! [B: num,C: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A)) ) ) ).

% max.bounded_iff
tff(fact_3543_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: 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,B),C)),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).

% max.bounded_iff
tff(fact_3544_max_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ord_max_int(B),C)),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A)) ) ) ).

% max.bounded_iff
tff(fact_3545_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_3546_Diff__cancel,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A2,A2) = bot_bo2099793752762293965at_nat ).

% Diff_cancel
tff(fact_3547_Diff__cancel,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,A2) = bot_bot_set_real ).

% Diff_cancel
tff(fact_3548_Diff__cancel,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,A2) = bot_bot_set_int ).

% Diff_cancel
tff(fact_3549_Diff__cancel,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(A2,A2) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_3550_empty__Diff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(bot_bo2099793752762293965at_nat,A2) = bot_bo2099793752762293965at_nat ).

% empty_Diff
tff(fact_3551_empty__Diff,axiom,
    ! [A2: set_real] : minus_minus_set_real(bot_bot_set_real,A2) = bot_bot_set_real ).

% empty_Diff
tff(fact_3552_empty__Diff,axiom,
    ! [A2: set_int] : minus_minus_set_int(bot_bot_set_int,A2) = bot_bot_set_int ).

% empty_Diff
tff(fact_3553_empty__Diff,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(bot_bot_set_nat,A2) = bot_bot_set_nat ).

% empty_Diff
tff(fact_3554_Diff__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A2,bot_bo2099793752762293965at_nat) = A2 ).

% Diff_empty
tff(fact_3555_Diff__empty,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,bot_bot_set_real) = A2 ).

% Diff_empty
tff(fact_3556_Diff__empty,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,bot_bot_set_int) = A2 ).

% Diff_empty
tff(fact_3557_Diff__empty,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(A2,bot_bot_set_nat) = A2 ).

% Diff_empty
tff(fact_3558_finite__Diff2,axiom,
    ! [B2: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A2,B2)))
      <=> pp(aa_set_int_bool(finite_finite_int,A2)) ) ) ).

% finite_Diff2
tff(fact_3559_finite__Diff2,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B2)))
      <=> pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ) ).

% finite_Diff2
tff(fact_3560_finite__Diff2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
     => ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A2,B2)))
      <=> pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2)) ) ) ).

% finite_Diff2
tff(fact_3561_finite__Diff2,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A2,B2)))
      <=> pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ) ).

% finite_Diff2
tff(fact_3562_finite__Diff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A2,B2))) ) ).

% finite_Diff
tff(fact_3563_finite__Diff,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B2))) ) ).

% finite_Diff
tff(fact_3564_finite__Diff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
     => pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A2,B2))) ) ).

% finite_Diff
tff(fact_3565_finite__Diff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A2,B2))) ) ).

% finite_Diff
tff(fact_3566_Diff__eq__empty__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat(A2,B2) = bot_bo2099793752762293965at_nat )
    <=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B2)) ) ).

% Diff_eq_empty_iff
tff(fact_3567_Diff__eq__empty__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( minus_minus_set_real(A2,B2) = bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2)) ) ).

% Diff_eq_empty_iff
tff(fact_3568_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( minus_minus_set_nat(A2,B2) = bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2)) ) ).

% Diff_eq_empty_iff
tff(fact_3569_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( minus_minus_set_int(A2,B2) = bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ).

% Diff_eq_empty_iff
tff(fact_3570_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(I),set_or4548717258645045905et_nat(L,U)))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,L),I))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_3571_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_3572_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_3573_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_3574_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_3575_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( pp(aa_set_int_bool(member_int(I),aa_int_set_int(aa_int8639817059373041269et_int(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_3576_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_3577_Icc__eq__Icc,axiom,
    ! [L: set_int,H: set_int,L2: set_int,H2: set_int] :
      ( ( set_or370866239135849197et_int(L,H) = set_or370866239135849197et_int(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(L),H))
          & ~ pp(aa_set_int_bool(ord_less_eq_set_int(L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_3578_Icc__eq__Icc,axiom,
    ! [L: rat,H: rat,L2: rat,H2: rat] :
      ( ( set_or633870826150836451st_rat(L,H) = set_or633870826150836451st_rat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ 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,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_3579_Icc__eq__Icc,axiom,
    ! [L: num,H: num,L2: num,H2: num] :
      ( ( set_or7049704709247886629st_num(L,H) = set_or7049704709247886629st_num(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ 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,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_3580_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L2: nat,H2: nat] :
      ( ( set_or1269000886237332187st_nat(L,H) = set_or1269000886237332187st_nat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ 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,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_3581_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L2: int,H2: int] :
      ( ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),H) = aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L2),H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ 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,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_3582_Icc__eq__Icc,axiom,
    ! [L: real,H: real,L2: real,H2: real] :
      ( ( set_or1222579329274155063t_real(L,H) = set_or1222579329274155063t_real(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),H))
          & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_3583_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)))
    <=> ( X = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_3584_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,set_or1269000886237332187st_nat(L,U))) ).

% finite_atLeastAtMost
tff(fact_3585_atLeastatMost__empty__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( set_or370866239135849197et_int(A,B) = bot_bot_set_set_int )
    <=> ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_3586_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat )
    <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_3587_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num )
    <=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_3588_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( set_or1269000886237332187st_nat(A,B) = bot_bot_set_nat )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_3589_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B) = bot_bot_set_int )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_3590_atLeastatMost__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_3591_atLeastatMost__empty__iff2,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_set_int = set_or370866239135849197et_int(A,B) )
    <=> ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_3592_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat = set_or633870826150836451st_rat(A,B) )
    <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_3593_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A,B) )
    <=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_3594_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = set_or1269000886237332187st_nat(A,B) )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_3595_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B) )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_3596_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(A,B) )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_3597_atLeastatMost__subset__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(A,B),set_or370866239135849197et_int(C,D))
    <=> ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
        | ( pp(aa_set_int_bool(ord_less_eq_set_int(C),A))
          & pp(aa_set_int_bool(ord_less_eq_set_int(B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3598_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C,D)))
    <=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3599_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C,D)))
    <=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        | ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
          & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3600_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_or1269000886237332187st_nat(A,B)),set_or1269000886237332187st_nat(C,D)))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3601_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B)),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,C),D)))
    <=> ( ~ 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_eq_int,C),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3602_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C,D)))
    <=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3603_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_3604_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_3605_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( set_or1269000886237332187st_nat(A,B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_3606_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_3607_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_3608_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_set_rat_bool(finite_finite_rat,set_or633870826150836451st_rat(A,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% infinite_Icc_iff
tff(fact_3609_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,set_or1222579329274155063t_real(A,B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% infinite_Icc_iff
tff(fact_3610_zle__diff1__eq,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),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,W2),Z)) ) ).

% zle_diff1_eq
tff(fact_3611_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N2))) ) ).

% real_arch_pow
tff(fact_3612_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N2)),Y)) ) ) ).

% real_arch_pow_inv
tff(fact_3613_less__eq__real__def,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | ( X = Y ) ) ) ).

% less_eq_real_def
tff(fact_3614_Diff__infinite__finite,axiom,
    ! [T2: set_int,S2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
       => ~ pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(S2,T2))) ) ) ).

% Diff_infinite_finite
tff(fact_3615_Diff__infinite__finite,axiom,
    ! [T2: set_complex,S2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(S2,T2))) ) ) ).

% Diff_infinite_finite
tff(fact_3616_Diff__infinite__finite,axiom,
    ! [T2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,T2))
     => ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,S2))
       => ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(S2,T2))) ) ) ).

% Diff_infinite_finite
tff(fact_3617_Diff__infinite__finite,axiom,
    ! [T2: set_nat,S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
       => ~ pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(S2,T2))) ) ) ).

% Diff_infinite_finite
tff(fact_3618_double__diff,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),C2))
       => ( minus_minus_set_nat(B2,minus_minus_set_nat(C2,A2)) = A2 ) ) ) ).

% double_diff
tff(fact_3619_double__diff,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))
       => ( minus_minus_set_int(B2,minus_minus_set_int(C2,A2)) = A2 ) ) ) ).

% double_diff
tff(fact_3620_Diff__subset,axiom,
    ! [A2: set_nat,B2: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,minus_minus_set_nat(A2,B2)),A2)) ).

% Diff_subset
tff(fact_3621_Diff__subset,axiom,
    ! [A2: set_int,B2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(minus_minus_set_int(A2,B2)),A2)) ).

% Diff_subset
tff(fact_3622_Diff__mono,axiom,
    ! [A2: set_nat,C2: set_nat,D6: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),C2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,D6),B2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,minus_minus_set_nat(A2,B2)),minus_minus_set_nat(C2,D6))) ) ) ).

% Diff_mono
tff(fact_3623_Diff__mono,axiom,
    ! [A2: set_int,C2: set_int,D6: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(D6),B2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(minus_minus_set_int(A2,B2)),minus_minus_set_int(C2,D6))) ) ) ).

% Diff_mono
tff(fact_3624_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_3625_psubset__imp__ex__mem,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A2),B2))
     => ? [B3: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B3),minus_1356011639430497352at_nat(B2,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_3626_psubset__imp__ex__mem,axiom,
    ! [A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A2),B2))
     => ? [B3: real] : pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B2,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_3627_psubset__imp__ex__mem,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A2),B2))
     => ? [B3: set_nat] : pp(aa_set_set_nat_bool(member_set_nat(B3),minus_2163939370556025621et_nat(B2,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_3628_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
     => ? [B3: int] : pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B2,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_3629_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B2))
     => ? [B3: nat] : pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(B2,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_3630_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ~ pp(aa_set_rat_bool(finite_finite_rat,set_or633870826150836451st_rat(A,B))) ) ).

% infinite_Icc
tff(fact_3631_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ~ pp(aa_set_real_bool(finite_finite_real,set_or1222579329274155063t_real(A,B))) ) ).

% infinite_Icc
tff(fact_3632_ex__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N))
          & pp(aa_nat_bool(P,M3)) )
    <=> ? [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_3633_all__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N))
         => pp(aa_nat_bool(P,M3)) )
    <=> ! [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_3634_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,N4),set_or1269000886237332187st_nat(zero_zero_nat,N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N4)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_3635_decr__mult__lemma,axiom,
    ! [D: int,P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [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),D))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X2: int] :
              ( pp(aa_int_bool(P,X2))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_3636_plusinfinity,axiom,
    ! [D: int,P3: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),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)))) )
       => ( ? [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_12: int] : pp(aa_int_bool(P3,X_12))
           => ? [X_1: int] : pp(aa_int_bool(P,X_1)) ) ) ) ) ).

% plusinfinity
tff(fact_3637_minusinfinity,axiom,
    ! [D: int,P1: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [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),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(P1,X4)) ) )
         => ( ? [X_12: int] : pp(aa_int_bool(P1,X_12))
           => ? [X_1: int] : pp(aa_int_bool(P,X_1)) ) ) ) ) ).

% minusinfinity
tff(fact_3638_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(divide_divide_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_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_3639_atLeastatMost__psubset__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ord_less_set_set_int(set_or370866239135849197et_int(A,B),set_or370866239135849197et_int(C,D))
    <=> ( ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
          | ( pp(aa_set_int_bool(ord_less_eq_set_int(C),A))
            & pp(aa_set_int_bool(ord_less_eq_set_int(B),D))
            & ( pp(aa_set_int_bool(ord_less_set_int(C),A))
              | pp(aa_set_int_bool(ord_less_set_int(B),D)) ) ) )
        & pp(aa_set_int_bool(ord_less_eq_set_int(C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3640_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_set_rat_bool(ord_less_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C,D)))
    <=> ( ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
          | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),D))
            & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A))
              | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),D)) ) ) )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3641_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( pp(aa_set_num_bool(ord_less_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C,D)))
    <=> ( ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
          | ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),D))
            & ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A))
              | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),D)) ) ) )
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3642_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,set_or1269000886237332187st_nat(A,B)),set_or1269000886237332187st_nat(C,D)))
    <=> ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),D))
            & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A))
              | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),D)) ) ) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3643_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B)),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,C),D)))
    <=> ( ( ~ 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_eq_int,C),A))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),D))
            & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A))
              | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),D)) ) ) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3644_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C,D)))
    <=> ( ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
          | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),D))
            & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A))
              | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),D)) ) ) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3645_max_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,D),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(C),D)),aa_rat_rat(ord_max_rat(A),B))) ) ) ).

% max.mono
tff(fact_3646_max_Omono,axiom,
    ! [C: num,A: num,D: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,D),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,C),D)),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ) ).

% max.mono
tff(fact_3647_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C),D)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ) ).

% max.mono
tff(fact_3648_max_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ord_max_int(C),D)),aa_int_int(ord_max_int(A),B))) ) ) ).

% max.mono
tff(fact_3649_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( A = aa_rat_rat(ord_max_rat(A),B) ) ) ).

% max.orderE
tff(fact_3650_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).

% max.orderE
tff(fact_3651_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.orderE
tff(fact_3652_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( A = aa_int_int(ord_max_int(A),B) ) ) ).

% max.orderE
tff(fact_3653_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(ord_max_rat(A),B) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% max.orderI
tff(fact_3654_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) )
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ).

% max.orderI
tff(fact_3655_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).

% max.orderI
tff(fact_3656_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(ord_max_int(A),B) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% max.orderI
tff(fact_3657_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),A))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A)) ) ) ).

% max.boundedE
tff(fact_3658_max_OboundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A))
     => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A)) ) ) ).

% max.boundedE
tff(fact_3659_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: 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,B),C)),A))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).

% max.boundedE
tff(fact_3660_max_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ord_max_int(B),C)),A))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A)) ) ) ).

% max.boundedE
tff(fact_3661_max_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),A)) ) ) ).

% max.boundedI
tff(fact_3662_max_OboundedI,axiom,
    ! [B: num,A: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)) ) ) ).

% max.boundedI
tff(fact_3663_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)) ) ) ).

% max.boundedI
tff(fact_3664_max_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ord_max_int(B),C)),A)) ) ) ).

% max.boundedI
tff(fact_3665_max_Oorder__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
    <=> ( A = aa_rat_rat(ord_max_rat(A),B) ) ) ).

% max.order_iff
tff(fact_3666_max_Oorder__iff,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
    <=> ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).

% max.order_iff
tff(fact_3667_max_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.order_iff
tff(fact_3668_max_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
    <=> ( A = aa_int_int(ord_max_int(A),B) ) ) ).

% max.order_iff
tff(fact_3669_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(ord_max_rat(A),B))) ).

% max.cobounded1
tff(fact_3670_max_Ocobounded1,axiom,
    ! [A: num,B: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ).

% max.cobounded1
tff(fact_3671_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ).

% max.cobounded1
tff(fact_3672_max_Ocobounded1,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(ord_max_int(A),B))) ).

% max.cobounded1
tff(fact_3673_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(ord_max_rat(A),B))) ).

% max.cobounded2
tff(fact_3674_max_Ocobounded2,axiom,
    ! [B: num,A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ).

% max.cobounded2
tff(fact_3675_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ).

% max.cobounded2
tff(fact_3676_max_Ocobounded2,axiom,
    ! [B: int,A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(ord_max_int(A),B))) ).

% max.cobounded2
tff(fact_3677_le__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),aa_rat_rat(ord_max_rat(X),Y)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_3678_le__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_3679_le__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: 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,X),Y)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_3680_le__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_int_int(ord_max_int(X),Y)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_3681_max_Oabsorb__iff1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
    <=> ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_3682_max_Oabsorb__iff1,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_3683_max_Oabsorb__iff1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_3684_max_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
    <=> ( aa_int_int(ord_max_int(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_3685_max_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_3686_max_Oabsorb__iff2,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_3687_max_Oabsorb__iff2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_3688_max_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
    <=> ( aa_int_int(ord_max_int(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_3689_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(ord_max_rat(A),B))) ) ).

% max.coboundedI1
tff(fact_3690_max_OcoboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).

% max.coboundedI1
tff(fact_3691_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.coboundedI1
tff(fact_3692_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(ord_max_int(A),B))) ) ).

% max.coboundedI1
tff(fact_3693_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(ord_max_rat(A),B))) ) ).

% max.coboundedI2
tff(fact_3694_max_OcoboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),B))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).

% max.coboundedI2
tff(fact_3695_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.coboundedI2
tff(fact_3696_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(ord_max_int(A),B))) ) ).

% max.coboundedI2
tff(fact_3697_less__max__iff__disj,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_3698_less__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),aa_rat_rat(ord_max_rat(X),Y)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_3699_less__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),X))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_3700_less__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: 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,X),Y)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_3701_less__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_int_int(ord_max_int(X),Y)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_3702_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,B),C)),A))
     => ~ ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
         => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_3703_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(ord_max_rat(B),C)),A))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_3704_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A))
     => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_3705_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_3706_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ord_max_int(B),C)),A))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_3707_max_Ostrict__order__iff,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
    <=> ( ( A = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_3708_max_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
    <=> ( ( A = aa_rat_rat(ord_max_rat(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_3709_max_Ostrict__order__iff,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
    <=> ( ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_3710_max_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_3711_max_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
    <=> ( ( A = aa_int_int(ord_max_int(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_3712_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_3713_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(ord_max_rat(A),B))) ) ).

% max.strict_coboundedI1
tff(fact_3714_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_3715_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_3716_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(ord_max_int(A),B))) ) ).

% max.strict_coboundedI1
tff(fact_3717_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_3718_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(ord_max_rat(A),B))) ) ).

% max.strict_coboundedI2
tff(fact_3719_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_3720_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_3721_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(ord_max_int(A),B))) ) ).

% max.strict_coboundedI2
tff(fact_3722_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q6: int,R4: int,Q4: 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,B),Q6)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),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,B),R2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q4),Q6)) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_3723_unique__quotient__lemma,axiom,
    ! [B: int,Q6: int,R4: int,Q4: 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,B),Q6)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),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),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q6),Q4)) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_3724_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q4: int,R2: int,B5: 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,B),Q4)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),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,B5),Q6)),R4)),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
         => ( 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),B5))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q6),Q4)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_3725_zdiv__mono2__lemma,axiom,
    ! [B: int,Q4: int,R2: int,B5: 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,B),Q4)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),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,B5),Q6)),R4)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B5))
         => ( 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),B5))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q4),Q6)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_3726_q__pos__lemma,axiom,
    ! [B5: 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,B5),Q6)),R4)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B5))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Q6)) ) ) ) ).

% q_pos_lemma
tff(fact_3727_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),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),B))
         => ( aa_int_int(divide_divide_int(A),B) = Q4 ) ) ) ) ).

% int_div_pos_eq
tff(fact_3728_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),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,B),R2))
         => ( aa_int_int(divide_divide_int(A),B) = Q4 ) ) ) ) ).

% int_div_neg_eq
tff(fact_3729_split__zdiv,axiom,
    ! [P: fun_int_bool,N: int,K: int] :
      ( pp(aa_int_bool(P,aa_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_3730_zdiv__mono1,axiom,
    ! [A: int,A5: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(A5),B))) ) ) ).

% zdiv_mono1
tff(fact_3731_zdiv__mono2,axiom,
    ! [A: int,B5: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B5))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(A),B5))) ) ) ) ).

% zdiv_mono2
tff(fact_3732_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_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_3733_zdiv__mono1__neg,axiom,
    ! [A: int,A5: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A5),B)),aa_int_int(divide_divide_int(A),B))) ) ) ).

% zdiv_mono1_neg
tff(fact_3734_zdiv__mono2__neg,axiom,
    ! [A: int,B5: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B5))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B5)),aa_int_int(divide_divide_int(A),B))) ) ) ) ).

% zdiv_mono2_neg
tff(fact_3735_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(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_3736_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(divide_divide_int(K),L))) ) ) ).

% div_positive_int
tff(fact_3737_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_3738_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_3739_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(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_3740_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_3741_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_3742_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(A),B)))
      <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_3743_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_3744_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_3745_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
     => ( 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(divide_divide_int(X),K)),X)) ) ) ).

% int_div_less_self
tff(fact_3746_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int)) ) ) ).

% div_neg_pos_less0
tff(fact_3747_div__positive,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),divide6298287555418463151nteger(A,B))) ) ) ).

% div_positive
tff(fact_3748_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
       => 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,A),B))) ) ) ).

% div_positive
tff(fact_3749_div__positive,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(A),B))) ) ) ).

% div_positive
tff(fact_3750_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
       => ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3751_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3752_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
       => ( aa_int_int(divide_divide_int(A),B) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3753_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3754_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3755_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3756_discrete,axiom,
    ! [A: nat,B: 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_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B)) ) ).

% discrete
tff(fact_3757_discrete,axiom,
    ! [A: int,B: 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_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B)) ) ).

% discrete
tff(fact_3758_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q4: 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(Q4),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))),Q4)),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(Q4),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))),Q4)),R2) ) ) ) ).

% divmod_step_eq
tff(fact_3759_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q4: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2))
       => ( aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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))),Q4)),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))
       => ( aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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))),Q4)),R2) ) ) ) ).

% divmod_step_eq
tff(fact_3760_divmod__step__eq,axiom,
    ! [L: num,R2: code_integer,Q4: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R2))
       => ( unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q4,R2)) = produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q4),one_one_Code_integer),minus_8373710615458151222nteger(R2,numera6620942414471956472nteger(L))) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R2))
       => ( unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q4,R2)) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q4),R2) ) ) ) ).

% divmod_step_eq
tff(fact_3761_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_3762_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_3763_unset__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),A) = 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,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_3764_unset__bit__0,axiom,
    ! [A: code_integer] : bit_se8260200283734997820nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_3765_unset__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se4203085406695923979it_int(zero_zero_nat),A) = 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(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_3766_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X: 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),X))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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,X),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,X),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(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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_3767_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X: 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),X))
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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,X),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(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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,X),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,X),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(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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_3768_pred__lesseq__max,axiom,
    ! [Deg: nat,X: 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,X),Ma))
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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,X),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(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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,X),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),X),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,X),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(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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_3769_pred__less__length__list,axiom,
    ! [Deg: nat,X: 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,X),Ma))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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,X),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),X),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,X),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(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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_3770_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_set_vebt(T) = collect_nat(vEBT_vebt_member(T)) ).

% set_vebt'_def
tff(fact_3771_Diff__idemp,axiom,
    ! [A2: set_nat,B2: set_nat] : minus_minus_set_nat(minus_minus_set_nat(A2,B2),B2) = minus_minus_set_nat(A2,B2) ).

% Diff_idemp
tff(fact_3772_Diff__iff,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),minus_1356011639430497352at_nat(A2,B2)))
    <=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A2))
        & ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B2)) ) ) ).

% Diff_iff
tff(fact_3773_Diff__iff,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A2,B2)))
    <=> ( pp(aa_set_real_bool(member_real(C),A2))
        & ~ pp(aa_set_real_bool(member_real(C),B2)) ) ) ).

% Diff_iff
tff(fact_3774_Diff__iff,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),minus_2163939370556025621et_nat(A2,B2)))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(C),A2))
        & ~ pp(aa_set_set_nat_bool(member_set_nat(C),B2)) ) ) ).

% Diff_iff
tff(fact_3775_Diff__iff,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),minus_minus_set_int(A2,B2)))
    <=> ( pp(aa_set_int_bool(member_int(C),A2))
        & ~ pp(aa_set_int_bool(member_int(C),B2)) ) ) ).

% Diff_iff
tff(fact_3776_Diff__iff,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),minus_minus_set_nat(A2,B2)))
    <=> ( pp(aa_set_nat_bool(member_nat(C),A2))
        & ~ pp(aa_set_nat_bool(member_nat(C),B2)) ) ) ).

% Diff_iff
tff(fact_3777_DiffI,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A2))
     => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B2))
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),minus_1356011639430497352at_nat(A2,B2))) ) ) ).

% DiffI
tff(fact_3778_DiffI,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),A2))
     => ( ~ pp(aa_set_real_bool(member_real(C),B2))
       => pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A2,B2))) ) ) ).

% DiffI
tff(fact_3779_DiffI,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),A2))
     => ( ~ pp(aa_set_set_nat_bool(member_set_nat(C),B2))
       => pp(aa_set_set_nat_bool(member_set_nat(C),minus_2163939370556025621et_nat(A2,B2))) ) ) ).

% DiffI
tff(fact_3780_DiffI,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),A2))
     => ( ~ pp(aa_set_int_bool(member_int(C),B2))
       => pp(aa_set_int_bool(member_int(C),minus_minus_set_int(A2,B2))) ) ) ).

% DiffI
tff(fact_3781_DiffI,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(C),B2))
       => pp(aa_set_nat_bool(member_nat(C),minus_minus_set_nat(A2,B2))) ) ) ).

% DiffI
tff(fact_3782_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_3783_finite__Collect__conjI,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( ( pp(aa_set_real_bool(finite_finite_real,collect_real(P)))
        | pp(aa_set_real_bool(finite_finite_real,collect_real(Q))) )
     => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_af(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_3784_finite__Collect__conjI,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ( finite8100373058378681591st_nat(collect_list_nat(P))
        | finite8100373058378681591st_nat(collect_list_nat(Q)) )
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_ag(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_3785_finite__Collect__conjI,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(P)))
        | pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(Q))) )
     => pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_ah(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_3786_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_ai(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_3787_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_aj(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_3788_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_ak(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_3789_finite__Collect__conjI,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,Q: fun_Pr7757163693017731481t_bool] :
      ( ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(P)))
        | pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Q))) )
     => pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun830836278467416944t_bool(aTP_Lamm_al(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_3790_finite__Collect__disjI,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_am(P),Q))))
    <=> ( pp(aa_set_real_bool(finite_finite_real,collect_real(P)))
        & pp(aa_set_real_bool(finite_finite_real,collect_real(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_3791_finite__Collect__disjI,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( finite8100373058378681591st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_an(P),Q)))
    <=> ( finite8100373058378681591st_nat(collect_list_nat(P))
        & finite8100373058378681591st_nat(collect_list_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_3792_finite__Collect__disjI,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_ao(P),Q))))
    <=> ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(P)))
        & pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_3793_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_ap(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_3794_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_aq(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_3795_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_ar(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_3796_finite__Collect__disjI,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,Q: fun_Pr7757163693017731481t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun830836278467416944t_bool(aTP_Lamm_as(P),Q))))
    <=> ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(P)))
        & pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_3797_finite__interval__int1,axiom,
    ! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_at(A),B)))) ).

% finite_interval_int1
tff(fact_3798_finite__interval__int4,axiom,
    ! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_au(A),B)))) ).

% finite_interval_int4
tff(fact_3799_pred__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_pred(T,X) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_av(T),X)) = bot_bot_set_nat ) ) ) ).

% pred_empty
tff(fact_3800_succ__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_succ(T,X) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_aw(T),X)) = bot_bot_set_nat ) ) ) ).

% succ_empty
tff(fact_3801_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se4203085406695923979it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% unset_bit_negative_int_iff
tff(fact_3802_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_ax,A2)))) ) ).

% finite_Collect_subsets
tff(fact_3803_finite__Collect__subsets,axiom,
    ! [A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => finite6551019134538273531omplex(collect_set_complex(aTP_Lamm_ay(A2))) ) ).

% finite_Collect_subsets
tff(fact_3804_finite__Collect__subsets,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
     => finite9047747110432174090at_nat(collec5514110066124741708at_nat(aTP_Lamm_az(A2))) ) ).

% finite_Collect_subsets
tff(fact_3805_finite__Collect__subsets,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => finite6197958912794628473et_int(collect_set_int(aTP_Lamm_ba(A2))) ) ).

% finite_Collect_subsets
tff(fact_3806_finite__interval__int2,axiom,
    ! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_bb(A),B)))) ).

% finite_interval_int2
tff(fact_3807_finite__interval__int3,axiom,
    ! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_bc(A),B)))) ).

% finite_interval_int3
tff(fact_3808_finite__Collect__less__nat,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bd,K)))) ).

% finite_Collect_less_nat
tff(fact_3809_finite__Collect__le__nat,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_be,K)))) ).

% finite_Collect_le_nat
tff(fact_3810_del__x__not__mi,axiom,
    ! [Mi: nat,X: 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),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),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,X),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,X),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 = vEBT_vebt_delete(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,X),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))
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(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_nat2(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),the_nat2(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat2(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))))))),Ma))),Deg,Newlist,vEBT_vebt_delete(Summary,H)) ) )
                      & ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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,X),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_nat2(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H))))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_3811_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X: 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),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),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,X),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,X),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 = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
               => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                 => ( ( Sn = 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,X),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_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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,X),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_nat2(vEBT_vebt_maxt(Sn)))),the_nat2(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat2(vEBT_vebt_maxt(Sn)))))))),Ma))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_3812_del__x__not__mia,axiom,
    ! [Mi: nat,X: 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),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),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,X),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,X),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,X),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_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),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,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(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_nat2(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),the_nat2(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),the_nat2(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),vEBT_vebt_delete(Summary,H)),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,X),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_nat2(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),H))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),Summary)) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_3813_del__in__range,axiom,
    ! [Mi: nat,X: 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),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),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))
         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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_nat2(vEBT_vebt_maxt(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),the_nat2(vEBT_vebt_maxt(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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_delete(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat2(vEBT_vebt_maxt(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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_delete(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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_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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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_nat2(vEBT_vebt_maxt(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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_delete(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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_delete(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)) ) ) ) ) ).

% del_in_range
tff(fact_3814_del__x__mi,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat] :
      ( ( ( X = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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)))
                 => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),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(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_nat2(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),the_nat2(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),the_nat2(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),vEBT_vebt_delete(Summary,H)),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_nat2(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),H))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),Summary)) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_3815_del__x__mi__lets__in,axiom,
    ! [X: 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] :
      ( ( ( X = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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 = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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(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_nat2(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),the_nat2(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat2(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))))))),Ma))),Deg,Newlist,vEBT_vebt_delete(Summary,H)) ) )
                        & ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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_nat2(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H))))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_3816_del__x__mi__lets__in__minNull,axiom,
    ! [X: 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] :
      ( ( ( X = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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 = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( ( Sn = vEBT_vebt_delete(Summary,H) )
                         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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_nat2(vEBT_vebt_maxt(Sn)))),the_nat2(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat2(vEBT_vebt_maxt(Sn)))))))),Ma))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_3817_del__x__mia,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),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))
         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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))))),size_s6755466524823107622T_VEBT(TreeList)),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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)))))),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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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))))))),none_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),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_nat2(vEBT_vebt_maxt(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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_nat2(vEBT_vebt_maxt(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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)))),vEBT_vebt_delete(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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_nat2(vEBT_vebt_maxt(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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)))),vEBT_vebt_delete(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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)))))),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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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)))))),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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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_nat2(vEBT_vebt_maxt(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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)))),vEBT_vebt_delete(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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)))),vEBT_vebt_delete(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(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)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)) ) ) ) ) ).

% del_x_mia
tff(fact_3818_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_3819_not__iless0,axiom,
    ! [N: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(N),zero_z5237406670263579293d_enat)) ).

% not_iless0
tff(fact_3820_enat__less__induct,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,N: extended_enat] :
      ( ! [N2: extended_enat] :
          ( ! [M2: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(M2),N2))
             => pp(aa_Ext2772979874139974381t_bool(P,M2)) )
         => pp(aa_Ext2772979874139974381t_bool(P,N2)) )
     => pp(aa_Ext2772979874139974381t_bool(P,N)) ) ).

% enat_less_induct
tff(fact_3821_less__set__def,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A2),B2))
    <=> pp(aa_fun8663595511526100224l_bool(ord_le7158584875777748973t_bool(aTP_Lamm_a(A2)),aTP_Lamm_a(B2))) ) ).

% less_set_def
tff(fact_3822_less__set__def,axiom,
    ! [A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A2),B2))
    <=> ord_le7519368973566061808l_bool(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2)) ) ).

% less_set_def
tff(fact_3823_less__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A2),B2))
    <=> ord_le4523237222650602006t_bool(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2)) ) ).

% less_set_def
tff(fact_3824_less__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B2))
    <=> ord_le6841235830588351564t_bool(aa_set3744802623701456604t_bool(aTP_Lamm_ad,A2),aa_set3744802623701456604t_bool(aTP_Lamm_ad,B2)) ) ).

% less_set_def
tff(fact_3825_less__set__def,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
    <=> ord_le3716944864119727344t_bool(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2)) ) ).

% less_set_def
tff(fact_3826_minus__set__def,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A2,B2) = collec3392354462482085612at_nat(minus_5597117856863188850t_bool(aTP_Lamm_a(A2),aTP_Lamm_a(B2))) ).

% minus_set_def
tff(fact_3827_minus__set__def,axiom,
    ! [A2: set_real,B2: set_real] : minus_minus_set_real(A2,B2) = collect_real(minus_6582619742941512235l_bool(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2))) ).

% minus_set_def
tff(fact_3828_minus__set__def,axiom,
    ! [A2: set_list_nat,B2: set_list_nat] : minus_7954133019191499631st_nat(A2,B2) = collect_list_nat(minus_8064998606961450743t_bool(aTP_Lamm_ab(A2),aTP_Lamm_ab(B2))) ).

% minus_set_def
tff(fact_3829_minus__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] : minus_2163939370556025621et_nat(A2,B2) = collect_set_nat(minus_8322047804725131473t_bool(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2))) ).

% minus_set_def
tff(fact_3830_minus__set__def,axiom,
    ! [A2: set_int,B2: set_int] : minus_minus_set_int(A2,B2) = collect_int(minus_7727386867957376939t_bool(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2))) ).

% minus_set_def
tff(fact_3831_minus__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] : minus_minus_set_nat(A2,B2) = collect_nat(minus_1628305797571225351t_bool(aa_set3744802623701456604t_bool(aTP_Lamm_ad,A2),aa_set3744802623701456604t_bool(aTP_Lamm_ad,B2))) ).

% minus_set_def
tff(fact_3832_set__diff__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A2,B2) = collec3392354462482085612at_nat(aa_set5223363028666124222t_bool(aTP_Lamm_bf(A2),B2)) ).

% set_diff_eq
tff(fact_3833_set__diff__eq,axiom,
    ! [A2: set_real,B2: set_real] : minus_minus_set_real(A2,B2) = collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_bg(A2),B2)) ).

% set_diff_eq
tff(fact_3834_set__diff__eq,axiom,
    ! [A2: set_list_nat,B2: set_list_nat] : minus_7954133019191499631st_nat(A2,B2) = collect_list_nat(aa_set257547205302447420t_bool(aTP_Lamm_bh(A2),B2)) ).

% set_diff_eq
tff(fact_3835_set__diff__eq,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] : minus_2163939370556025621et_nat(A2,B2) = collect_set_nat(aa_set9121396911425987952t_bool(aTP_Lamm_bi(A2),B2)) ).

% set_diff_eq
tff(fact_3836_set__diff__eq,axiom,
    ! [A2: set_int,B2: set_int] : minus_minus_set_int(A2,B2) = collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_bj(A2),B2)) ).

% set_diff_eq
tff(fact_3837_set__diff__eq,axiom,
    ! [A2: set_nat,B2: set_nat] : minus_minus_set_nat(A2,B2) = collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_bk(A2),B2)) ).

% set_diff_eq
tff(fact_3838_DiffD2,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),minus_1356011639430497352at_nat(A2,B2)))
     => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B2)) ) ).

% DiffD2
tff(fact_3839_DiffD2,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A2,B2)))
     => ~ pp(aa_set_real_bool(member_real(C),B2)) ) ).

% DiffD2
tff(fact_3840_DiffD2,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),minus_2163939370556025621et_nat(A2,B2)))
     => ~ pp(aa_set_set_nat_bool(member_set_nat(C),B2)) ) ).

% DiffD2
tff(fact_3841_DiffD2,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),minus_minus_set_int(A2,B2)))
     => ~ pp(aa_set_int_bool(member_int(C),B2)) ) ).

% DiffD2
tff(fact_3842_DiffD2,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),minus_minus_set_nat(A2,B2)))
     => ~ pp(aa_set_nat_bool(member_nat(C),B2)) ) ).

% DiffD2
tff(fact_3843_DiffD1,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),minus_1356011639430497352at_nat(A2,B2)))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A2)) ) ).

% DiffD1
tff(fact_3844_DiffD1,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A2,B2)))
     => pp(aa_set_real_bool(member_real(C),A2)) ) ).

% DiffD1
tff(fact_3845_DiffD1,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),minus_2163939370556025621et_nat(A2,B2)))
     => pp(aa_set_set_nat_bool(member_set_nat(C),A2)) ) ).

% DiffD1
tff(fact_3846_DiffD1,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),minus_minus_set_int(A2,B2)))
     => pp(aa_set_int_bool(member_int(C),A2)) ) ).

% DiffD1
tff(fact_3847_DiffD1,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),minus_minus_set_nat(A2,B2)))
     => pp(aa_set_nat_bool(member_nat(C),A2)) ) ).

% DiffD1
tff(fact_3848_DiffE,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),minus_1356011639430497352at_nat(A2,B2)))
     => ~ ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A2))
         => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B2)) ) ) ).

% DiffE
tff(fact_3849_DiffE,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A2,B2)))
     => ~ ( pp(aa_set_real_bool(member_real(C),A2))
         => pp(aa_set_real_bool(member_real(C),B2)) ) ) ).

% DiffE
tff(fact_3850_DiffE,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),minus_2163939370556025621et_nat(A2,B2)))
     => ~ ( pp(aa_set_set_nat_bool(member_set_nat(C),A2))
         => pp(aa_set_set_nat_bool(member_set_nat(C),B2)) ) ) ).

% DiffE
tff(fact_3851_DiffE,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),minus_minus_set_int(A2,B2)))
     => ~ ( pp(aa_set_int_bool(member_int(C),A2))
         => pp(aa_set_int_bool(member_int(C),B2)) ) ) ).

% DiffE
tff(fact_3852_DiffE,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),minus_minus_set_nat(A2,B2)))
     => ~ ( pp(aa_set_nat_bool(member_nat(C),A2))
         => pp(aa_set_nat_bool(member_nat(C),B2)) ) ) ).

% DiffE
tff(fact_3853_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Xa3: nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa3)),R))
        <=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa3)),S2)) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_3854_pred__equals__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Xa3: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Xa3),R)
        <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_3855_pred__equals__eq2,axiom,
    ! [R: set_Pr7592006421409863191nteger,S2: set_Pr7592006421409863191nteger] :
      ( ! [X3: fun_Co6545587727236015215e_term,Xa3: produc8923325533196201883nteger] :
          ( member4581088367259704696nteger(aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,X3),Xa3),R)
        <=> member4581088367259704696nteger(aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,X3),Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_3856_pred__equals__eq2,axiom,
    ! [R: set_Pr5663546166630115725nteger,S2: set_Pr5663546166630115725nteger] :
      ( ! [X3: fun_Pr272337284026993017e_term,Xa3: produc8923325533196201883nteger] :
          ( member6423679177046879214nteger(aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,X3),Xa3),R)
        <=> member6423679177046879214nteger(aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,X3),Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_3857_pred__equals__eq2,axiom,
    ! [R: set_Pr233528076387193594nt_int,S2: set_Pr233528076387193594nt_int] :
      ( ! [X3: fun_Pr9164807099015279978e_term,Xa3: product_prod_int_int] :
          ( member690604328116145115nt_int(aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,X3),Xa3),R)
        <=> member690604328116145115nt_int(aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,X3),Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_3858_pred__equals__eq2,axiom,
    ! [R: set_Pr5033440245338114598nt_int,S2: set_Pr5033440245338114598nt_int] :
      ( ! [X3: fun_in6429348457993719614e_term,Xa3: product_prod_int_int] :
          ( member500491123954006279nt_int(aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,X3),Xa3),R)
        <=> member500491123954006279nt_int(aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,X3),Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_3859_Collect__restrict,axiom,
    ! [X6: set_Pr1261947904930325089at_nat,P: fun_Pr7757163693017731481t_bool] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(collec3392354462482085612at_nat(aa_fun830836278467416944t_bool(aTP_Lamm_bl(X6),P))),X6)) ).

% Collect_restrict
tff(fact_3860_Collect__restrict,axiom,
    ! [X6: set_real,P: fun_real_bool] : pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bm(X6),P))),X6)) ).

% Collect_restrict
tff(fact_3861_Collect__restrict,axiom,
    ! [X6: set_list_nat,P: fun_list_nat_bool] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_bn(X6),P)),X6) ).

% Collect_restrict
tff(fact_3862_Collect__restrict,axiom,
    ! [X6: set_set_nat,P: fun_set_nat_bool] : pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_bo(X6),P))),X6)) ).

% Collect_restrict
tff(fact_3863_Collect__restrict,axiom,
    ! [X6: set_nat,P: fun_nat_bool] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bp(X6),P))),X6)) ).

% Collect_restrict
tff(fact_3864_Collect__restrict,axiom,
    ! [X6: set_int,P: fun_int_bool] : pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bq(X6),P))),X6)) ).

% Collect_restrict
tff(fact_3865_prop__restrict,axiom,
    ! [X: product_prod_nat_nat,Z5: set_Pr1261947904930325089at_nat,X6: set_Pr1261947904930325089at_nat,P: fun_Pr7757163693017731481t_bool] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),Z5))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(Z5),collec3392354462482085612at_nat(aa_fun830836278467416944t_bool(aTP_Lamm_bl(X6),P))))
       => pp(aa_Pro2232559814647311250t_bool(P,X)) ) ) ).

% prop_restrict
tff(fact_3866_prop__restrict,axiom,
    ! [X: real,Z5: set_real,X6: set_real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(member_real(X),Z5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(Z5),collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bm(X6),P))))
       => pp(aa_real_bool(P,X)) ) ) ).

% prop_restrict
tff(fact_3867_prop__restrict,axiom,
    ! [X: list_nat,Z5: set_list_nat,X6: set_list_nat,P: fun_list_nat_bool] :
      ( member_list_nat(X,Z5)
     => ( ord_le6045566169113846134st_nat(Z5,collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_bn(X6),P)))
       => pp(aa_list_nat_bool(P,X)) ) ) ).

% prop_restrict
tff(fact_3868_prop__restrict,axiom,
    ! [X: set_nat,Z5: set_set_nat,X6: set_set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),Z5))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(Z5),collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_bo(X6),P))))
       => pp(aa_set_nat_bool(P,X)) ) ) ).

% prop_restrict
tff(fact_3869_prop__restrict,axiom,
    ! [X: nat,Z5: set_nat,X6: set_nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(member_nat(X),Z5))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Z5),collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bp(X6),P))))
       => pp(aa_nat_bool(P,X)) ) ) ).

% prop_restrict
tff(fact_3870_prop__restrict,axiom,
    ! [X: int,Z5: set_int,X6: set_int,P: fun_int_bool] :
      ( pp(aa_set_int_bool(member_int(X),Z5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Z5),collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bq(X6),P))))
       => pp(aa_int_bool(P,X)) ) ) ).

% prop_restrict
tff(fact_3871_pred__subset__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_fun8663595511526100224l_bool(ord_le2468614981408567801t_bool(aTP_Lamm_a(R)),aTP_Lamm_a(S2)))
    <=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(R),S2)) ) ).

% pred_subset_eq
tff(fact_3872_pred__subset__eq,axiom,
    ! [R: set_real,S2: set_real] :
      ( ord_le5573776112191778788l_bool(aTP_Lamm_aa(R),aTP_Lamm_aa(S2))
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(R),S2)) ) ).

% pred_subset_eq
tff(fact_3873_pred__subset__eq,axiom,
    ! [R: set_set_nat,S2: set_set_nat] :
      ( ord_le1461730252015488778t_bool(aTP_Lamm_ac(R),aTP_Lamm_ac(S2))
    <=> pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(R),S2)) ) ).

% pred_subset_eq
tff(fact_3874_pred__subset__eq,axiom,
    ! [R: set_nat,S2: set_nat] :
      ( ord_le8840051076831633728t_bool(aa_set3744802623701456604t_bool(aTP_Lamm_ad,R),aa_set3744802623701456604t_bool(aTP_Lamm_ad,S2))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,R),S2)) ) ).

% pred_subset_eq
tff(fact_3875_pred__subset__eq,axiom,
    ! [R: set_int,S2: set_int] :
      ( ord_le5715760110363009508t_bool(aTP_Lamm_ae(R),aTP_Lamm_ae(S2))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(R),S2)) ) ).

% pred_subset_eq
tff(fact_3876_less__eq__set__def,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B2))
    <=> pp(aa_fun8663595511526100224l_bool(ord_le2468614981408567801t_bool(aTP_Lamm_a(A2)),aTP_Lamm_a(B2))) ) ).

% less_eq_set_def
tff(fact_3877_less__eq__set__def,axiom,
    ! [A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
    <=> ord_le5573776112191778788l_bool(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2)) ) ).

% less_eq_set_def
tff(fact_3878_less__eq__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B2))
    <=> ord_le1461730252015488778t_bool(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2)) ) ).

% less_eq_set_def
tff(fact_3879_less__eq__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
    <=> ord_le8840051076831633728t_bool(aa_set3744802623701456604t_bool(aTP_Lamm_ad,A2),aa_set3744802623701456604t_bool(aTP_Lamm_ad,B2)) ) ).

% less_eq_set_def
tff(fact_3880_less__eq__set__def,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
    <=> ord_le5715760110363009508t_bool(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2)) ) ).

% less_eq_set_def
tff(fact_3881_Collect__subset,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: fun_Pr7757163693017731481t_bool] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(collec3392354462482085612at_nat(aa_fun830836278467416944t_bool(aTP_Lamm_bl(A2),P))),A2)) ).

% Collect_subset
tff(fact_3882_Collect__subset,axiom,
    ! [A2: set_real,P: fun_real_bool] : pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bm(A2),P))),A2)) ).

% Collect_subset
tff(fact_3883_Collect__subset,axiom,
    ! [A2: set_list_nat,P: fun_list_nat_bool] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_bn(A2),P)),A2) ).

% Collect_subset
tff(fact_3884_Collect__subset,axiom,
    ! [A2: set_set_nat,P: fun_set_nat_bool] : pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_bo(A2),P))),A2)) ).

% Collect_subset
tff(fact_3885_Collect__subset,axiom,
    ! [A2: set_nat,P: fun_nat_bool] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bp(A2),P))),A2)) ).

% Collect_subset
tff(fact_3886_Collect__subset,axiom,
    ! [A2: set_int,P: fun_int_bool] : pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bq(A2),P))),A2)) ).

% Collect_subset
tff(fact_3887_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B2: set_nat,R: fun_re9121486324101951821t_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B2))
                  & pp(aa_nat_bool(aa_real_fun_nat_bool(R,X4),Xa)) ) )
         => ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),B2))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_br(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3888_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B2: set_int,R: fun_re8661203372455305201t_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
             => ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),B2))
                  & pp(aa_int_bool(aa_real_fun_int_bool(R,X4),Xa)) ) )
         => ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),B2))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_bs(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3889_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B2: set_complex,R: fun_re3010880487650781423x_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
             => ? [Xa: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa),B2))
                  & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(R,X4),Xa)) ) )
         => ? [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),B2))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_bt(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3890_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_nat,R: fun_nat_fun_nat_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A2))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B2))
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Xa)) ) )
         => ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),B2))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_bu(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3891_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_int,R: fun_nat_fun_int_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A2))
             => ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),B2))
                  & pp(aa_int_bool(aa_nat_fun_int_bool(R,X4),Xa)) ) )
         => ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),B2))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_int_fun_nat_bool(aa_fun8184653438340336124t_bool(aTP_Lamm_bv(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3892_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_complex,R: fun_na5941084329111710283x_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A2))
             => ? [Xa: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa),B2))
                  & pp(aa_complex_bool(aa_nat5545486172511849908x_bool(R,X4),Xa)) ) )
         => ? [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),B2))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_bw(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3893_pigeonhole__infinite__rel,axiom,
    ! [A2: set_int,B2: set_nat,R: fun_int_fun_nat_bool] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A2))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B2))
                  & pp(aa_nat_bool(aa_int_fun_nat_bool(R,X4),Xa)) ) )
         => ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),B2))
              & ~ pp(aa_set_int_bool(finite_finite_int,collect_int(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_bx(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3894_pigeonhole__infinite__rel,axiom,
    ! [A2: set_int,B2: set_int,R: fun_int_fun_int_bool] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A2))
             => ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),B2))
                  & pp(aa_int_bool(aa_int_fun_int_bool(R,X4),Xa)) ) )
         => ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),B2))
              & ~ pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aa_fun8168818243683596540t_bool(aTP_Lamm_by(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3895_pigeonhole__infinite__rel,axiom,
    ! [A2: set_int,B2: set_complex,R: fun_in5750378413879446255x_bool] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A2))
             => ? [Xa: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa),B2))
                  & pp(aa_complex_bool(aa_int5354780257279585880x_bool(R,X4),Xa)) ) )
         => ? [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),B2))
              & ~ pp(aa_set_int_bool(finite_finite_int,collect_int(aa_com2211550117870696024t_bool(aa_fun3087709329828230012t_bool(aTP_Lamm_bz(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3896_pigeonhole__infinite__rel,axiom,
    ! [A2: set_complex,B2: set_nat,R: fun_co2841405903858895691t_bool] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A2))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B2))
                  & pp(aa_nat_bool(aa_com5335841084339320244t_bool(R,X4),Xa)) ) )
         => ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),B2))
              & ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_ca(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3897_not__finite__existsD,axiom,
    ! [P: fun_real_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,collect_real(P)))
     => ? [X_1: real] : pp(aa_real_bool(P,X_1)) ) ).

% not_finite_existsD
tff(fact_3898_not__finite__existsD,axiom,
    ! [P: fun_list_nat_bool] :
      ( ~ finite8100373058378681591st_nat(collect_list_nat(P))
     => ? [X_1: list_nat] : pp(aa_list_nat_bool(P,X_1)) ) ).

% not_finite_existsD
tff(fact_3899_not__finite__existsD,axiom,
    ! [P: fun_set_nat_bool] :
      ( ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(P)))
     => ? [X_1: set_nat] : pp(aa_set_nat_bool(P,X_1)) ) ).

% not_finite_existsD
tff(fact_3900_not__finite__existsD,axiom,
    ! [P: fun_nat_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => ? [X_1: nat] : pp(aa_nat_bool(P,X_1)) ) ).

% not_finite_existsD
tff(fact_3901_not__finite__existsD,axiom,
    ! [P: fun_int_bool] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
     => ? [X_1: int] : pp(aa_int_bool(P,X_1)) ) ).

% not_finite_existsD
tff(fact_3902_not__finite__existsD,axiom,
    ! [P: fun_complex_bool] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(P)))
     => ? [X_1: complex] : pp(aa_complex_bool(P,X_1)) ) ).

% not_finite_existsD
tff(fact_3903_not__finite__existsD,axiom,
    ! [P: fun_Pr7757163693017731481t_bool] :
      ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(P)))
     => ? [X_1: product_prod_nat_nat] : pp(aa_Pro2232559814647311250t_bool(P,X_1)) ) ).

% not_finite_existsD
tff(fact_3904_empty__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(aTP_Lamm_cb) ).

% empty_def
tff(fact_3905_empty__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(aTP_Lamm_cc) ).

% empty_def
tff(fact_3906_empty__def,axiom,
    bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(aTP_Lamm_cd) ).

% empty_def
tff(fact_3907_empty__def,axiom,
    bot_bot_set_real = collect_real(aTP_Lamm_ce) ).

% empty_def
tff(fact_3908_empty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_cf) ).

% empty_def
tff(fact_3909_empty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_cg) ).

% empty_def
tff(fact_3910_lambda__zero,axiom,
    aTP_Lamm_ch = times_times_complex(zero_zero_complex) ).

% lambda_zero
tff(fact_3911_lambda__zero,axiom,
    aTP_Lamm_ci = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_3912_lambda__zero,axiom,
    aTP_Lamm_cj = aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat) ).

% lambda_zero
tff(fact_3913_lambda__zero,axiom,
    aTP_Lamm_ck = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_3914_lambda__zero,axiom,
    aTP_Lamm_cl = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_3915_max__def__raw,axiom,
    ! [X2: set_int,Xa: set_int] :
      ( ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Xa))
       => ( ord_max_set_int(X2,Xa) = Xa ) )
      & ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(X2),Xa))
       => ( ord_max_set_int(X2,Xa) = X2 ) ) ) ).

% max_def_raw
tff(fact_3916_max__def__raw,axiom,
    ! [X2: rat,Xa: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Xa))
       => ( aa_rat_rat(ord_max_rat(X2),Xa) = Xa ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Xa))
       => ( aa_rat_rat(ord_max_rat(X2),Xa) = X2 ) ) ) ).

% max_def_raw
tff(fact_3917_max__def__raw,axiom,
    ! [X2: num,Xa: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Xa))
       => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X2),Xa) = Xa ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Xa))
       => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X2),Xa) = X2 ) ) ) ).

% max_def_raw
tff(fact_3918_max__def__raw,axiom,
    ! [X2: nat,Xa: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Xa))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Xa) = Xa ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Xa))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Xa) = X2 ) ) ) ).

% max_def_raw
tff(fact_3919_max__def__raw,axiom,
    ! [X2: int,Xa: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Xa))
       => ( aa_int_int(ord_max_int(X2),Xa) = Xa ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Xa))
       => ( aa_int_int(ord_max_int(X2),Xa) = X2 ) ) ) ).

% max_def_raw
tff(fact_3920_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ord_le4899916226251588169t_bool(aTP_Lamm_cm(R),aTP_Lamm_cm(S2))
    <=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(R),S2)) ) ).

% pred_subset_eq2
tff(fact_3921_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ord_le4546832483993983377t_bool(aTP_Lamm_cn(R),aTP_Lamm_cn(S2))
    <=> ord_le2843351958646193337nt_int(R,S2) ) ).

% pred_subset_eq2
tff(fact_3922_pred__subset__eq2,axiom,
    ! [R: set_Pr7592006421409863191nteger,S2: set_Pr7592006421409863191nteger] :
      ( ord_le7456723041722595787r_bool(aTP_Lamm_co(R),aTP_Lamm_co(S2))
    <=> ord_le5347933966117430903nteger(R,S2) ) ).

% pred_subset_eq2
tff(fact_3923_pred__subset__eq2,axiom,
    ! [R: set_Pr5663546166630115725nteger,S2: set_Pr5663546166630115725nteger] :
      ( ord_le1120571524945282261r_bool(aTP_Lamm_cp(R),aTP_Lamm_cp(S2))
    <=> ord_le3337582088779512301nteger(R,S2) ) ).

% pred_subset_eq2
tff(fact_3924_pred__subset__eq2,axiom,
    ! [R: set_Pr233528076387193594nt_int,S2: set_Pr233528076387193594nt_int] :
      ( ord_le1486933126236225064t_bool(aTP_Lamm_cq(R),aTP_Lamm_cq(S2))
    <=> ord_le5172437211506631514nt_int(R,S2) ) ).

% pred_subset_eq2
tff(fact_3925_pred__subset__eq2,axiom,
    ! [R: set_Pr5033440245338114598nt_int,S2: set_Pr5033440245338114598nt_int] :
      ( ord_le4148652360786930684t_bool(aTP_Lamm_cr(R),aTP_Lamm_cr(S2))
    <=> ord_le172202941055982214nt_int(R,S2) ) ).

% pred_subset_eq2
tff(fact_3926_bot__empty__eq2,axiom,
    ! [X2: int,Xa: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(bot_bo6207810721336151133t_bool,X2),Xa))
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Xa),bot_bo1796632182523588997nt_int) ) ).

% bot_empty_eq2
tff(fact_3927_bot__empty__eq2,axiom,
    ! [X2: fun_Co6545587727236015215e_term,Xa: produc8923325533196201883nteger] :
      ( pp(aa_Pro5843519176478241656r_bool(aa_fun5646951941296221740r_bool(bot_bo1209944405316066303r_bool,X2),Xa))
    <=> member4581088367259704696nteger(aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,X2),Xa),bot_bo360956144901883563nteger) ) ).

% bot_empty_eq2
tff(fact_3928_bot__empty__eq2,axiom,
    ! [X2: fun_Pr272337284026993017e_term,Xa: produc8923325533196201883nteger] :
      ( pp(aa_Pro5843519176478241656r_bool(aa_fun5449300841655810358r_bool(bot_bo7238443650746976009r_bool,X2),Xa))
    <=> member6423679177046879214nteger(aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,X2),Xa),bot_bo5920137642514184737nteger) ) ).

% bot_empty_eq2
tff(fact_3929_bot__empty__eq2,axiom,
    ! [X2: fun_Pr9164807099015279978e_term,Xa: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_fun5300525129335604105t_bool(bot_bo9220524098407837788t_bool,X2),Xa))
    <=> member690604328116145115nt_int(aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,X2),Xa),bot_bo5853814528170464142nt_int) ) ).

% bot_empty_eq2
tff(fact_3930_bot__empty__eq2,axiom,
    ! [X2: fun_in6429348457993719614e_term,Xa: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_fun4446166836006121821t_bool(bot_bo6140462223577857584t_bool,X2),Xa))
    <=> member500491123954006279nt_int(aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,X2),Xa),bot_bo1891714543702108858nt_int) ) ).

% bot_empty_eq2
tff(fact_3931_bot__empty__eq2,axiom,
    ! [X2: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bot_bo6560894463593755925t_bool,X2),Xa))
    <=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Xa)),bot_bo2099793752762293965at_nat)) ) ).

% bot_empty_eq2
tff(fact_3932_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_cs(P),I)))) ).

% finite_M_bounded_by_nat
tff(fact_3933_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),aa_nat_nat(F,N2)))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ct(F),U)))) ) ).

% finite_less_ub
tff(fact_3934_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_set_vebt(T) = collect_nat(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T)) ).

% set_vebt_def
tff(fact_3935_finite__lists__length__eq,axiom,
    ! [A2: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_cu(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_3936_finite__lists__length__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,N: nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
     => finite500796754983035824at_nat(collec3343600615725829874at_nat(aa_nat1772366067986465313t_bool(aTP_Lamm_cv(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_3937_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_cw(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_3938_finite__lists__length__eq,axiom,
    ! [A2: set_bool,N: nat] :
      ( finite_finite_bool(A2)
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_cx(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_3939_finite__lists__length__eq,axiom,
    ! [A2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_cy(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_3940_finite__lists__length__eq,axiom,
    ! [A2: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_cz(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_3941_finite__lists__length__le,axiom,
    ! [A2: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_da(A2),N))) ) ).

% finite_lists_length_le
tff(fact_3942_finite__lists__length__le,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,N: nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
     => finite500796754983035824at_nat(collec3343600615725829874at_nat(aa_nat1772366067986465313t_bool(aTP_Lamm_db(A2),N))) ) ).

% finite_lists_length_le
tff(fact_3943_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_dc(A2),N))) ) ).

% finite_lists_length_le
tff(fact_3944_finite__lists__length__le,axiom,
    ! [A2: set_bool,N: nat] :
      ( finite_finite_bool(A2)
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_dd(A2),N))) ) ).

% finite_lists_length_le
tff(fact_3945_finite__lists__length__le,axiom,
    ! [A2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_de(A2),N))) ) ).

% finite_lists_length_le
tff(fact_3946_finite__lists__length__le,axiom,
    ! [A2: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_df(A2),N))) ) ).

% finite_lists_length_le
tff(fact_3947_periodic__finite__ex,axiom,
    ! [D: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [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)))) )
       => ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
        <=> ? [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D)))
              & pp(aa_int_bool(P,X3)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_3948_aset_I7_J,axiom,
    ! [D6: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ! [X2: int] :
          ( ! [Xa4: int] :
              ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D6)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),A2))
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D6))) ) ) ) ).

% aset(7)
tff(fact_3949_aset_I5_J,axiom,
    ! [D6: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ( pp(aa_set_int_bool(member_int(T),A2))
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D6)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A2))
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D6)),T)) ) ) ) ) ).

% aset(5)
tff(fact_3950_aset_I4_J,axiom,
    ! [D6: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ( pp(aa_set_int_bool(member_int(T),A2))
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D6)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A2))
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( X2 != T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D6) != T ) ) ) ) ) ).

% aset(4)
tff(fact_3951_aset_I3_J,axiom,
    ! [D6: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A2))
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D6)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A2))
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( X2 = T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D6) = T ) ) ) ) ) ).

% aset(3)
tff(fact_3952_bset_I7_J,axiom,
    ! [D6: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ( pp(aa_set_int_bool(member_int(T),B2))
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D6)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B2))
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D6))) ) ) ) ) ).

% bset(7)
tff(fact_3953_bset_I5_J,axiom,
    ! [D6: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ! [X2: int] :
          ( ! [Xa4: int] :
              ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D6)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),B2))
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D6)),T)) ) ) ) ).

% bset(5)
tff(fact_3954_bset_I4_J,axiom,
    ! [D6: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ( pp(aa_set_int_bool(member_int(T),B2))
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D6)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B2))
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( X2 != T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D6) != T ) ) ) ) ) ).

% bset(4)
tff(fact_3955_bset_I3_J,axiom,
    ! [D6: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B2))
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D6)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B2))
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( X2 = T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D6) = T ) ) ) ) ) ).

% bset(3)
tff(fact_3956_aset_I8_J,axiom,
    ! [D6: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ! [X2: int] :
          ( ! [Xa4: int] :
              ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D6)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),A2))
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X2))
           => 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,X2),D6))) ) ) ) ).

% aset(8)
tff(fact_3957_aset_I6_J,axiom,
    ! [D6: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A2))
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D6)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A2))
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),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,X2),D6)),T)) ) ) ) ) ).

% aset(6)
tff(fact_3958_bset_I8_J,axiom,
    ! [D6: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B2))
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D6)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B2))
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X2))
             => 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,X2),D6))) ) ) ) ) ).

% bset(8)
tff(fact_3959_bset_I6_J,axiom,
    ! [D6: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ! [X2: int] :
          ( ! [Xa4: int] :
              ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D6)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),B2))
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),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,X2),D6)),T)) ) ) ) ).

% bset(6)
tff(fact_3960_cpmi,axiom,
    ! [D6: int,P: fun_int_bool,P3: fun_int_bool,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ( ? [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] :
              ( ! [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D6)))
                 => ! [Xb: int] :
                      ( pp(aa_set_int_bool(member_int(Xb),B2))
                     => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
             => ( pp(aa_int_bool(P,X4))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D6))) ) )
         => ( ! [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),D6)))) )
           => ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
            <=> ( ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D6)))
                    & pp(aa_int_bool(P3,X3)) )
                | ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D6)))
                    & ? [Xa3: int] :
                        ( pp(aa_set_int_bool(member_int(Xa3),B2))
                        & pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa3),X3))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_3961_cppi,axiom,
    ! [D6: int,P: fun_int_bool,P3: fun_int_bool,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ( ? [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] :
              ( ! [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D6)))
                 => ! [Xb: int] :
                      ( pp(aa_set_int_bool(member_int(Xb),A2))
                     => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
             => ( pp(aa_int_bool(P,X4))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D6))) ) )
         => ( ! [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),D6)))) )
           => ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
            <=> ( ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D6)))
                    & pp(aa_int_bool(P3,X3)) )
                | ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D6)))
                    & ? [Xa3: int] :
                        ( pp(aa_set_int_bool(member_int(Xa3),A2))
                        & pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa3),X3))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_3962_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList,S)),X))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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,X),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_3963_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd)),X))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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,X),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_3964_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(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)),X))
    <=> ( ( X != Mi )
       => ( ( X != Ma )
         => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
                 => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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,X),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_3965_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList,Vc)),X))
    <=> ( ( X = Mi )
        | ( X = Ma )
        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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,X),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_3966_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( ( ( Xa2 = zero_zero_nat )
               => pp(A3) )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => pp(B3) )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S4: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S4)
               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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,Xa2),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,Xa2),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_3967_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => pp(A3) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => pp(B3) )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S4: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S4)
             => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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,Xa2),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,Xa2),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_3968_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
      <=> pp(Y) )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa2 = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B3) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
           => pp(Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S4: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S4)
               => ( pp(Y)
                <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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,Xa2),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,Xa2),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_3969_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( ! [Mi3: nat,Ma3: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,Va3,Vb2)
           => ~ ( ( Xa2 = Mi3 )
                | ( Xa2 = Ma3 ) ) )
       => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList2,Vc2)
             => ~ ( ( Xa2 = Mi3 )
                  | ( Xa2 = Ma3 )
                  | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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,Xa2),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,Xa2),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] : X = 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,Xa2),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(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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,Xa2),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,Xa2),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_3970_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => pp(A3) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => pp(B3) )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary3: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3)
             => ~ ( ( Xa2 != Mi3 )
                 => ( ( Xa2 != Ma3 )
                   => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi3))
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi3))
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),Xa2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),Xa2))
                           => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                               => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_3971_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( ! [Uu2: bool,Uv2: bool] : X != vEBT_Leaf(Uu2,Uv2)
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
         => ( ! [Mi3: nat,Ma3: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,Va3,Vb2)
               => ( ( Xa2 = Mi3 )
                  | ( Xa2 = Ma3 ) ) )
           => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList2,Vc2)
                 => ( ( Xa2 = Mi3 )
                    | ( Xa2 = Ma3 )
                    | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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,Xa2),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,Xa2),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] : X = 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,Xa2),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(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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,Xa2),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,Xa2),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_3972_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => pp(Y) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
           => pp(Y) )
         => ( ! [Mi3: nat,Ma3: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,Va3,Vb2)
               => ( pp(Y)
                <=> ~ ( ( Xa2 = Mi3 )
                      | ( Xa2 = Ma3 ) ) ) )
           => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList2,Vc2)
                 => ( pp(Y)
                  <=> ~ ( ( Xa2 = Mi3 )
                        | ( Xa2 = Ma3 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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,Xa2),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,Xa2),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] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2)
                   => ( pp(Y)
                    <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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,Xa2),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,Xa2),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_3973_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(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),X) = 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,X),Mi)),Mi),X)),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,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)))),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,X),Mi)),X),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,X),Mi)),Mi),X)),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,X),Mi)),Mi),X)),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_insert(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,X),Mi)),Mi),X)),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,X),Mi)),Mi),X)),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_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,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,X),Mi)),Mi),X)),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_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,X),Mi)),Mi),X)),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)),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_3974_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( ( ( Xa2 = zero_zero_nat )
               => pp(A3) )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => pp(B3) )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
             => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary3: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3)
                   => ( ( Xa2 != Mi3 )
                     => ( ( Xa2 != Ma3 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi3))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi3))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),Xa2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),Xa2))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                   => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_3975_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
      <=> pp(Y) )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa2 = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B3) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => pp(Y) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
             => pp(Y) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => pp(Y) )
             => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary3: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3)
                   => ( pp(Y)
                    <=> ~ ( ( Xa2 != Mi3 )
                         => ( ( Xa2 != Ma3 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi3))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi3))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),Xa2))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),Xa2))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                       => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_3976_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa2) = Y )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => ( Y = vEBT_Leaf(fTrue,B3) ) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => ( Y = vEBT_Leaf(A3,fTrue) ) )
                    & ( ( Xa2 != one_one_nat )
                     => ( Y = vEBT_Leaf(A3,B3) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,zero_zero_nat,Ts2,S4) )
             => ( Y != vEBT_Node(Info2,zero_zero_nat,Ts2,S4) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S4) )
               => ( Y != vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S4) ) )
           => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary3) )
                 => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),Xa2)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary3) ) )
             => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3) )
                   => ( Y != 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,Xa2),Mi3)),Mi3),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Xa2),Mi3)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma3)))),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,Xa2),Mi3)),Xa2),Mi3)),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,Xa2),Mi3)),Mi3),Xa2)),Ma3))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),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,Xa2),Mi3)),Mi3),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(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,Xa2),Mi3)),Mi3),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Xa2),Mi3)),Mi3),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,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,Xa2),Mi3)),Mi3),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary3,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,Xa2),Mi3)),Mi3),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary3)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3)) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_3977_vebt__succ_Osimps_I6_J,axiom,
    ! [X: 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,X),Mi))
       => ( vEBT_vebt_succ(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),X) = some_nat(Mi) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
       => ( vEBT_vebt_succ(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),X) = 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,X),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(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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,X),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,X),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(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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_3978_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X: 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),X))
       => ( vEBT_vebt_pred(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),X) = some_nat(Ma) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
       => ( vEBT_vebt_pred(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),X) = 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,X),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(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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,X),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),X),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,X),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(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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_3979_vebt__delete_Osimps_I7_J,axiom,
    ! [X: 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,X),Mi))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X)) )
       => ( vEBT_vebt_delete(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),X) = 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,X),Mi))
            | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X)) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( vEBT_vebt_delete(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),X) = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary) ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( vEBT_vebt_delete(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),X) = 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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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_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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat2(vEBT_vebt_maxt(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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_nat2(vEBT_vebt_maxt(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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_delete(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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_nat2(vEBT_vebt_maxt(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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))))))))))))),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(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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_delete(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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_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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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_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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),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_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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(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_nat2(vEBT_vebt_maxt(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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_delete(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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))))))))),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(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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_delete(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(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_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat2(vEBT_vebt_mint(Summary))))))),X)),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)),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_3980_vebt__delete_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(X,Xa2) = Y )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( ( Xa2 = zero_zero_nat )
             => ( Y != vEBT_Leaf(fFalse,B3) ) ) )
       => ( ! [A3: bool] :
              ( ? [B3: bool] : X = vEBT_Leaf(A3,B3)
             => ( ( Xa2 = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != vEBT_Leaf(A3,fFalse) ) ) )
         => ( ! [A3: bool,B3: bool] :
                ( ( X = vEBT_Leaf(A3,B3) )
               => ( ? [N2: nat] : Xa2 = aa_nat_nat(suc,aa_nat_nat(suc,N2))
                 => ( Y != vEBT_Leaf(A3,B3) ) ) )
           => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary3) )
                 => ( Y != vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary3) ) )
             => ( ! [Mi3: nat,Ma3: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,TrLst2,Smry2) )
                   => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,TrLst2,Smry2) ) )
               => ( ! [Mi3: nat,Ma3: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) )
                     => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) ) )
                 => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3) )
                       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi3))
                                | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),Xa2)) )
                             => ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3) ) )
                            & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi3))
                                  | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),Xa2)) )
                             => ( ( ( ( Xa2 = Mi3 )
                                    & ( Xa2 = Ma3 ) )
                                 => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3) ) )
                                & ( ~ ( ( Xa2 = Mi3 )
                                      & ( Xa2 = Ma3 ) )
                                 => ( Y = 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_numeral_nat,aa_num_num(bit0,one)))))),the_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat2(vEBT_vebt_mint(Summary3))))))),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Xa2),Mi3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(vEBT_vebt_mint(Summary3))),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat2(vEBT_vebt_mint(Summary3))))))),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary3)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_3981_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(X,Xa2) = Y )
     => ( ! [Uu2: bool,B3: bool] :
            ( ( X = vEBT_Leaf(Uu2,B3) )
           => ( ( Xa2 = zero_zero_nat )
             => ~ ( ( pp(B3)
                   => ( Y = some_nat(one_one_nat) ) )
                  & ( ~ pp(B3)
                   => ( Y = none_nat ) ) ) ) )
       => ( ( ? [Uv2: bool,Uw2: bool] : X = vEBT_Leaf(Uv2,Uw2)
           => ( ? [N2: nat] : Xa2 = aa_nat_nat(suc,N2)
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2)
             => ( Y != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2)
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2)
                 => ( Y != none_nat ) )
               => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3) )
                     => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi3))
                           => ( Y = some_nat(Mi3) ) )
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi3))
                           => ( Y = 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,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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(Summary3,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary3,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat2(vEBT_vebt_succ(Summary3,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_3982_vebt__pred_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(X,Xa2) = Y )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( ( Xa2 = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A3: bool] :
              ( ? [Uw2: bool] : X = vEBT_Leaf(A3,Uw2)
             => ( ( Xa2 = aa_nat_nat(suc,zero_zero_nat) )
               => ~ ( ( pp(A3)
                     => ( Y = some_nat(zero_zero_nat) ) )
                    & ( ~ pp(A3)
                     => ( Y = none_nat ) ) ) ) )
         => ( ! [A3: bool,B3: bool] :
                ( ( X = vEBT_Leaf(A3,B3) )
               => ( ? [Va2: nat] : Xa2 = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
                 => ~ ( ( pp(B3)
                       => ( Y = some_nat(one_one_nat) ) )
                      & ( ~ pp(B3)
                       => ( ( pp(A3)
                           => ( Y = some_nat(zero_zero_nat) ) )
                          & ( ~ pp(A3)
                           => ( Y = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3)
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2)
                 => ( Y != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2)
                   => ( Y != none_nat ) )
                 => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3) )
                       => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),Xa2))
                             => ( Y = some_nat(Ma3) ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),Xa2))
                             => ( Y = 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,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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(Summary3,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Mi3),Xa2),some_nat(Mi3),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary3,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,the_nat2(vEBT_vebt_pred(Summary3,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_3983_finite__nth__roots,axiom,
    ! [N: nat,C: 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_dg(N),C)))) ) ).

% finite_nth_roots
tff(fact_3984_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_dh,N)))) ) ).

% finite_roots_unity
tff(fact_3985_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_di,N)))) ) ).

% finite_roots_unity
tff(fact_3986_vebt__succ_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(X,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,B3: bool] :
              ( ( X = vEBT_Leaf(Uu2,B3) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( ( pp(B3)
                     => ( Y = some_nat(one_one_nat) ) )
                    & ( ~ pp(B3)
                     => ( Y = none_nat ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,B3),zero_zero_nat)) ) ) )
         => ( ! [Uv2: bool,Uw2: bool] :
                ( ( X = vEBT_Leaf(Uv2,Uw2) )
               => ! [N2: nat] :
                    ( ( Xa2 = aa_nat_nat(suc,N2) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uv2,Uw2),aa_nat_nat(suc,N2))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2) )
                 => ( ( Y = none_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Xa2)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2),Xa2)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Xa2)) ) )
                 => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3) )
                       => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi3))
                             => ( Y = some_nat(Mi3) ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi3))
                             => ( Y = 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,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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(Summary3,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary3,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat2(vEBT_vebt_succ(Summary3,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3),Xa2)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_3987_vebt__pred_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(X,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),zero_zero_nat)) ) ) )
         => ( ! [A3: bool,Uw2: bool] :
                ( ( X = vEBT_Leaf(A3,Uw2) )
               => ( ( Xa2 = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( ( pp(A3)
                       => ( Y = some_nat(zero_zero_nat) ) )
                      & ( ~ pp(A3)
                       => ( Y = none_nat ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,Uw2),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: bool,B3: bool] :
                  ( ( X = vEBT_Leaf(A3,B3) )
                 => ! [Va2: nat] :
                      ( ( Xa2 = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                     => ( ( ( pp(B3)
                           => ( Y = some_nat(one_one_nat) ) )
                          & ( ~ pp(B3)
                           => ( ( pp(A3)
                               => ( Y = some_nat(zero_zero_nat) ) )
                              & ( ~ pp(A3)
                               => ( Y = none_nat ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Xa2)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2),Xa2)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2) )
                       => ( ( Y = none_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Xa2)) ) )
                   => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                          ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3) )
                         => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),Xa2))
                               => ( Y = some_nat(Ma3) ) )
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),Xa2))
                               => ( Y = 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,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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(Summary3,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Mi3),Xa2),some_nat(Mi3),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary3,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,the_nat2(vEBT_vebt_pred(Summary3,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3),Xa2)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_3988_vebt__delete_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(X,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = vEBT_Leaf(fFalse,B3) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),zero_zero_nat)) ) ) )
         => ( ! [A3: bool,B3: bool] :
                ( ( X = vEBT_Leaf(A3,B3) )
               => ( ( Xa2 = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = vEBT_Leaf(A3,fFalse) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: bool,B3: bool] :
                  ( ( X = vEBT_Leaf(A3,B3) )
                 => ! [N2: nat] :
                      ( ( Xa2 = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                     => ( ( Y = vEBT_Leaf(A3,B3) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),aa_nat_nat(suc,aa_nat_nat(suc,N2)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary3) )
                   => ( ( Y = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary3) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary3),Xa2)) ) )
               => ( ! [Mi3: nat,Ma3: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,TrLst2,Smry2) )
                     => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,TrLst2,Smry2) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,TrLst2,Smry2),Xa2)) ) )
                 => ( ! [Mi3: nat,Ma3: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) )
                       => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2),Xa2)) ) )
                   => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                          ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3) )
                         => ( ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi3))
                                  | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),Xa2)) )
                               => ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3) ) )
                              & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi3))
                                    | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),Xa2)) )
                               => ( ( ( ( Xa2 = Mi3 )
                                      & ( Xa2 = Ma3 ) )
                                   => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3) ) )
                                  & ( ~ ( ( Xa2 = Mi3 )
                                        & ( Xa2 = Ma3 ) )
                                   => ( Y = 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,Xa2),Mi3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(vEBT_vebt_mint(Summary3))),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat2(vEBT_vebt_mint(Summary3))))))),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(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,Xa2),Mi3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(vEBT_vebt_mint(Summary3))),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat2(vEBT_vebt_mint(Summary3))))))),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Xa2),Mi3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(vEBT_vebt_mint(Summary3))),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat2(vEBT_vebt_mint(Summary3))))))),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,Xa2),Mi3)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(vEBT_vebt_mint(Summary3))),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat2(vEBT_vebt_mint(Summary3))))))),Xa2)),Mi3)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi3)),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_nat2(vEBT_vebt_mint(Summary3))),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat2(vEBT_vebt_mint(Summary3))))))),Ma3))),aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi3))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma3)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary3,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,Xa2),Mi3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(vEBT_vebt_mint(Summary3))),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat2(vEBT_vebt_mint(Summary3))))))),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Xa2),Mi3)),aa_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_numeral_nat,aa_num_num(bit0,one)))))),the_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat2(vEBT_vebt_mint(Summary3))))))),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Xa2),Mi3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat2(vEBT_vebt_mint(Summary3))),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat2(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat2(vEBT_vebt_mint(Summary3))))))),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary3)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3)) ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3),Xa2)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_3989_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( ( ( Xa2 = zero_zero_nat )
                   => ( Y = vEBT_Leaf(fTrue,B3) ) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => ( Y = vEBT_Leaf(A3,fTrue) ) )
                      & ( ( Xa2 != one_one_nat )
                       => ( Y = vEBT_Leaf(A3,B3) ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,zero_zero_nat,Ts2,S4) )
               => ( ( Y = vEBT_Node(Info2,zero_zero_nat,Ts2,S4) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S4),Xa2)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S4) )
                 => ( ( Y = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S4) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S4),Xa2)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary3) )
                   => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),Xa2)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary3) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary3),Xa2)) ) )
               => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3) )
                     => ( ( Y = 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,Xa2),Mi3)),Mi3),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Xa2),Mi3)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma3)))),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,Xa2),Mi3)),Xa2),Mi3)),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,Xa2),Mi3)),Mi3),Xa2)),Ma3))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),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,Xa2),Mi3)),Mi3),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(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,Xa2),Mi3)),Mi3),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Xa2),Mi3)),Mi3),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,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,Xa2),Mi3)),Mi3),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary3,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,Xa2),Mi3)),Mi3),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary3)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3)) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_3990_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),U))) ).

% finite_atLeastAtMost_int
tff(fact_3991_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_3992_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( pp(Y)
                <=> ( ( ( Xa2 = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B3) )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xa2)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ( ~ pp(Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xa2)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ~ pp(Y)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa2)) ) )
               => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3) )
                     => ( ( pp(Y)
                        <=> ( ( Xa2 != Mi3 )
                           => ( ( Xa2 != Ma3 )
                             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi3))
                                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi3))
                                 => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),Xa2))
                                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),Xa2))
                                     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                         => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_3993_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2))
               => ( ( ( Xa2 = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B3) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xa2)) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xa2)) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa2)) )
               => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3),Xa2))
                       => ( ( Xa2 != Mi3 )
                         => ( ( Xa2 != Ma3 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi3))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi3))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),Xa2))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),Xa2))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                       => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_3994_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2))
               => ( ( ( Xa2 = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B3) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xa2)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S4: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S4) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S4),Xa2))
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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,Xa2),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,Xa2),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_3995_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2))
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B3) )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S4: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S4) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S4),Xa2))
                 => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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,Xa2),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,Xa2),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_3996_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( pp(Y)
                <=> ( ( ( Xa2 = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B3) )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xa2)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S4: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S4) )
                 => ( ( pp(Y)
                    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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,Xa2),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,Xa2),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(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S4),Xa2)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_3997_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2))
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B3) )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary3),Xa2))
                 => ~ ( ( Xa2 != Mi3 )
                     => ( ( Xa2 != Ma3 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi3))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi3))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),Xa2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),Xa2))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                   => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_3998_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ~ pp(Y)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xa2)) ) )
           => ( ! [Mi3: nat,Ma3: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,Va3,Vb2) )
                 => ( ( pp(Y)
                    <=> ( ( Xa2 = Mi3 )
                        | ( Xa2 = Ma3 ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,Va3,Vb2),Xa2)) ) )
             => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList2,Vc2) )
                   => ( ( pp(Y)
                      <=> ( ( Xa2 = Mi3 )
                          | ( Xa2 = Ma3 )
                          | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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,Xa2),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,Xa2),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(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList2,Vc2),Xa2)) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2) )
                     => ( ( pp(Y)
                        <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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,Xa2),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,Xa2),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(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2),Xa2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_3999_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xa2)) )
           => ( ! [Mi3: nat,Ma3: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,Va3,Vb2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,Va3,Vb2),Xa2))
                   => ( ( Xa2 = Mi3 )
                      | ( Xa2 = Ma3 ) ) ) )
             => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList2,Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList2,Vc2),Xa2))
                     => ( ( Xa2 = Mi3 )
                        | ( Xa2 = Ma3 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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,Xa2),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,Xa2),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] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2),Xa2))
                       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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,Xa2),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,Xa2),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_4000_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Mi3: nat,Ma3: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,Va3,Vb2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),zero_zero_nat,Va3,Vb2),Xa2))
               => ~ ( ( Xa2 = Mi3 )
                    | ( Xa2 = Ma3 ) ) ) )
         => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList2,Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList2,Vc2),Xa2))
                 => ~ ( ( Xa2 = Mi3 )
                      | ( Xa2 = Ma3 )
                      | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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,Xa2),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,Xa2),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] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2),Xa2))
                   => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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,Xa2),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,Xa2),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_4001_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_4002_mult__le__cancel__iff1,axiom,
    ! [Z: real,X: real,Y: 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,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_4003_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: rat,Y: 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(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_4004_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: 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,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_4005_mult__le__cancel__iff2,axiom,
    ! [Z: real,X: real,Y: 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),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_4006_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: rat,Y: 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(aa_rat_fun_rat_rat(times_times_rat,Z),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_4007_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: 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),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Y)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_4008_divides__aux__eq,axiom,
    ! [Q4: nat,R2: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q4),R2)))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_4009_divides__aux__eq,axiom,
    ! [Q4: int,R2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(unique6319869463603278526ux_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R2)))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_4010_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_4011_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_4012_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
     => ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),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))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),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_4013_low__def,axiom,
    ! [X: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),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_4014_mod__self,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),A) = zero_zero_nat ).

% mod_self
tff(fact_4015_mod__self,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),A) = zero_zero_int ).

% mod_self
tff(fact_4016_mod__self,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,A) = zero_z3403309356797280102nteger ).

% mod_self
tff(fact_4017_mod__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),zero_zero_nat) = A ).

% mod_by_0
tff(fact_4018_mod__by__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),zero_zero_int) = A ).

% mod_by_0
tff(fact_4019_mod__by__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,zero_z3403309356797280102nteger) = A ).

% mod_by_0
tff(fact_4020_mod__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).

% mod_0
tff(fact_4021_mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_4022_mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% mod_0
tff(fact_4023_bits__mod__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).

% bits_mod_0
tff(fact_4024_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_4025_bits__mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% bits_mod_0
tff(fact_4026_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_4027_mod__mult__self2__is__0,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = zero_zero_nat ).

% mod_mult_self2_is_0
tff(fact_4028_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_4029_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),B) = zero_z3403309356797280102nteger ).

% mod_mult_self2_is_0
tff(fact_4030_mod__mult__self1__is__0,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),B) = zero_zero_nat ).

% mod_mult_self1_is_0
tff(fact_4031_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),B) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_4032_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),B) = zero_z3403309356797280102nteger ).

% mod_mult_self1_is_0
tff(fact_4033_mod__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).

% mod_by_1
tff(fact_4034_mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_4035_mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_4036_bits__mod__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).

% bits_mod_by_1
tff(fact_4037_bits__mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_4038_bits__mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_4039_mod__div__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).

% mod_div_trivial
tff(fact_4040_mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),B)),B) = zero_zero_int ).

% mod_div_trivial
tff(fact_4041_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_4042_bits__mod__div__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).

% bits_mod_div_trivial
tff(fact_4043_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),B)),B) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_4044_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_4045_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_4046_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),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,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4047_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4048_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4049_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),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,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4050_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4051_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4052_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_4053_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_4054_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_4055_unique__quotient,axiom,
    ! [A: int,B: int,Q4: int,R2: int,Q6: int,R4: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R2)))
     => ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q6),R4)))
       => ( Q4 = Q6 ) ) ) ).

% unique_quotient
tff(fact_4056_unique__remainder,axiom,
    ! [A: int,B: int,Q4: int,R2: int,Q6: int,R4: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R2)))
     => ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q6),R4)))
       => ( R2 = R4 ) ) ) ).

% unique_remainder
tff(fact_4057_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_4058_eucl__rel__int__by0,axiom,
    ! [K: int] : pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K))) ).

% eucl_rel_int_by0
tff(fact_4059_div__int__unique,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R2)))
     => ( aa_int_int(divide_divide_int(K),L) = Q4 ) ) ).

% div_int_unique
tff(fact_4060_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,modulo364778990260209775nteger(A,B)),A)) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4061_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),A)) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4062_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(A),B)),A)) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4063_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4064_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),B)),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4065_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,B)),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4066_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_4067_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = A )
    <=> ( aa_int_int(divide_divide_int(A),B) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_4068_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = A )
    <=> ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_4069_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_4070_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))
         => ( ! [N2: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),P2))
               => ( pp(aa_nat_bool(P,N2))
                 => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N2)),P2))) ) )
           => pp(aa_nat_bool(P,M)) ) ) ) ) ).

% mod_induct
tff(fact_4071_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_4072_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,N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,N2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M5),N2)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M5),N2)) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ).

% gcd_nat_induct
tff(fact_4073_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_4074_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),D) = zero_zero_nat )
     => ? [Q3: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q3) ) ).

% mod_eq_0D
tff(fact_4075_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_4076_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_4077_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_4078_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),modulo364778990260209775nteger(A,B))) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4079_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => 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,A),B))) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4080_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),B))) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4081_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
       => ( modulo364778990260209775nteger(A,B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4082_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4083_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
       => ( aa_int_int(modulo_modulo_int(A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4084_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q4: 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,Q4))) = 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,Q4)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_4085_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q4: num] :
      ( ( aa_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,Q4))) = zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q4)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_4086_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q4: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) = zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(Q4)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_4087_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_4088_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_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_4089_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_4090_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_4091_div__less__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_nat,zero_zero_nat),N))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),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,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),N))) ) ) ) ) ).

% div_less_mono
tff(fact_4092_mod__eq__nat1E,axiom,
    ! [M: nat,Q4: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q4) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ~ ! [S4: 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,Q4),S4)) ) ) ).

% mod_eq_nat1E
tff(fact_4093_mod__eq__nat2E,axiom,
    ! [M: nat,Q4: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q4) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ~ ! [S4: 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,Q4),S4)) ) ) ).

% mod_eq_nat2E
tff(fact_4094_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
       => ? [Q3: nat] : X = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_4095_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q4: int] :
      ( ( L != zero_zero_int )
     => ( ( K = aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L) )
       => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),zero_zero_int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_4096_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_4097_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
     => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),modulo364778990260209775nteger(divide6298287555418463151nteger(A,B),C)),modulo364778990260209775nteger(A,B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4098_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4099_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4100_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_4101_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_4102_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B))
       => ( aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_4103_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),B))
       => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) = modulo364778990260209775nteger(A,B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_4104_bits__stable__imp__add__self,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
tff(fact_4105_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_4106_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_4107_verit__le__mono__div,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_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,A2),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,B2),N)),zero_zero_nat)),one_one_nat),zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),N))) ) ) ).

% verit_le_mono_div
tff(fact_4108_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B))
       => ( 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,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_4109_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B))
       => ( 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(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))) = aa_int_int(divide_divide_int(A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_4110_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),B))
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))) = divide6298287555418463151nteger(A,B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_4111_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: 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,A),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,A),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_4112_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_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(modulo_modulo_int(A),aa_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(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4113_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,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_4114_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),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),Q4)),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))
             => ( Q4 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_4115_mod__double__modulus,axiom,
    ! [M: code_integer,X: 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),X))
       => ( ( modulo364778990260209775nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) = modulo364778990260209775nteger(X,M) )
          | ( modulo364778990260209775nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(X,M),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_4116_mod__double__modulus,axiom,
    ! [M: nat,X: 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),X))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),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,X),M) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),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,X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_4117_mod__double__modulus,axiom,
    ! [M: int,X: 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),X))
       => ( ( aa_int_int(modulo_modulo_int(X),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(modulo_modulo_int(X),M) )
          | ( aa_int_int(modulo_modulo_int(X),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(modulo_modulo_int(X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_4118_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))))
         => ( minus_8373710615458151222nteger(modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)),B) = modulo364778990260209775nteger(A,B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4119_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4120_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4121_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))))
         => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),one_one_Code_integer) = divide6298287555418463151nteger(A,B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_4122_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))))
         => ( 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,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_4123_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))))
         => ( 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(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))),one_one_int) = aa_int_int(divide_divide_int(A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_4124_mult__less__iff1,axiom,
    ! [Z: real,X: real,Y: 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,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_4125_mult__less__iff1,axiom,
    ! [Z: rat,X: rat,Y: 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(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_4126_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: 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,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_4127_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
     => ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),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))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),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_4128_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(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_4129_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(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_4130_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(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_nat(Ys))),aa_nat_nat(aa_lis1299702591413342011at_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_4131_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,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_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys))))
     => ( nth_Pr6837108013167703752BT_int(produc7292646706713671643BT_int(Xs,Ys),N) = produc736041933913180425BT_int(nth_VEBT_VEBT(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_4132_product__nth,axiom,
    ! [N: nat,Xs: list_bool,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_bool(Xs)),size_s6755466524823107622T_VEBT(Ys))))
     => ( nth_Pr3250499561091541003T_VEBT(produc2282853104913622818T_VEBT(Xs,Ys),N) = produc9074556814250467700T_VEBT(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_4133_product__nth,axiom,
    ! [N: nat,Xs: list_bool,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_size_list_bool(Xs)),size_size_list_bool(Ys))))
     => ( nth_Pr2856333732135533678l_bool(product_bool_bool(Xs,Ys),N) = produc8297744024737662807l_bool(aa_nat_bool(nth_bool(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_4134_product__nth,axiom,
    ! [N: nat,Xs: list_bool,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_size_list_bool(Xs)),size_size_list_nat(Ys))))
     => ( nth_Pr2186422209488468047ol_nat(product_bool_nat(Xs,Ys),N) = produc8874003862241262736ol_nat(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_nat(Ys))),aa_nat_nat(aa_lis1299702591413342011at_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_4135_product__nth,axiom,
    ! [N: nat,Xs: list_bool,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_bool(Xs)),size_size_list_int(Ys))))
     => ( nth_Pr7231943226834047147ol_int(product_bool_int(Xs,Ys),N) = produc8871513391732212460ol_int(aa_nat_bool(nth_bool(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_4136_product__nth,axiom,
    ! [N: nat,Xs: list_nat,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_nat(Xs)),size_s6755466524823107622T_VEBT(Ys))))
     => ( nth_Pr744662078594809490T_VEBT(produc7156399406898700509T_VEBT(Xs,Ys),N) = produc599794634098209291T_VEBT(aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_4137_product__nth,axiom,
    ! [N: nat,Xs: list_nat,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_size_list_nat(Xs)),size_size_list_bool(Ys))))
     => ( nth_Pr396318242284072693t_bool(product_nat_bool(Xs,Ys),N) = produc3913204875133826030t_bool(aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,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_4138_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: fun_real_complex,Y: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_dj(I6),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_dj(I6),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_dk(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_4139_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: fun_nat_complex,Y: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_dl(I6),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_dl(I6),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_dm(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_4140_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: fun_int_complex,Y: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_dn(I6),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_dn(I6),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_do(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_4141_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: fun_complex_complex,Y: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_dp(I6),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_dp(I6),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_dq(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_4142_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: fun_real_real,Y: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_dr(I6),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_dr(I6),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_ds(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_4143_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: fun_nat_real,Y: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_dt(I6),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_dt(I6),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_du(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_4144_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: fun_int_real,Y: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_dv(I6),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_dv(I6),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_dw(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_4145_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: fun_complex_real,Y: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_dx(I6),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_dx(I6),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_dy(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_4146_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: fun_real_rat,Y: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_dz(I6),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_dz(I6),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_ea(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_4147_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: fun_nat_rat,Y: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_eb(I6),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_eb(I6),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_ec(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_4148_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: fun_real_complex,Y: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_ed(I6),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_ed(I6),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_ee(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_4149_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: fun_nat_complex,Y: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_ef(I6),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_ef(I6),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_eg(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_4150_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: fun_int_complex,Y: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_eh(I6),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_eh(I6),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_ei(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_4151_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: fun_complex_complex,Y: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_ej(I6),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_ej(I6),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_ek(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_4152_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: fun_real_real,Y: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_el(I6),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_el(I6),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_em(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_4153_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: fun_nat_real,Y: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_en(I6),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_en(I6),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_eo(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_4154_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: fun_int_real,Y: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_ep(I6),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_ep(I6),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_eq(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_4155_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: fun_complex_real,Y: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_er(I6),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_er(I6),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_es(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_4156_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: fun_real_rat,Y: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_et(I6),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_et(I6),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_eu(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_4157_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: fun_nat_rat,Y: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_ev(I6),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_ev(I6),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_ew(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_4158_even__succ__mod__exp,axiom,
    ! [A: 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))),A))
     => ( 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),A)),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,A),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_4159_even__succ__mod__exp,axiom,
    ! [A: 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))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_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(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_4160_even__succ__mod__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_4161_Bolzano,axiom,
    ! [A: real,B: real,P: fun_re1255401033344146673l_bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ! [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,A),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
               => ? [D3: real] :
                    ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D3))
                    & ! [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)),D3)) )
                       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B3)) ) ) ) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),B)) ) ) ) ).

% Bolzano
tff(fact_4162_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_4163_dvd__0__right,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),zero_z3403309356797280102nteger)) ).

% dvd_0_right
tff(fact_4164_dvd__0__right,axiom,
    ! [A: complex] : pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),zero_zero_complex)) ).

% dvd_0_right
tff(fact_4165_dvd__0__right,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),zero_zero_real)) ).

% dvd_0_right
tff(fact_4166_dvd__0__right,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),zero_zero_rat)) ).

% dvd_0_right
tff(fact_4167_dvd__0__right,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat)) ).

% dvd_0_right
tff(fact_4168_dvd__0__right,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),zero_zero_int)) ).

% dvd_0_right
tff(fact_4169_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A))
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
tff(fact_4170_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,zero_zero_complex),A))
    <=> ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
tff(fact_4171_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,zero_zero_real),A))
    <=> ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
tff(fact_4172_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,zero_zero_rat),A))
    <=> ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
tff(fact_4173_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
    <=> ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
tff(fact_4174_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),A))
    <=> ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
tff(fact_4175_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_4176_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_4177_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_4178_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_4179_dvd__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)))
    <=> ( ( C = zero_zero_complex )
        | pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_4180_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( C = zero_zero_real )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_4181_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
    <=> ( ( C = zero_zero_rat )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_4182_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( C = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_4183_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_4184_dvd__mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)))
    <=> ( ( C = zero_zero_complex )
        | pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_4185_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( C = zero_zero_real )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_4186_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
    <=> ( ( C = zero_zero_rat )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_4187_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( C = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_4188_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),C)) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_4189_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != 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,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_4190_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != 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,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_4191_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),C)) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_4192_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != 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,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_4193_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != 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,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_4194_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_imp_mod_0
tff(fact_4195_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( aa_int_int(modulo_modulo_int(B),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_4196_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
     => ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
tff(fact_4197_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(modulo_modulo_int(K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_4198_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(modulo_modulo_int(K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_4199_pow__divides__pow__iff,axiom,
    ! [N: nat,A: nat,B: 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,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ) ).

% pow_divides_pow_iff
tff(fact_4200_pow__divides__pow__iff,axiom,
    ! [N: nat,A: int,B: 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(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% pow_divides_pow_iff
tff(fact_4201_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_4202_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_4203_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_4204_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int] : size_s3661962791536183091BT_int(produc7292646706713671643BT_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_4205_length__product,axiom,
    ! [Xs: list_bool,Ys: list_VEBT_VEBT] : size_s7211631430880329078T_VEBT(produc2282853104913622818T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_4206_length__product,axiom,
    ! [Xs: list_bool,Ys: list_bool] : size_s6988337448923680857l_bool(product_bool_bool(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_bool(Ys)) ).

% length_product
tff(fact_4207_length__product,axiom,
    ! [Xs: list_bool,Ys: list_nat] : size_s3243772863075988794ol_nat(product_bool_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_4208_length__product,axiom,
    ! [Xs: list_bool,Ys: list_int] : size_s753689718144262038ol_int(product_bool_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_4209_length__product,axiom,
    ! [Xs: list_nat,Ys: list_VEBT_VEBT] : size_s4762443039079500285T_VEBT(produc7156399406898700509T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_4210_length__product,axiom,
    ! [Xs: list_nat,Ys: list_bool] : size_s783543119881238752t_bool(product_nat_bool(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_bool(Ys)) ).

% length_product
tff(fact_4211_zero__le__power__eq__numeral,axiom,
    ! [A: real,W2: 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,A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( 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,W2)))
        | ( ~ 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,W2)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_4212_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( 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,W2)))
        | ( ~ 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,W2)))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_4213_zero__le__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( 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,W2)))
        | ( ~ 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,W2)))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_4214_power__less__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W2))),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,W2)))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_4215_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2))),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,W2)))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_4216_power__less__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))),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,W2)))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_4217_power__less__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),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,A),zero_zero_real)) ) ) ).

% power_less_zero_eq
tff(fact_4218_power__less__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),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,A),zero_zero_rat)) ) ) ).

% power_less_zero_eq
tff(fact_4219_power__less__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),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,A),zero_zero_int)) ) ) ).

% power_less_zero_eq
tff(fact_4220_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_4221_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_4222_zero__less__power__eq__numeral,axiom,
    ! [A: real,W2: 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,A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = 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,W2)))
          & ( A != 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,W2)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_4223_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = 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,W2)))
          & ( A != 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,W2)))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_4224_zero__less__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = 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,W2)))
          & ( A != 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,W2)))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_4225_even__power,axiom,
    ! [A: 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(A),N)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% even_power
tff(fact_4226_even__power,axiom,
    ! [A: 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,A),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% even_power
tff(fact_4227_even__power,axiom,
    ! [A: 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(power_power_int(A),N)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% even_power
tff(fact_4228_power__le__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_real))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2)))
        & ( ( ~ 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,W2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),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,W2)))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_4229_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_rat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2)))
        & ( ( ~ 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,W2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),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,W2)))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_4230_power__le__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_int))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2)))
        & ( ( ~ 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,W2)))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),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,W2)))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_4231_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))),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_4232_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_4233_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(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_4234_even__succ__div__exp,axiom,
    ! [A: 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))),A))
     => ( 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),A)),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,A),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_4235_even__succ__div__exp,axiom,
    ! [A: 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))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_4236_even__succ__div__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_4237_dvd__field__iff,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B))
    <=> ( ( A = zero_zero_complex )
       => ( B = zero_zero_complex ) ) ) ).

% dvd_field_iff
tff(fact_4238_dvd__field__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
    <=> ( ( A = zero_zero_real )
       => ( B = zero_zero_real ) ) ) ).

% dvd_field_iff
tff(fact_4239_dvd__field__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
    <=> ( ( A = zero_zero_rat )
       => ( B = zero_zero_rat ) ) ) ).

% dvd_field_iff
tff(fact_4240_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A))
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
tff(fact_4241_dvd__0__left,axiom,
    ! [A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,zero_zero_complex),A))
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
tff(fact_4242_dvd__0__left,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,zero_zero_real),A))
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
tff(fact_4243_dvd__0__left,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,zero_zero_rat),A))
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
tff(fact_4244_dvd__0__left,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
tff(fact_4245_dvd__0__left,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),A))
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
tff(fact_4246_gcd__nat_Oextremum,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat)) ).

% gcd_nat.extremum
tff(fact_4247_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
tff(fact_4248_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
    <=> ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
tff(fact_4249_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat))
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_4250_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_4251_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_4252_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(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_4253_subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ex,A))),collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ex,B))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% subset_divisors_dvd
tff(fact_4254_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ey,A))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ey,B))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% subset_divisors_dvd
tff(fact_4255_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ord_le7084787975880047091nteger(collect_Code_integer(aa_Cod7250484176784686200r_bool(aTP_Lamm_ez,A)),collect_Code_integer(aa_Cod7250484176784686200r_bool(aTP_Lamm_ez,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B)) ) ).

% subset_divisors_dvd
tff(fact_4256_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_fa,A))),collect_int(aa_int_fun_int_bool(aTP_Lamm_fa,B))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% subset_divisors_dvd
tff(fact_4257_strict__subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ex,A))),collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ex,B))))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_4258_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ey,A))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ey,B))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_4259_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_fa,A))),collect_int(aa_int_fun_int_bool(aTP_Lamm_fa,B))))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_4260_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ord_le1307284697595431911nteger(collect_Code_integer(aa_Cod7250484176784686200r_bool(aTP_Lamm_ez,A)),collect_Code_integer(aa_Cod7250484176784686200r_bool(aTP_Lamm_ez,B)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
        & ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_4261_not__is__unit__0,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),one_one_Code_integer)) ).

% not_is_unit_0
tff(fact_4262_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_4263_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_4264_pinf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z3: code_integer] :
    ! [X2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z3),X2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S))) ) ) ).

% pinf(9)
tff(fact_4265_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S))) ) ) ).

% pinf(9)
tff(fact_4266_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X2),S)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X2),S))) ) ) ).

% pinf(9)
tff(fact_4267_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S))) ) ) ).

% pinf(9)
tff(fact_4268_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S))) ) ) ).

% pinf(9)
tff(fact_4269_pinf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z3: code_integer] :
    ! [X2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z3),X2))
     => ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S)))
      <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S))) ) ) ).

% pinf(10)
tff(fact_4270_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X2))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S)))
      <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S))) ) ) ).

% pinf(10)
tff(fact_4271_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X2))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X2),S)))
      <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X2),S))) ) ) ).

% pinf(10)
tff(fact_4272_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X2))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S)))
      <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S))) ) ) ).

% pinf(10)
tff(fact_4273_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X2))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S)))
      <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S))) ) ) ).

% pinf(10)
tff(fact_4274_minf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z3: code_integer] :
    ! [X2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Z3))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S))) ) ) ).

% minf(9)
tff(fact_4275_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S))) ) ) ).

% minf(9)
tff(fact_4276_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X2),S)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X2),S))) ) ) ).

% minf(9)
tff(fact_4277_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S))) ) ) ).

% minf(9)
tff(fact_4278_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S))) ) ) ).

% minf(9)
tff(fact_4279_minf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z3: code_integer] :
    ! [X2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Z3))
     => ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S)))
      <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S))) ) ) ).

% minf(10)
tff(fact_4280_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z3))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S)))
      <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S))) ) ) ).

% minf(10)
tff(fact_4281_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z3))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X2),S)))
      <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X2),S))) ) ) ).

% minf(10)
tff(fact_4282_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z3))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S)))
      <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S))) ) ) ).

% minf(10)
tff(fact_4283_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z3))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S)))
      <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S))) ) ) ).

% minf(10)
tff(fact_4284_dvd__div__eq__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B),A))
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = zero_zero_complex )
      <=> ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_4285_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_4286_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ( ( aa_int_int(divide_divide_int(A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_4287_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_4288_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_4289_dvd__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
     => ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_4290_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% mod_0_imp_dvd
tff(fact_4291_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% mod_0_imp_dvd
tff(fact_4292_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A)) ) ).

% mod_0_imp_dvd
tff(fact_4293_dvd__eq__mod__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_eq_mod_eq_0
tff(fact_4294_dvd__eq__mod__eq__0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
    <=> ( aa_int_int(modulo_modulo_int(B),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_4295_dvd__eq__mod__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
    <=> ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_eq_mod_eq_0
tff(fact_4296_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% mod_eq_0_iff_dvd
tff(fact_4297_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% mod_eq_0_iff_dvd
tff(fact_4298_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A)) ) ).

% mod_eq_0_iff_dvd
tff(fact_4299_dvd__power__le,axiom,
    ! [X: code_integer,Y: code_integer,N: nat,M: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Y))
     => ( 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(X),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),M))) ) ) ).

% dvd_power_le
tff(fact_4300_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),Y))
     => ( 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,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),M))) ) ) ).

% dvd_power_le
tff(fact_4301_dvd__power__le,axiom,
    ! [X: real,Y: real,N: nat,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y))
     => ( 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,X),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),M))) ) ) ).

% dvd_power_le
tff(fact_4302_dvd__power__le,axiom,
    ! [X: int,Y: int,N: nat,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
     => ( 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(power_power_int(X),N)),aa_nat_int(power_power_int(Y),M))) ) ) ).

% dvd_power_le
tff(fact_4303_dvd__power__le,axiom,
    ! [X: complex,Y: complex,N: nat,M: nat] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),Y))
     => ( 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(power_power_complex(X),N)),aa_nat_complex(power_power_complex(Y),M))) ) ) ).

% dvd_power_le
tff(fact_4304_power__le__dvd,axiom,
    ! [A: code_integer,N: nat,B: code_integer,M: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),B))
     => ( 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(A),M)),B)) ) ) ).

% power_le_dvd
tff(fact_4305_power__le__dvd,axiom,
    ! [A: nat,N: nat,B: 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,A),N)),B))
     => ( 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,A),M)),B)) ) ) ).

% power_le_dvd
tff(fact_4306_power__le__dvd,axiom,
    ! [A: real,N: nat,B: real,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),B))
     => ( 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,A),M)),B)) ) ) ).

% power_le_dvd
tff(fact_4307_power__le__dvd,axiom,
    ! [A: int,N: nat,B: int,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),B))
     => ( 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(power_power_int(A),M)),B)) ) ) ).

% power_le_dvd
tff(fact_4308_power__le__dvd,axiom,
    ! [A: complex,N: nat,B: complex,M: nat] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),N)),B))
     => ( 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(power_power_complex(A),M)),B)) ) ) ).

% power_le_dvd
tff(fact_4309_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: 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(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ).

% le_imp_power_dvd
tff(fact_4310_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: 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,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% le_imp_power_dvd
tff(fact_4311_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: 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,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ).

% le_imp_power_dvd
tff(fact_4312_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: 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(power_power_int(A),M)),aa_nat_int(power_power_int(A),N))) ) ).

% le_imp_power_dvd
tff(fact_4313_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: 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(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N))) ) ).

% le_imp_power_dvd
tff(fact_4314_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(modulo_modulo_int(K),L)),L)) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_4315_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(modulo_modulo_int(K),L))) ) ).

% neg_mod_bound
tff(fact_4316_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_4317_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_4318_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_4319_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_4320_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_4321_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_4322_zdvd__not__zless,axiom,
    ! [M: int,N: 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_int,M),N))
       => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N),M)) ) ) ).

% zdvd_not_zless
tff(fact_4323_mod__int__unique,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R2)))
     => ( aa_int_int(modulo_modulo_int(K),L) = R2 ) ) ).

% mod_int_unique
tff(fact_4324_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_fa,I)))) ) ).

% finite_divisors_int
tff(fact_4325_unity__coeff__ex,axiom,
    ! [P: fun_Co8822979462182099027r_bool,L: code_integer] :
      ( ? [X3: code_integer] : pp(aa_Code_integer_bool(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,L),X3)))
    <=> ? [X3: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,L),plus_p5714425477246183910nteger(X3,zero_z3403309356797280102nteger)))
          & pp(aa_Code_integer_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_4326_unity__coeff__ex,axiom,
    ! [P: fun_complex_bool,L: complex] :
      ( ? [X3: complex] : pp(aa_complex_bool(P,aa_complex_complex(times_times_complex(L),X3)))
    <=> ? [X3: complex] :
          ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,L),aa_complex_complex(plus_plus_complex(X3),zero_zero_complex)))
          & pp(aa_complex_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_4327_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_4328_unity__coeff__ex,axiom,
    ! [P: fun_rat_bool,L: rat] :
      ( ? [X3: rat] : pp(aa_rat_bool(P,aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(plus_plus_rat,X3),zero_zero_rat)))
          & pp(aa_rat_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_4329_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_4330_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_4331_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C3: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C3) ) ) ).

% unit_dvdE
tff(fact_4332_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ~ ( ( A != zero_zero_nat )
         => ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C3) ) ) ).

% unit_dvdE
tff(fact_4333_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ~ ( ( A != zero_zero_int )
         => ! [C3: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C3) ) ) ).

% unit_dvdE
tff(fact_4334_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),D))
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D),C) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_4335_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),D))
           => ( ( aa_int_int(divide_divide_int(B),A) = aa_int_int(divide_divide_int(D),C) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_4336_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),D))
           => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(D,C) )
            <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_4337_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_4338_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(divide_divide_int(B),C)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_4339_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C)))
        <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_4340_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_4341_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),C))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_4342_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C))
        <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_4343_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_4344_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
       => ( ( aa_int_int(divide_divide_int(B),A) = C )
        <=> ( B = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_4345_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
       => ( ( divide6298287555418463151nteger(B,A) = C )
        <=> ( B = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_4346_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_4347_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( ( aa_int_int(divide_divide_int(A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_4348_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
tff(fact_4349_is__unit__power__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),one_one_Code_integer))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_4350_is__unit__power__iff,axiom,
    ! [A: 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,A),N)),one_one_nat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_4351_is__unit__power__iff,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),one_one_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_4352_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_4353_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_4354_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_4355_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(modulo_modulo_int(K),L)),zero_zero_int)) ) ).

% neg_mod_sign
tff(fact_4356_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(modulo_modulo_int(K),L))) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_4357_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_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_4358_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),B)))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),B)),B)) ) ) ).

% pos_mod_conj
tff(fact_4359_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(A),B)),zero_zero_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(modulo_modulo_int(A),B))) ) ) ).

% neg_mod_conj
tff(fact_4360_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_4361_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_4362_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_4363_zdiv__mono__strict,axiom,
    ! [A2: int,B2: int,N: 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),N))
       => ( ( aa_int_int(modulo_modulo_int(A2),N) = zero_zero_int )
         => ( ( aa_int_int(modulo_modulo_int(B2),N) = zero_zero_int )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(A2),N)),aa_int_int(divide_divide_int(B2),N))) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_4364_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D4: nat,X4: nat,Y2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D4),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D4),B))
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),D4) ) ) ) ).

% bezout_add_strong_nat
tff(fact_4365_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_4366_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_4367_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q4: 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),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q4) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Q4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_4368_eucl__rel__int,axiom,
    ! [K: int,L: int] : pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(divide_divide_int(K),L)),aa_int_int(modulo_modulo_int(K),L)))) ).

% eucl_rel_int
tff(fact_4369_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K2: nat,M5: nat] : X != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K2),M5) ).

% prod_decode_aux.cases
tff(fact_4370_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_ey,M)))) ) ).

% finite_divisors_nat
tff(fact_4371_full__exhaustive__int_H_Ocases,axiom,
    ! [X: produc4441460161354545860nt_int] :
      ~ ! [F2: fun_Pr9164807099015279978e_term,D4: int,I2: int] : X != aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,F2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D4),I2)) ).

% full_exhaustive_int'.cases
tff(fact_4372_exhaustive__int_H_Ocases,axiom,
    ! [X: produc6768802789813714928nt_int] :
      ~ ! [F2: fun_in6429348457993719614e_term,D4: int,I2: int] : X != aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,F2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D4),I2)) ).

% exhaustive_int'.cases
tff(fact_4373_small__lazy_H_Ocases,axiom,
    ! [X: product_prod_int_int] :
      ~ ! [D4: int,I2: int] : X != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D4),I2) ).

% small_lazy'.cases
tff(fact_4374_even__zero,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_z3403309356797280102nteger)) ).

% even_zero
tff(fact_4375_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_4376_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_4377_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4378_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) = aa_int_int(divide_divide_int(one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4379_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4380_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4381_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(divide_divide_int(one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4382_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4383_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ~ ( ( A != 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),A) = B3 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B3) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B3) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4384_is__unitE,axiom,
    ! [A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ~ ( ( A != 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(divide_divide_int(one_one_int),A) = B3 )
                 => ( ( aa_int_int(divide_divide_int(one_one_int),B3) = A )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B3) = one_one_int )
                     => ( aa_int_int(divide_divide_int(C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4385_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B3: code_integer] :
              ( ( B3 != zero_z3403309356797280102nteger )
             => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B3),one_one_Code_integer))
               => ( ( divide6298287555418463151nteger(one_one_Code_integer,A) = B3 )
                 => ( ( divide6298287555418463151nteger(one_one_Code_integer,B3) = A )
                   => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B3) = one_one_Code_integer )
                     => ( divide6298287555418463151nteger(C,A) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4386_dvd__power__iff,axiom,
    ! [X: code_integer,M: nat,N: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),M)),aa_nat_Code_integer(power_8256067586552552935nteger(X),N)))
      <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),one_one_Code_integer))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% dvd_power_iff
tff(fact_4387_dvd__power__iff,axiom,
    ! [X: nat,M: nat,N: nat] :
      ( ( X != 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,X),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)))
      <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),one_one_nat))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% dvd_power_iff
tff(fact_4388_dvd__power__iff,axiom,
    ! [X: int,M: nat,N: nat] :
      ( ( X != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(X),M)),aa_nat_int(power_power_int(X),N)))
      <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),one_one_int))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% dvd_power_iff
tff(fact_4389_dvd__power,axiom,
    ! [N: nat,X: code_integer] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_Code_integer ) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),aa_nat_Code_integer(power_8256067586552552935nteger(X),N))) ) ).

% dvd_power
tff(fact_4390_dvd__power,axiom,
    ! [N: nat,X: rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_rat ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),aa_nat_rat(power_power_rat(X),N))) ) ).

% dvd_power
tff(fact_4391_dvd__power,axiom,
    ! [N: nat,X: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_nat ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N))) ) ).

% dvd_power
tff(fact_4392_dvd__power,axiom,
    ! [N: nat,X: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_real ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ).

% dvd_power
tff(fact_4393_dvd__power,axiom,
    ! [N: nat,X: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_int ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_nat_int(power_power_int(X),N))) ) ).

% dvd_power
tff(fact_4394_dvd__power,axiom,
    ! [N: nat,X: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_complex ) )
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),aa_nat_complex(power_power_complex(X),N))) ) ).

% dvd_power
tff(fact_4395_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(modulo_modulo_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_4396_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(modulo_modulo_int(K),L) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_4397_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_4398_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_4399_dvd__minus__add,axiom,
    ! [Q4: nat,N: nat,R2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q4),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q4),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),Q4)))
        <=> 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)),Q4)))) ) ) ) ).

% dvd_minus_add
tff(fact_4400_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_4401_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_4402_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_4403_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_4404_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_4405_power__mono__odd,axiom,
    ! [N: nat,A: real,B: 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,A),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N))) ) ) ).

% power_mono_odd
tff(fact_4406_power__mono__odd,axiom,
    ! [N: nat,A: rat,B: 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,A),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ).

% power_mono_odd
tff(fact_4407_power__mono__odd,axiom,
    ! [N: nat,A: int,B: 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,A),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ).

% power_mono_odd
tff(fact_4408_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_4409_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_4410_split__zmod,axiom,
    ! [P: fun_int_bool,N: int,K: int] :
      ( pp(aa_int_bool(P,aa_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_4411_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),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,B),R2))
         => ( aa_int_int(modulo_modulo_int(A),B) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_4412_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),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),B))
         => ( aa_int_int(modulo_modulo_int(A),B) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_4413_even__unset__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(M,A)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_4414_even__unset__bit__iff,axiom,
    ! [M: nat,A: 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),A)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_4415_even__unset__bit__iff,axiom,
    ! [M: nat,A: 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_se4203085406695923979it_int(M),A)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_4416_even__set__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(M,A)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_4417_even__set__bit__iff,axiom,
    ! [M: nat,A: 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),A)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_4418_even__set__bit__iff,axiom,
    ! [M: nat,A: 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_se7879613467334960850it_int(M),A)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_4419_mod2__eq__if,axiom,
    ! [A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),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))),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ) ).

% mod2_eq_if
tff(fact_4420_mod2__eq__if,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
       => ( aa_int_int(modulo_modulo_int(A),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))),A))
       => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ) ).

% mod2_eq_if
tff(fact_4421_mod2__eq__if,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
       => ( modulo364778990260209775nteger(A,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))),A))
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ) ).

% mod2_eq_if
tff(fact_4422_parity__cases,axiom,
    ! [A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),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))),A))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_4423_parity__cases,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
       => ( aa_int_int(modulo_modulo_int(A),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))),A))
         => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_4424_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
       => ( modulo364778990260209775nteger(A,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))),A))
         => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_4425_zero__le__even__power,axiom,
    ! [N: nat,A: 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,A),N))) ) ).

% zero_le_even_power
tff(fact_4426_zero__le__even__power,axiom,
    ! [N: nat,A: 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(A),N))) ) ).

% zero_le_even_power
tff(fact_4427_zero__le__even__power,axiom,
    ! [N: nat,A: 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(power_power_int(A),N))) ) ).

% zero_le_even_power
tff(fact_4428_zero__le__odd__power,axiom,
    ! [N: nat,A: 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,A),N)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% zero_le_odd_power
tff(fact_4429_zero__le__odd__power,axiom,
    ! [N: nat,A: 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(A),N)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% zero_le_odd_power
tff(fact_4430_zero__le__odd__power,axiom,
    ! [N: nat,A: 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(power_power_int(A),N)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).

% zero_le_odd_power
tff(fact_4431_zero__le__power__eq,axiom,
    ! [A: 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,A),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),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_4432_zero__le__power__eq,axiom,
    ! [A: 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(A),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),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_4433_zero__le__power__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),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),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_4434_verit__le__mono__div__int,axiom,
    ! [A2: int,B2: int,N: 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),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(divide_divide_int(A2),N)),aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(B2),N)),zero_zero_int),one_one_int),zero_zero_int))),aa_int_int(divide_divide_int(B2),N))) ) ) ).

% verit_le_mono_div_int
tff(fact_4435_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(divide_divide_int(N),K)),aa_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_4436_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(divide_divide_int(N),K)),aa_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_4437_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N2: nat] : X != aa_nat_nat(suc,N2) ) ).

% list_decode.cases
tff(fact_4438_zero__less__power__eq,axiom,
    ! [A: 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,A),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))
          & ( A != 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),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_4439_zero__less__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),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))
          & ( A != 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),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_4440_zero__less__power__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),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))
          & ( A != 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),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_4441_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_4442_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(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_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_4443_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))),divide6298287555418463151nteger(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_4444_power__le__zero__eq,axiom,
    ! [A: 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,A),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,A),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))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_4445_power__le__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),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,A),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))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_4446_power__le__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),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,A),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))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_4447_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_4448_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_4449_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(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
    <=> ( ( aa_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_4450_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))),divide6298287555418463151nteger(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_4451_even__mult__exp__div__exp__iff,axiom,
    ! [A: 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,A),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,A),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_4452_even__mult__exp__div__exp__iff,axiom,
    ! [A: 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(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_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(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(divide_divide_int(A),aa_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_4453_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),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))),divide6298287555418463151nteger(A,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_4454_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
         => ~ ! [Va2: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ~ ( ( 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,Va2))))
                     => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Va2))))
                     => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_4455_option_Osize__gen_I2_J,axiom,
    ! [X: fun_nat_nat,X22: nat] : size_option_nat(X,some_nat(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(X,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_4456_option_Osize__gen_I2_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat,X22: product_prod_nat_nat] : size_o8335143837870341156at_nat(X,some_P7363390416028606310at_nat(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(X,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_4457_option_Osize__gen_I2_J,axiom,
    ! [X: fun_num_nat,X22: num] : size_option_num(X,some_num(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(X,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_4458_diff__shunt__var,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat(X,Y) = bot_bo2099793752762293965at_nat )
    <=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X),Y)) ) ).

% diff_shunt_var
tff(fact_4459_diff__shunt__var,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( minus_minus_set_real(X,Y) = bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(X),Y)) ) ).

% diff_shunt_var
tff(fact_4460_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( minus_minus_set_nat(X,Y) = bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y)) ) ).

% diff_shunt_var
tff(fact_4461_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( minus_minus_set_int(X,Y) = bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(X),Y)) ) ).

% diff_shunt_var
tff(fact_4462_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_fb(L)),Qr) ).

% divmod_step_def
tff(fact_4463_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_fc(L)),Qr) ).

% divmod_step_def
tff(fact_4464_divmod__step__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_fd(L),Qr) ).

% divmod_step_def
tff(fact_4465_take__bit__rec,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( N = zero_zero_nat )
       => ( bit_se1745604003318907178nteger(N,A) = zero_z3403309356797280102nteger ) )
      & ( ( N != zero_zero_nat )
       => ( bit_se1745604003318907178nteger(N,A) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))),numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).

% take_bit_rec
tff(fact_4466_take__bit__rec,axiom,
    ! [N: nat,A: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),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(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),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(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% take_bit_rec
tff(fact_4467_take__bit__rec,axiom,
    ! [N: nat,A: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_int_int(bit_se2923211474154528505it_int(N),A) = zero_zero_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_int_int(bit_se2923211474154528505it_int(N),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(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) ) ) ).

% take_bit_rec
tff(fact_4468_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_4469_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))))
       => ( vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va))) = 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)))),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))))),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))))
       => ( vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va))) = 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))))),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))))),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_4470_even__flip__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(M,A)))
    <=> ~ ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_4471_even__flip__bit__iff,axiom,
    ! [M: nat,A: 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),A)))
    <=> ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_4472_even__flip__bit__iff,axiom,
    ! [M: nat,A: 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_se2159334234014336723it_int(M),A)))
    <=> ~ ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_4473_intind,axiom,
    ! [I: nat,N: nat,P: fun_nat_bool,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_nat_bool(P,X))
       => pp(aa_nat_bool(P,aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,replicate_nat(N,X)),I))) ) ) ).

% intind
tff(fact_4474_intind,axiom,
    ! [I: nat,N: nat,P: fun_int_bool,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_int_bool(P,X))
       => pp(aa_int_bool(P,aa_nat_int(nth_int(replicate_int(N,X)),I))) ) ) ).

% intind
tff(fact_4475_intind,axiom,
    ! [I: nat,N: nat,P: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_VEBT_VEBT_bool(P,X))
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(replicate_VEBT_VEBT(N,X),I))) ) ) ).

% intind
tff(fact_4476_take__bit__of__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),zero_zero_nat) = zero_zero_nat ).

% take_bit_of_0
tff(fact_4477_take__bit__of__0,axiom,
    ! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(N),zero_zero_int) = zero_zero_int ).

% take_bit_of_0
tff(fact_4478_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2159334234014336723it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% flip_bit_negative_int_iff
tff(fact_4479_replicate__eq__replicate,axiom,
    ! [M: nat,X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(M,X) = replicate_VEBT_VEBT(N,Y) )
    <=> ( ( M = N )
        & ( ( M != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_4480_length__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(N,X)) = N ).

% length_replicate
tff(fact_4481_length__replicate,axiom,
    ! [N: nat,X: bool] : size_size_list_bool(replicate_bool(N,X)) = N ).

% length_replicate
tff(fact_4482_length__replicate,axiom,
    ! [N: nat,X: nat] : size_size_list_nat(replicate_nat(N,X)) = N ).

% length_replicate
tff(fact_4483_length__replicate,axiom,
    ! [N: nat,X: int] : size_size_list_int(replicate_int(N,X)) = N ).

% length_replicate
tff(fact_4484_case__prod__conv,axiom,
    ! [F: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(F,A),B)) ) ).

% case_prod_conv
tff(fact_4485_case__prod__conv,axiom,
    ! [F: fun_na1668100129766719005at_nat,A: nat,B: nat] : aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,A),B) ).

% case_prod_conv
tff(fact_4486_case__prod__conv,axiom,
    ! [F: fun_na4786964484885125043t_bool,A: nat,B: nat] : aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(F,A),B) ).

% case_prod_conv
tff(fact_4487_case__prod__conv,axiom,
    ! [F: fun_in1696471567231710611nt_int,A: int,B: int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,A),B) ).

% case_prod_conv
tff(fact_4488_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_bool,A: int,B: int] :
      ( pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(F,A),B)) ) ).

% case_prod_conv
tff(fact_4489_take__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,zero_zero_nat),A) = zero_zero_nat ).

% take_bit_0
tff(fact_4490_take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(zero_zero_nat),A) = zero_zero_int ).

% take_bit_0
tff(fact_4491_in__set__replicate,axiom,
    ! [X: product_prod_nat_nat,N: nat,Y: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4492_in__set__replicate,axiom,
    ! [X: real,N: nat,Y: real] :
      ( pp(aa_set_real_bool(member_real(X),set_real2(replicate_real(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4493_in__set__replicate,axiom,
    ! [X: set_nat,N: nat,Y: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(replicate_set_nat(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4494_in__set__replicate,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa_set_int_bool(member_int(X),set_int2(replicate_int(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4495_in__set__replicate,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(replicate_nat(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4496_in__set__replicate,axiom,
    ! [X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4497_Bex__set__replicate,axiom,
    ! [N: nat,A: int,P: fun_int_bool] :
      ( ? [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),set_int2(replicate_int(N,A))))
          & pp(aa_int_bool(P,X3)) )
    <=> ( pp(aa_int_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_4498_Bex__set__replicate,axiom,
    ! [N: nat,A: nat,P: fun_nat_bool] :
      ( ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(replicate_nat(N,A))))
          & pp(aa_nat_bool(P,X3)) )
    <=> ( pp(aa_nat_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_4499_Bex__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ? [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A))))
          & pp(aa_VEBT_VEBT_bool(P,X3)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_4500_Ball__set__replicate,axiom,
    ! [N: nat,A: int,P: fun_int_bool] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),set_int2(replicate_int(N,A))))
         => pp(aa_int_bool(P,X3)) )
    <=> ( pp(aa_int_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_4501_Ball__set__replicate,axiom,
    ! [N: nat,A: nat,P: fun_nat_bool] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(replicate_nat(N,A))))
         => pp(aa_nat_bool(P,X3)) )
    <=> ( pp(aa_nat_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_4502_Ball__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A))))
         => pp(aa_VEBT_VEBT_bool(P,X3)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_4503_nth__replicate,axiom,
    ! [I: nat,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,replicate_nat(N,X)),I) = X ) ) ).

% nth_replicate
tff(fact_4504_nth__replicate,axiom,
    ! [I: nat,N: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_int(nth_int(replicate_int(N,X)),I) = X ) ) ).

% nth_replicate
tff(fact_4505_nth__replicate,axiom,
    ! [I: nat,N: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( nth_VEBT_VEBT(replicate_VEBT_VEBT(N,X),I) = X ) ) ).

% nth_replicate
tff(fact_4506_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),one_one_nat) = zero_zero_nat )
    <=> ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_4507_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),one_one_int) = zero_zero_int )
    <=> ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_4508_even__take__bit__eq,axiom,
    ! [N: nat,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1745604003318907178nteger(N,A)))
    <=> ( ( N = zero_zero_nat )
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)) ) ) ).

% even_take_bit_eq
tff(fact_4509_even__take__bit__eq,axiom,
    ! [N: nat,A: 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_se2925701944663578781it_nat,N),A)))
    <=> ( ( 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))),A)) ) ) ).

% even_take_bit_eq
tff(fact_4510_even__take__bit__eq,axiom,
    ! [N: nat,A: 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_se2923211474154528505it_int(N),A)))
    <=> ( ( N = zero_zero_nat )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ) ).

% even_take_bit_eq
tff(fact_4511_take__bit__Suc__0,axiom,
    ! [A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,zero_zero_nat),A) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_4512_take__bit__Suc__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,zero_zero_nat)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_4513_take__bit__Suc__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,zero_zero_nat)),A) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_4514_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_4515_take__bit__flip__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) ) ) ) ).

% take_bit_flip_bit_eq
tff(fact_4516_take__bit__flip__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_se2159334234014336723it_int(M),A)) = aa_int_int(bit_se2923211474154528505it_int(N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_se2159334234014336723it_int(M),A)) = aa_int_int(bit_se2159334234014336723it_int(M),aa_int_int(bit_se2923211474154528505it_int(N),A)) ) ) ) ).

% take_bit_flip_bit_eq
tff(fact_4517_take__bit__tightened,axiom,
    ! [N: nat,A: nat,B: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),B) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),B) ) ) ) ).

% take_bit_tightened
tff(fact_4518_take__bit__tightened,axiom,
    ! [N: nat,A: int,B: int,M: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),A) = aa_int_int(bit_se2923211474154528505it_int(N),B) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_int_int(bit_se2923211474154528505it_int(M),A) = aa_int_int(bit_se2923211474154528505it_int(M),B) ) ) ) ).

% take_bit_tightened
tff(fact_4519_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q4: 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),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),Q4))) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_4520_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_4521_old_Oprod_Ocase,axiom,
    ! [F: fun_nat_fun_nat_bool,X1: nat,X22: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X22)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(F,X1),X22)) ) ).

% old.prod.case
tff(fact_4522_old_Oprod_Ocase,axiom,
    ! [F: fun_na1668100129766719005at_nat,X1: nat,X22: nat] : aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X22)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X1),X22) ).

% old.prod.case
tff(fact_4523_old_Oprod_Ocase,axiom,
    ! [F: fun_na4786964484885125043t_bool,X1: nat,X22: nat] : aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X22)) = aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(F,X1),X22) ).

% old.prod.case
tff(fact_4524_old_Oprod_Ocase,axiom,
    ! [F: fun_in1696471567231710611nt_int,X1: int,X22: int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X1),X22) ).

% old.prod.case
tff(fact_4525_old_Oprod_Ocase,axiom,
    ! [F: fun_int_fun_int_bool,X1: int,X22: int] :
      ( pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(F,X1),X22)) ) ).

% old.prod.case
tff(fact_4526_split__cong,axiom,
    ! [Q4: product_prod_nat_nat,F: fun_nat_fun_nat_bool,G: fun_nat_fun_nat_bool,P2: product_prod_nat_nat] :
      ( ! [X4: nat,Y2: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y2) = Q4 )
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(F,X4),Y2))
          <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(G,X4),Y2)) ) )
     => ( ( P2 = Q4 )
       => ( pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(F),P2))
        <=> pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(G),Q4)) ) ) ) ).

% split_cong
tff(fact_4527_split__cong,axiom,
    ! [Q4: product_prod_nat_nat,F: fun_na1668100129766719005at_nat,G: fun_na1668100129766719005at_nat,P2: product_prod_nat_nat] :
      ( ! [X4: nat,Y2: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y2) = Q4 )
         => ( aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X4),Y2) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(G,X4),Y2) ) )
     => ( ( P2 = Q4 )
       => ( aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),P2) = aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(G),Q4) ) ) ) ).

% split_cong
tff(fact_4528_split__cong,axiom,
    ! [Q4: product_prod_nat_nat,F: fun_na4786964484885125043t_bool,G: fun_na4786964484885125043t_bool,P2: product_prod_nat_nat] :
      ( ! [X4: nat,Y2: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y2) = Q4 )
         => ( aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(F,X4),Y2) = aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(G,X4),Y2) ) )
     => ( ( P2 = Q4 )
       => ( aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(F),P2) = aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(G),Q4) ) ) ) ).

% split_cong
tff(fact_4529_split__cong,axiom,
    ! [Q4: product_prod_int_int,F: fun_in1696471567231710611nt_int,G: fun_in1696471567231710611nt_int,P2: product_prod_int_int] :
      ( ! [X4: int,Y2: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y2) = Q4 )
         => ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X4),Y2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(G,X4),Y2) ) )
     => ( ( P2 = Q4 )
       => ( aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),P2) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(G),Q4) ) ) ) ).

% split_cong
tff(fact_4530_split__cong,axiom,
    ! [Q4: product_prod_int_int,F: fun_int_fun_int_bool,G: fun_int_fun_int_bool,P2: product_prod_int_int] :
      ( ! [X4: int,Y2: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y2) = Q4 )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(F,X4),Y2))
          <=> pp(aa_int_bool(aa_int_fun_int_bool(G,X4),Y2)) ) )
     => ( ( P2 = Q4 )
       => ( pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(F),P2))
        <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(G),Q4)) ) ) ) ).

% split_cong
tff(fact_4531_case__prodE2,axiom,
    ! [Q: fun_bool_bool,P: fun_nat_fun_nat_bool,Z: product_prod_nat_nat] :
      ( pp(aa_bool_bool(Q,aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(P),Z)))
     => ~ ! [X4: nat,Y2: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y2) )
           => ~ pp(aa_bool_bool(Q,aa_nat_bool(aa_nat_fun_nat_bool(P,X4),Y2))) ) ) ).

% case_prodE2
tff(fact_4532_case__prodE2,axiom,
    ! [Q: fun_fu7919257609373989805t_bool,P: fun_na1668100129766719005at_nat,Z: product_prod_nat_nat] :
      ( pp(aa_fun4519828053337421206t_bool(Q,aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(P),Z)))
     => ~ ! [X4: nat,Y2: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y2) )
           => ~ pp(aa_fun4519828053337421206t_bool(Q,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(P,X4),Y2))) ) ) ).

% case_prodE2
tff(fact_4533_case__prodE2,axiom,
    ! [Q: fun_fu5641360819561228055l_bool,P: fun_na4786964484885125043t_bool,Z: product_prod_nat_nat] :
      ( pp(aa_fun8663595511526100224l_bool(Q,aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(P),Z)))
     => ~ ! [X4: nat,Y2: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y2) )
           => ~ pp(aa_fun8663595511526100224l_bool(Q,aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(P,X4),Y2))) ) ) ).

% case_prodE2
tff(fact_4534_case__prodE2,axiom,
    ! [Q: fun_Pr803113274115125473t_bool,P: fun_in1696471567231710611nt_int,Z: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(Q,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(P),Z)))
     => ~ ! [X4: int,Y2: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y2) )
           => ~ pp(aa_Pro4501881432599481050t_bool(Q,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(P,X4),Y2))) ) ) ).

% case_prodE2
tff(fact_4535_case__prodE2,axiom,
    ! [Q: fun_bool_bool,P: fun_int_fun_int_bool,Z: product_prod_int_int] :
      ( pp(aa_bool_bool(Q,aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(P),Z)))
     => ~ ! [X4: int,Y2: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y2) )
           => ~ pp(aa_bool_bool(Q,aa_int_bool(aa_int_fun_int_bool(P,X4),Y2))) ) ) ).

% case_prodE2
tff(fact_4536_case__prod__eta,axiom,
    ! [F: fun_Pr7757163693017731481t_bool] : produc8955392012077580105t_bool(aTP_Lamm_fe(F)) = F ).

% case_prod_eta
tff(fact_4537_case__prod__eta,axiom,
    ! [F: fun_Pr2477239776880934807at_nat] : produc4528391901023682385at_nat(aTP_Lamm_ff(F)) = F ).

% case_prod_eta
tff(fact_4538_case__prod__eta,axiom,
    ! [F: fun_Pr248683899324676269t_bool] : produc5023667602222594023t_bool(aTP_Lamm_fg(F)) = F ).

% case_prod_eta
tff(fact_4539_case__prod__eta,axiom,
    ! [F: fun_Pr2478283820680522243nt_int] : produc4245557441103728435nt_int(aTP_Lamm_fh(F)) = F ).

% case_prod_eta
tff(fact_4540_case__prod__eta,axiom,
    ! [F: fun_Pr803113274115125473t_bool] : produc8560705615207772305t_bool(aTP_Lamm_fi(F)) = F ).

% case_prod_eta
tff(fact_4541_cond__case__prod__eta,axiom,
    ! [F: fun_nat_fun_nat_bool,G: fun_Pr7757163693017731481t_bool] :
      ( ! [X4: nat,Y2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(F,X4),Y2))
        <=> pp(aa_Pro2232559814647311250t_bool(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y2))) )
     => ( produc8955392012077580105t_bool(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_4542_cond__case__prod__eta,axiom,
    ! [F: fun_na1668100129766719005at_nat,G: fun_Pr2477239776880934807at_nat] :
      ( ! [X4: nat,Y2: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X4),Y2) = aa_Pro8023792847356852296at_nat(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y2))
     => ( produc4528391901023682385at_nat(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_4543_cond__case__prod__eta,axiom,
    ! [F: fun_na4786964484885125043t_bool,G: fun_Pr248683899324676269t_bool] :
      ( ! [X4: nat,Y2: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(F,X4),Y2) = aa_Pro595088089689268830t_bool(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y2))
     => ( produc5023667602222594023t_bool(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_4544_cond__case__prod__eta,axiom,
    ! [F: fun_in1696471567231710611nt_int,G: fun_Pr2478283820680522243nt_int] :
      ( ! [X4: int,Y2: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X4),Y2) = aa_Pro6970504127699319548nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y2))
     => ( produc4245557441103728435nt_int(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_4545_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_bool,G: fun_Pr803113274115125473t_bool] :
      ( ! [X4: int,Y2: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(F,X4),Y2))
        <=> pp(aa_Pro4501881432599481050t_bool(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y2))) )
     => ( produc8560705615207772305t_bool(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_4546_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(bit_se2923211474154528505it_int(M),K)),aa_int_int(bit_se2923211474154528505it_int(N),K))) ) ).

% take_bit_tightened_less_eq_int
tff(fact_4547_not__take__bit__negative,axiom,
    ! [N: nat,K: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),zero_zero_int)) ).

% not_take_bit_negative
tff(fact_4548_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(bit_se2923211474154528505it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_4549_take__bit__unset__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) ) ) ) ).

% take_bit_unset_bit_eq
tff(fact_4550_take__bit__unset__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_se4203085406695923979it_int(M),A)) = aa_int_int(bit_se2923211474154528505it_int(N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_se4203085406695923979it_int(M),A)) = aa_int_int(bit_se4203085406695923979it_int(M),aa_int_int(bit_se2923211474154528505it_int(N),A)) ) ) ) ).

% take_bit_unset_bit_eq
tff(fact_4551_take__bit__set__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) ) ) ) ).

% take_bit_set_bit_eq
tff(fact_4552_take__bit__set__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_se7879613467334960850it_int(M),A)) = aa_int_int(bit_se2923211474154528505it_int(N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_se7879613467334960850it_int(M),A)) = aa_int_int(bit_se7879613467334960850it_int(M),aa_int_int(bit_se2923211474154528505it_int(N),A)) ) ) ) ).

% take_bit_set_bit_eq
tff(fact_4553_replicate__eqI,axiom,
    ! [Xs: list_P6011104703257516679at_nat,N: nat,X: product_prod_nat_nat] :
      ( ( size_s5460976970255530739at_nat(Xs) = N )
     => ( ! [Y2: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y2),set_Pr5648618587558075414at_nat(Xs)))
           => ( Y2 = X ) )
       => ( Xs = replic4235873036481779905at_nat(N,X) ) ) ) ).

% replicate_eqI
tff(fact_4554_replicate__eqI,axiom,
    ! [Xs: list_real,N: nat,X: real] :
      ( ( size_size_list_real(Xs) = N )
     => ( ! [Y2: real] :
            ( pp(aa_set_real_bool(member_real(Y2),set_real2(Xs)))
           => ( Y2 = X ) )
       => ( Xs = replicate_real(N,X) ) ) ) ).

% replicate_eqI
tff(fact_4555_replicate__eqI,axiom,
    ! [Xs: list_set_nat,N: nat,X: set_nat] :
      ( ( size_s3254054031482475050et_nat(Xs) = N )
     => ( ! [Y2: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(Y2),set_set_nat2(Xs)))
           => ( Y2 = X ) )
       => ( Xs = replicate_set_nat(N,X) ) ) ) ).

% replicate_eqI
tff(fact_4556_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = N )
     => ( ! [Y2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y2),set_VEBT_VEBT2(Xs)))
           => ( Y2 = X ) )
       => ( Xs = replicate_VEBT_VEBT(N,X) ) ) ) ).

% replicate_eqI
tff(fact_4557_replicate__eqI,axiom,
    ! [Xs: list_bool,N: nat,X: bool] :
      ( ( size_size_list_bool(Xs) = N )
     => ( ! [Y2: bool] :
            ( member_bool(Y2,set_bool2(Xs))
           => ( pp(Y2)
            <=> pp(X) ) )
       => ( Xs = replicate_bool(N,X) ) ) ) ).

% replicate_eqI
tff(fact_4558_replicate__eqI,axiom,
    ! [Xs: list_nat,N: nat,X: nat] :
      ( ( size_size_list_nat(Xs) = N )
     => ( ! [Y2: nat] :
            ( pp(aa_set_nat_bool(member_nat(Y2),set_nat2(Xs)))
           => ( Y2 = X ) )
       => ( Xs = replicate_nat(N,X) ) ) ) ).

% replicate_eqI
tff(fact_4559_replicate__eqI,axiom,
    ! [Xs: list_int,N: nat,X: int] :
      ( ( size_size_list_int(Xs) = N )
     => ( ! [Y2: int] :
            ( pp(aa_set_int_bool(member_int(Y2),set_int2(Xs)))
           => ( Y2 = X ) )
       => ( Xs = replicate_int(N,X) ) ) ) ).

% replicate_eqI
tff(fact_4560_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Xs)))
         => ( X4 = X ) )
     => ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_4561_replicate__length__same,axiom,
    ! [Xs: list_bool,X: bool] :
      ( ! [X4: bool] :
          ( member_bool(X4,set_bool2(Xs))
         => ( pp(X4)
          <=> pp(X) ) )
     => ( replicate_bool(size_size_list_bool(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_4562_replicate__length__same,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
         => ( X4 = X ) )
     => ( replicate_nat(size_size_list_nat(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_4563_replicate__length__same,axiom,
    ! [Xs: list_int,X: int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs)))
         => ( X4 = X ) )
     => ( replicate_int(size_size_list_int(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_4564_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_4565_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_4566_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_4567_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(bit_se2923211474154528505it_int(N),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% take_bit_int_less_exp
tff(fact_4568_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_4569_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger(N,A) = zero_z3403309356797280102nteger )
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)),A)) ) ).

% take_bit_eq_0_iff
tff(fact_4570_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = 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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),A)) ) ).

% take_bit_eq_0_iff
tff(fact_4571_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),A) = zero_zero_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),A)) ) ).

% take_bit_eq_0_iff
tff(fact_4572_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_4573_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(bit_se2923211474154528505it_int(N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_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_4574_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(bit_se2923211474154528505it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_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_4575_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(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_4576_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_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(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_4577_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(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(bit_se2923211474154528505it_int(N),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ) ).

% take_bit_int_less_eq
tff(fact_4578_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(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_int_int(bit_se2923211474154528505it_int(N),K))) ) ).

% take_bit_int_greater_eq
tff(fact_4579_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
         => ( bit_se1745604003318907178nteger(N,A) = zero_z3403309356797280102nteger ) )
        & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
         => ( bit_se1745604003318907178nteger(N,A) = minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N),one_one_Code_integer) ) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_4580_stable__imp__take__bit__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = 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))),A))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = 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) ) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_4581_stable__imp__take__bit__eq,axiom,
    ! [A: int,N: nat] :
      ( ( aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
         => ( aa_int_int(bit_se2923211474154528505it_int(N),A) = 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))),A))
         => ( aa_int_int(bit_se2923211474154528505it_int(N),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) ) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_4582_option_Osize__gen_I1_J,axiom,
    ! [X: fun_nat_nat] : size_option_nat(X,none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_4583_option_Osize__gen_I1_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(X,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_4584_option_Osize__gen_I1_J,axiom,
    ! [X: fun_num_nat] : size_option_num(X,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_4585_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_fb(L)),Qr) ).

% divmod_step_nat_def
tff(fact_4586_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_fc(L)),Qr) ).

% divmod_step_int_def
tff(fact_4587_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,N)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_fj),unique5052692396658037445od_int(M,N)) ).

% divmod_algorithm_code(5)
tff(fact_4588_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] : unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,N)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_fk),unique5055182867167087721od_nat(M,N)) ).

% divmod_algorithm_code(5)
tff(fact_4589_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] : unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,N)) = produc6916734918728496179nteger(aTP_Lamm_fl,unique3479559517661332726nteger(M,N)) ).

% divmod_algorithm_code(5)
tff(fact_4590_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) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_fm),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_4591_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),X))
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat)) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [Va2: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( ( 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,Va2))))
                       => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,Va2))))
                       => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),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,Va2))),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,Va2)))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_4592_flip__bit__0,axiom,
    ! [A: code_integer] : bit_se1345352211410354436nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(zero_n356916108424825756nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_4593_flip__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_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))),A))),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,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_4594_flip__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2159334234014336723it_int(zero_zero_nat),A) = 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))),A))),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(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_4595_set__decode__0,axiom,
    ! [X: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),nat_set_decode(X)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X)) ) ).

% set_decode_0
tff(fact_4596_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( 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,A2)))
      <=> ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) ) ) ).

% even_set_encode_iff
tff(fact_4597_one__mod__2__pow__eq,axiom,
    ! [N: nat] : modulo364778990260209775nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_4598_one__mod__2__pow__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,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)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_4599_one__mod__2__pow__eq,axiom,
    ! [N: nat] : aa_int_int(modulo_modulo_int(one_one_int),aa_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_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_4600_case__prodI2,axiom,
    ! [P2: produc5207004643709237345nteger,C: fun_fu6012121808138353003r_bool] :
      ( ! [A3: fun_Co6545587727236015215e_term,B3: produc8923325533196201883nteger] :
          ( ( P2 = aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,A3),B3) )
         => pp(aa_Pro5843519176478241656r_bool(aa_fun5646951941296221740r_bool(C,A3),B3)) )
     => pp(aa_Pro7150874649148492216r_bool(produc401098500374025047r_bool(C),P2)) ) ).

% case_prodI2
tff(fact_4601_case__prodI2,axiom,
    ! [P2: produc1616345218596728535nteger,C: fun_fu1674507805476766837r_bool] :
      ( ! [A3: fun_Pr272337284026993017e_term,B3: produc8923325533196201883nteger] :
          ( ( P2 = aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,A3),B3) )
         => pp(aa_Pro5843519176478241656r_bool(aa_fun5449300841655810358r_bool(C,A3),B3)) )
     => pp(aa_Pro3640759179027349698r_bool(produc1236900166980094561r_bool(C),P2)) ) ).

% case_prodI2
tff(fact_4602_case__prodI2,axiom,
    ! [P2: produc4441460161354545860nt_int,C: fun_fu4196931294019812808t_bool] :
      ( ! [A3: fun_Pr9164807099015279978e_term,B3: product_prod_int_int] :
          ( ( P2 = aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,A3),B3) )
         => pp(aa_Pro4501881432599481050t_bool(aa_fun5300525129335604105t_bool(C,A3),B3)) )
     => pp(aa_Pro5031108791230793749t_bool(produc625993674133193396t_bool(C),P2)) ) ).

% case_prodI2
tff(fact_4603_case__prodI2,axiom,
    ! [P2: produc6768802789813714928nt_int,C: fun_fu5117548769634986908t_bool] :
      ( ! [A3: fun_in6429348457993719614e_term,B3: product_prod_int_int] :
          ( ( P2 = aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,A3),B3) )
         => pp(aa_Pro4501881432599481050t_bool(aa_fun4446166836006121821t_bool(C,A3),B3)) )
     => pp(aa_Pro7591400126937138665t_bool(produc6627716702706253960t_bool(C),P2)) ) ).

% case_prodI2
tff(fact_4604_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C: fun_nat_fun_nat_bool] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(C,A3),B3)) )
     => pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(C),P2)) ) ).

% case_prodI2
tff(fact_4605_case__prodI2,axiom,
    ! [P2: product_prod_int_int,C: fun_int_fun_int_bool] :
      ( ! [A3: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => pp(aa_int_bool(aa_int_fun_int_bool(C,A3),B3)) )
     => pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(C),P2)) ) ).

% case_prodI2
tff(fact_4606_case__prodI,axiom,
    ! [F: fun_fu6012121808138353003r_bool,A: fun_Co6545587727236015215e_term,B: produc8923325533196201883nteger] :
      ( pp(aa_Pro5843519176478241656r_bool(aa_fun5646951941296221740r_bool(F,A),B))
     => pp(aa_Pro7150874649148492216r_bool(produc401098500374025047r_bool(F),aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,A),B))) ) ).

% case_prodI
tff(fact_4607_case__prodI,axiom,
    ! [F: fun_fu1674507805476766837r_bool,A: fun_Pr272337284026993017e_term,B: produc8923325533196201883nteger] :
      ( pp(aa_Pro5843519176478241656r_bool(aa_fun5449300841655810358r_bool(F,A),B))
     => pp(aa_Pro3640759179027349698r_bool(produc1236900166980094561r_bool(F),aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,A),B))) ) ).

% case_prodI
tff(fact_4608_case__prodI,axiom,
    ! [F: fun_fu4196931294019812808t_bool,A: fun_Pr9164807099015279978e_term,B: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_fun5300525129335604105t_bool(F,A),B))
     => pp(aa_Pro5031108791230793749t_bool(produc625993674133193396t_bool(F),aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,A),B))) ) ).

% case_prodI
tff(fact_4609_case__prodI,axiom,
    ! [F: fun_fu5117548769634986908t_bool,A: fun_in6429348457993719614e_term,B: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_fun4446166836006121821t_bool(F,A),B))
     => pp(aa_Pro7591400126937138665t_bool(produc6627716702706253960t_bool(F),aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,A),B))) ) ).

% case_prodI
tff(fact_4610_case__prodI,axiom,
    ! [F: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(F,A),B))
     => pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B))) ) ).

% case_prodI
tff(fact_4611_case__prodI,axiom,
    ! [F: fun_int_fun_int_bool,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(F,A),B))
     => pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% case_prodI
tff(fact_4612_mem__case__prodI2,axiom,
    ! [P2: product_prod_int_int,Z: real,C: fun_in6553099857819052212t_real] :
      ( ! [A3: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => pp(aa_set_real_bool(member_real(Z),aa_int_set_real(aa_int4847878404602763637t_real(C,A3),B3))) )
     => pp(aa_set_real_bool(member_real(Z),produc6452406959799940328t_real(C,P2))) ) ).

% mem_case_prodI2
tff(fact_4613_mem__case__prodI2,axiom,
    ! [P2: product_prod_int_int,Z: nat,C: fun_in2630332840903238488et_nat] :
      ( ! [A3: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => pp(aa_set_nat_bool(member_nat(Z),aa_int_set_nat(aa_int1906528167449992217et_nat(C,A3),B3))) )
     => pp(aa_set_nat_bool(member_nat(Z),produc4251311855443802252et_nat(C,P2))) ) ).

% mem_case_prodI2
tff(fact_4614_mem__case__prodI2,axiom,
    ! [P2: product_prod_int_int,Z: int,C: fun_in7040156458970095540et_int] :
      ( ! [A3: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => pp(aa_set_int_bool(member_int(Z),aa_int_set_int(aa_int8639817059373041269et_int(C,A3),B3))) )
     => pp(aa_set_int_bool(member_int(Z),produc73460835934605544et_int(C,P2))) ) ).

% mem_case_prodI2
tff(fact_4615_mem__case__prodI2,axiom,
    ! [P2: product_prod_int_int,Z: set_nat,C: fun_in7207726656141197838et_nat] :
      ( ! [A3: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => pp(aa_set_set_nat_bool(member_set_nat(Z),aa_int_set_set_nat(aa_int1843896668418082255et_nat(C,A3),B3))) )
     => pp(aa_set_set_nat_bool(member_set_nat(Z),produc5233655623923918146et_nat(C,P2))) ) ).

% mem_case_prodI2
tff(fact_4616_mem__case__prodI2,axiom,
    ! [P2: product_prod_int_int,Z: product_prod_nat_nat,C: fun_in1904380326158164027at_nat] :
      ( ! [A3: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Z),aa_int8822742982810151799at_nat(aa_int6362027936895332260at_nat(C,A3),B3))) )
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Z),produc1656060378719767003at_nat(C,P2))) ) ).

% mem_case_prodI2
tff(fact_4617_mem__case__prodI2,axiom,
    ! [P2: produc5207004643709237345nteger,Z: real,C: fun_fu8939667915452561338t_real] :
      ( ! [A3: fun_Co6545587727236015215e_term,B3: produc8923325533196201883nteger] :
          ( ( P2 = aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,A3),B3) )
         => pp(aa_set_real_bool(member_real(Z),aa_Pro6515153133580421057t_real(aa_fun4299200212449711651t_real(C,A3),B3))) )
     => pp(aa_set_real_bool(member_real(Z),produc6662958528406444962t_real(C,P2))) ) ).

% mem_case_prodI2
tff(fact_4618_mem__case__prodI2,axiom,
    ! [P2: produc5207004643709237345nteger,Z: nat,C: fun_fu3482081750474265950et_nat] :
      ( ! [A3: fun_Co6545587727236015215e_term,B3: produc8923325533196201883nteger] :
          ( ( P2 = aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,A3),B3) )
         => pp(aa_set_nat_bool(member_nat(Z),aa_Pro381016180347496421et_nat(aa_fun3173333638319427527et_nat(C,A3),B3))) )
     => pp(aa_set_nat_bool(member_nat(Z),produc3005202867389586502et_nat(C,P2))) ) ).

% mem_case_prodI2
tff(fact_4619_mem__case__prodI2,axiom,
    ! [P2: produc5207004643709237345nteger,Z: int,C: fun_fu7891905368541123002et_int] :
      ( ! [A3: fun_Co6545587727236015215e_term,B3: produc8923325533196201883nteger] :
          ( ( P2 = aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,A3),B3) )
         => pp(aa_set_int_bool(member_int(Z),aa_Pro5426537197693075521et_int(aa_fun683250493387700771et_int(C,A3),B3))) )
     => pp(aa_set_int_bool(member_int(Z),produc8050723884735165602et_int(C,P2))) ) ).

% mem_case_prodI2
tff(fact_4620_mem__case__prodI2,axiom,
    ! [P2: produc6768802789813714928nt_int,Z: real,C: fun_fu1570079290182735305t_real] :
      ( ! [A3: fun_in6429348457993719614e_term,B3: product_prod_int_int] :
          ( ( P2 = aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,A3),B3) )
         => pp(aa_set_real_bool(member_real(Z),aa_Pro7989103684195733727t_real(aa_fun2257173110403687218t_real(C,A3),B3))) )
     => pp(aa_set_real_bool(member_real(Z),produc4003755410105663153t_real(C,P2))) ) ).

% mem_case_prodI2
tff(fact_4621_mem__case__prodI2,axiom,
    ! [P2: produc6768802789813714928nt_int,Z: nat,C: fun_fu4770337263308229357et_nat] :
      ( ! [A3: fun_in6429348457993719614e_term,B3: product_prod_int_int] :
          ( ( P2 = aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,A3),B3) )
         => pp(aa_set_nat_bool(member_nat(Z),aa_Pro3599192995643711491et_nat(aa_fun1858489612225131606et_nat(C,A3),B3))) )
     => pp(aa_set_nat_bool(member_nat(Z),produc2360586433946867413et_nat(C,P2))) ) ).

% mem_case_prodI2
tff(fact_4622_mem__case__prodI,axiom,
    ! [Z: real,C: fun_in6553099857819052212t_real,A: int,B: int] :
      ( pp(aa_set_real_bool(member_real(Z),aa_int_set_real(aa_int4847878404602763637t_real(C,A),B)))
     => pp(aa_set_real_bool(member_real(Z),produc6452406959799940328t_real(C,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)))) ) ).

% mem_case_prodI
tff(fact_4623_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_in2630332840903238488et_nat,A: int,B: int] :
      ( pp(aa_set_nat_bool(member_nat(Z),aa_int_set_nat(aa_int1906528167449992217et_nat(C,A),B)))
     => pp(aa_set_nat_bool(member_nat(Z),produc4251311855443802252et_nat(C,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)))) ) ).

% mem_case_prodI
tff(fact_4624_mem__case__prodI,axiom,
    ! [Z: int,C: fun_in7040156458970095540et_int,A: int,B: int] :
      ( pp(aa_set_int_bool(member_int(Z),aa_int_set_int(aa_int8639817059373041269et_int(C,A),B)))
     => pp(aa_set_int_bool(member_int(Z),produc73460835934605544et_int(C,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)))) ) ).

% mem_case_prodI
tff(fact_4625_mem__case__prodI,axiom,
    ! [Z: set_nat,C: fun_in7207726656141197838et_nat,A: int,B: int] :
      ( pp(aa_set_set_nat_bool(member_set_nat(Z),aa_int_set_set_nat(aa_int1843896668418082255et_nat(C,A),B)))
     => pp(aa_set_set_nat_bool(member_set_nat(Z),produc5233655623923918146et_nat(C,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)))) ) ).

% mem_case_prodI
tff(fact_4626_mem__case__prodI,axiom,
    ! [Z: product_prod_nat_nat,C: fun_in1904380326158164027at_nat,A: int,B: int] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Z),aa_int8822742982810151799at_nat(aa_int6362027936895332260at_nat(C,A),B)))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Z),produc1656060378719767003at_nat(C,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)))) ) ).

% mem_case_prodI
tff(fact_4627_mem__case__prodI,axiom,
    ! [Z: real,C: fun_fu8939667915452561338t_real,A: fun_Co6545587727236015215e_term,B: produc8923325533196201883nteger] :
      ( pp(aa_set_real_bool(member_real(Z),aa_Pro6515153133580421057t_real(aa_fun4299200212449711651t_real(C,A),B)))
     => pp(aa_set_real_bool(member_real(Z),produc6662958528406444962t_real(C,aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,A),B)))) ) ).

% mem_case_prodI
tff(fact_4628_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_fu3482081750474265950et_nat,A: fun_Co6545587727236015215e_term,B: produc8923325533196201883nteger] :
      ( pp(aa_set_nat_bool(member_nat(Z),aa_Pro381016180347496421et_nat(aa_fun3173333638319427527et_nat(C,A),B)))
     => pp(aa_set_nat_bool(member_nat(Z),produc3005202867389586502et_nat(C,aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,A),B)))) ) ).

% mem_case_prodI
tff(fact_4629_mem__case__prodI,axiom,
    ! [Z: int,C: fun_fu7891905368541123002et_int,A: fun_Co6545587727236015215e_term,B: produc8923325533196201883nteger] :
      ( pp(aa_set_int_bool(member_int(Z),aa_Pro5426537197693075521et_int(aa_fun683250493387700771et_int(C,A),B)))
     => pp(aa_set_int_bool(member_int(Z),produc8050723884735165602et_int(C,aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,A),B)))) ) ).

% mem_case_prodI
tff(fact_4630_mem__case__prodI,axiom,
    ! [Z: real,C: fun_fu1570079290182735305t_real,A: fun_in6429348457993719614e_term,B: product_prod_int_int] :
      ( pp(aa_set_real_bool(member_real(Z),aa_Pro7989103684195733727t_real(aa_fun2257173110403687218t_real(C,A),B)))
     => pp(aa_set_real_bool(member_real(Z),produc4003755410105663153t_real(C,aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,A),B)))) ) ).

% mem_case_prodI
tff(fact_4631_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_fu4770337263308229357et_nat,A: fun_in6429348457993719614e_term,B: product_prod_int_int] :
      ( pp(aa_set_nat_bool(member_nat(Z),aa_Pro3599192995643711491et_nat(aa_fun1858489612225131606et_nat(C,A),B)))
     => pp(aa_set_nat_bool(member_nat(Z),produc2360586433946867413et_nat(C,aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,A),B)))) ) ).

% mem_case_prodI
tff(fact_4632_case__prodI2_H,axiom,
    ! [P2: product_prod_nat_nat,C: fun_na4786964484885125043t_bool,X: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) = P2 )
         => pp(aa_Pro2232559814647311250t_bool(aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(C,A3),B3),X)) )
     => pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(C),P2),X)) ) ).

% case_prodI2'
tff(fact_4633_of__bool__less__eq__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_n2052037380579107095ol_rat(P)),zero_n2052037380579107095ol_rat(Q)))
    <=> ( pp(P)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_4634_of__bool__less__eq__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_n2687167440665602831ol_nat(P)),zero_n2687167440665602831ol_nat(Q)))
    <=> ( pp(P)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_4635_of__bool__less__eq__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),aa_bool_int(zero_n2684676970156552555ol_int,Q)))
    <=> ( pp(P)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_4636_of__bool__eq_I1_J,axiom,
    zero_n1201886186963655149omplex(fFalse) = zero_zero_complex ).

% of_bool_eq(1)
tff(fact_4637_of__bool__eq_I1_J,axiom,
    zero_n3304061248610475627l_real(fFalse) = zero_zero_real ).

% of_bool_eq(1)
tff(fact_4638_of__bool__eq_I1_J,axiom,
    zero_n2052037380579107095ol_rat(fFalse) = zero_zero_rat ).

% of_bool_eq(1)
tff(fact_4639_of__bool__eq_I1_J,axiom,
    zero_n2687167440665602831ol_nat(fFalse) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_4640_of__bool__eq_I1_J,axiom,
    aa_bool_int(zero_n2684676970156552555ol_int,fFalse) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_4641_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n1201886186963655149omplex(P) = zero_zero_complex )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_4642_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n3304061248610475627l_real(P) = zero_zero_real )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_4643_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n2052037380579107095ol_rat(P) = zero_zero_rat )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_4644_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n2687167440665602831ol_nat(P) = zero_zero_nat )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_4645_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( aa_bool_int(zero_n2684676970156552555ol_int,P) = zero_zero_int )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_4646_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_n3304061248610475627l_real(P)),zero_n3304061248610475627l_real(Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_4647_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_n2052037380579107095ol_rat(P)),zero_n2052037380579107095ol_rat(Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_4648_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_n2687167440665602831ol_nat(P)),zero_n2687167440665602831ol_nat(Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_4649_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),aa_bool_int(zero_n2684676970156552555ol_int,Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_4650_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),zero_n3304061248610475627l_real(P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_4651_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),zero_n2052037380579107095ol_rat(P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_4652_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),zero_n2687167440665602831ol_nat(P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_4653_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_bool_int(zero_n2684676970156552555ol_int,P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_4654_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_n3304061248610475627l_real(P)),one_one_real))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_4655_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_n2052037380579107095ol_rat(P)),one_one_rat))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_4656_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_n2687167440665602831ol_nat(P)),one_one_nat))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_4657_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),one_one_int))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_4658_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) = 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_4659_set__decode__zero,axiom,
    nat_set_decode(zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_4660_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_4661_set__encode__inverse,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( nat_set_decode(aa_set_nat_nat(nat_set_encode,A2)) = A2 ) ) ).

% set_encode_inverse
tff(fact_4662_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)) = 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_4663_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5052692396658037445od_int(M,one) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,M)),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_4664_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_4665_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique3479559517661332726nteger(M,one) = produc1086072967326762835nteger(numera6620942414471956472nteger(M),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_4666_take__bit__of__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),one_one_nat) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_1
tff(fact_4667_take__bit__of__1,axiom,
    ! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(N),one_one_int) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_1
tff(fact_4668_of__bool__half__eq__0,axiom,
    ! [B: bool] : divide6298287555418463151nteger(zero_n356916108424825756nteger(B),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% of_bool_half_eq_0
tff(fact_4669_of__bool__half__eq__0,axiom,
    ! [B: bool] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_n2687167440665602831ol_nat(B)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% of_bool_half_eq_0
tff(fact_4670_of__bool__half__eq__0,axiom,
    ! [B: bool] : aa_int_int(divide_divide_int(aa_bool_int(zero_n2684676970156552555ol_int,B)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% of_bool_half_eq_0
tff(fact_4671_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,N)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_4672_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_4673_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_4674_bits__1__div__exp,axiom,
    ! [N: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% bits_1_div_exp
tff(fact_4675_bits__1__div__exp,axiom,
    ! [N: nat] : 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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% bits_1_div_exp
tff(fact_4676_bits__1__div__exp,axiom,
    ! [N: nat] : aa_int_int(divide_divide_int(one_one_int),aa_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_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% bits_1_div_exp
tff(fact_4677_one__div__2__pow__eq,axiom,
    ! [N: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% one_div_2_pow_eq
tff(fact_4678_one__div__2__pow__eq,axiom,
    ! [N: nat] : 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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% one_div_2_pow_eq
tff(fact_4679_one__div__2__pow__eq,axiom,
    ! [N: nat] : aa_int_int(divide_divide_int(one_one_int),aa_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_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% one_div_2_pow_eq
tff(fact_4680_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_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(times_times_nat,zero_n2687167440665602831ol_nat(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)) ).

% take_bit_of_exp
tff(fact_4681_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] : aa_int_int(bit_se2923211474154528505it_int(M),aa_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_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% take_bit_of_exp
tff(fact_4682_take__bit__of__2,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(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_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_4683_take__bit__of__2,axiom,
    ! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(N),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,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_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_4684_mem__case__prodE,axiom,
    ! [Z: real,C: fun_in6553099857819052212t_real,P2: product_prod_int_int] :
      ( pp(aa_set_real_bool(member_real(Z),produc6452406959799940328t_real(C,P2)))
     => ~ ! [X4: int,Y2: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y2) )
           => ~ pp(aa_set_real_bool(member_real(Z),aa_int_set_real(aa_int4847878404602763637t_real(C,X4),Y2))) ) ) ).

% mem_case_prodE
tff(fact_4685_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_in2630332840903238488et_nat,P2: product_prod_int_int] :
      ( pp(aa_set_nat_bool(member_nat(Z),produc4251311855443802252et_nat(C,P2)))
     => ~ ! [X4: int,Y2: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y2) )
           => ~ pp(aa_set_nat_bool(member_nat(Z),aa_int_set_nat(aa_int1906528167449992217et_nat(C,X4),Y2))) ) ) ).

% mem_case_prodE
tff(fact_4686_mem__case__prodE,axiom,
    ! [Z: int,C: fun_in7040156458970095540et_int,P2: product_prod_int_int] :
      ( pp(aa_set_int_bool(member_int(Z),produc73460835934605544et_int(C,P2)))
     => ~ ! [X4: int,Y2: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y2) )
           => ~ pp(aa_set_int_bool(member_int(Z),aa_int_set_int(aa_int8639817059373041269et_int(C,X4),Y2))) ) ) ).

% mem_case_prodE
tff(fact_4687_mem__case__prodE,axiom,
    ! [Z: set_nat,C: fun_in7207726656141197838et_nat,P2: product_prod_int_int] :
      ( pp(aa_set_set_nat_bool(member_set_nat(Z),produc5233655623923918146et_nat(C,P2)))
     => ~ ! [X4: int,Y2: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y2) )
           => ~ pp(aa_set_set_nat_bool(member_set_nat(Z),aa_int_set_set_nat(aa_int1843896668418082255et_nat(C,X4),Y2))) ) ) ).

% mem_case_prodE
tff(fact_4688_mem__case__prodE,axiom,
    ! [Z: product_prod_nat_nat,C: fun_in1904380326158164027at_nat,P2: product_prod_int_int] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Z),produc1656060378719767003at_nat(C,P2)))
     => ~ ! [X4: int,Y2: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y2) )
           => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Z),aa_int8822742982810151799at_nat(aa_int6362027936895332260at_nat(C,X4),Y2))) ) ) ).

% mem_case_prodE
tff(fact_4689_mem__case__prodE,axiom,
    ! [Z: real,C: fun_fu8939667915452561338t_real,P2: produc5207004643709237345nteger] :
      ( pp(aa_set_real_bool(member_real(Z),produc6662958528406444962t_real(C,P2)))
     => ~ ! [X4: fun_Co6545587727236015215e_term,Y2: produc8923325533196201883nteger] :
            ( ( P2 = aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,X4),Y2) )
           => ~ pp(aa_set_real_bool(member_real(Z),aa_Pro6515153133580421057t_real(aa_fun4299200212449711651t_real(C,X4),Y2))) ) ) ).

% mem_case_prodE
tff(fact_4690_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_fu3482081750474265950et_nat,P2: produc5207004643709237345nteger] :
      ( pp(aa_set_nat_bool(member_nat(Z),produc3005202867389586502et_nat(C,P2)))
     => ~ ! [X4: fun_Co6545587727236015215e_term,Y2: produc8923325533196201883nteger] :
            ( ( P2 = aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,X4),Y2) )
           => ~ pp(aa_set_nat_bool(member_nat(Z),aa_Pro381016180347496421et_nat(aa_fun3173333638319427527et_nat(C,X4),Y2))) ) ) ).

% mem_case_prodE
tff(fact_4691_mem__case__prodE,axiom,
    ! [Z: int,C: fun_fu7891905368541123002et_int,P2: produc5207004643709237345nteger] :
      ( pp(aa_set_int_bool(member_int(Z),produc8050723884735165602et_int(C,P2)))
     => ~ ! [X4: fun_Co6545587727236015215e_term,Y2: produc8923325533196201883nteger] :
            ( ( P2 = aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,X4),Y2) )
           => ~ pp(aa_set_int_bool(member_int(Z),aa_Pro5426537197693075521et_int(aa_fun683250493387700771et_int(C,X4),Y2))) ) ) ).

% mem_case_prodE
tff(fact_4692_mem__case__prodE,axiom,
    ! [Z: real,C: fun_fu1570079290182735305t_real,P2: produc6768802789813714928nt_int] :
      ( pp(aa_set_real_bool(member_real(Z),produc4003755410105663153t_real(C,P2)))
     => ~ ! [X4: fun_in6429348457993719614e_term,Y2: product_prod_int_int] :
            ( ( P2 = aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,X4),Y2) )
           => ~ pp(aa_set_real_bool(member_real(Z),aa_Pro7989103684195733727t_real(aa_fun2257173110403687218t_real(C,X4),Y2))) ) ) ).

% mem_case_prodE
tff(fact_4693_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_fu4770337263308229357et_nat,P2: produc6768802789813714928nt_int] :
      ( pp(aa_set_nat_bool(member_nat(Z),produc2360586433946867413et_nat(C,P2)))
     => ~ ! [X4: fun_in6429348457993719614e_term,Y2: product_prod_int_int] :
            ( ( P2 = aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,X4),Y2) )
           => ~ pp(aa_set_nat_bool(member_nat(Z),aa_Pro3599192995643711491et_nat(aa_fun1858489612225131606et_nat(C,X4),Y2))) ) ) ).

% mem_case_prodE
tff(fact_4694_case__prodE,axiom,
    ! [C: fun_fu6012121808138353003r_bool,P2: produc5207004643709237345nteger] :
      ( pp(aa_Pro7150874649148492216r_bool(produc401098500374025047r_bool(C),P2))
     => ~ ! [X4: fun_Co6545587727236015215e_term,Y2: produc8923325533196201883nteger] :
            ( ( P2 = aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,X4),Y2) )
           => ~ pp(aa_Pro5843519176478241656r_bool(aa_fun5646951941296221740r_bool(C,X4),Y2)) ) ) ).

% case_prodE
tff(fact_4695_case__prodE,axiom,
    ! [C: fun_fu1674507805476766837r_bool,P2: produc1616345218596728535nteger] :
      ( pp(aa_Pro3640759179027349698r_bool(produc1236900166980094561r_bool(C),P2))
     => ~ ! [X4: fun_Pr272337284026993017e_term,Y2: produc8923325533196201883nteger] :
            ( ( P2 = aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,X4),Y2) )
           => ~ pp(aa_Pro5843519176478241656r_bool(aa_fun5449300841655810358r_bool(C,X4),Y2)) ) ) ).

% case_prodE
tff(fact_4696_case__prodE,axiom,
    ! [C: fun_fu4196931294019812808t_bool,P2: produc4441460161354545860nt_int] :
      ( pp(aa_Pro5031108791230793749t_bool(produc625993674133193396t_bool(C),P2))
     => ~ ! [X4: fun_Pr9164807099015279978e_term,Y2: product_prod_int_int] :
            ( ( P2 = aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,X4),Y2) )
           => ~ pp(aa_Pro4501881432599481050t_bool(aa_fun5300525129335604105t_bool(C,X4),Y2)) ) ) ).

% case_prodE
tff(fact_4697_case__prodE,axiom,
    ! [C: fun_fu5117548769634986908t_bool,P2: produc6768802789813714928nt_int] :
      ( pp(aa_Pro7591400126937138665t_bool(produc6627716702706253960t_bool(C),P2))
     => ~ ! [X4: fun_in6429348457993719614e_term,Y2: product_prod_int_int] :
            ( ( P2 = aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,X4),Y2) )
           => ~ pp(aa_Pro4501881432599481050t_bool(aa_fun4446166836006121821t_bool(C,X4),Y2)) ) ) ).

% case_prodE
tff(fact_4698_case__prodE,axiom,
    ! [C: fun_nat_fun_nat_bool,P2: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(C),P2))
     => ~ ! [X4: nat,Y2: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y2) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(C,X4),Y2)) ) ) ).

% case_prodE
tff(fact_4699_case__prodE,axiom,
    ! [C: fun_int_fun_int_bool,P2: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(C),P2))
     => ~ ! [X4: int,Y2: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y2) )
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(C,X4),Y2)) ) ) ).

% case_prodE
tff(fact_4700_case__prodD,axiom,
    ! [F: fun_fu6012121808138353003r_bool,A: fun_Co6545587727236015215e_term,B: produc8923325533196201883nteger] :
      ( pp(aa_Pro7150874649148492216r_bool(produc401098500374025047r_bool(F),aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,A),B)))
     => pp(aa_Pro5843519176478241656r_bool(aa_fun5646951941296221740r_bool(F,A),B)) ) ).

% case_prodD
tff(fact_4701_case__prodD,axiom,
    ! [F: fun_fu1674507805476766837r_bool,A: fun_Pr272337284026993017e_term,B: produc8923325533196201883nteger] :
      ( pp(aa_Pro3640759179027349698r_bool(produc1236900166980094561r_bool(F),aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,A),B)))
     => pp(aa_Pro5843519176478241656r_bool(aa_fun5449300841655810358r_bool(F,A),B)) ) ).

% case_prodD
tff(fact_4702_case__prodD,axiom,
    ! [F: fun_fu4196931294019812808t_bool,A: fun_Pr9164807099015279978e_term,B: product_prod_int_int] :
      ( pp(aa_Pro5031108791230793749t_bool(produc625993674133193396t_bool(F),aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,A),B)))
     => pp(aa_Pro4501881432599481050t_bool(aa_fun5300525129335604105t_bool(F,A),B)) ) ).

% case_prodD
tff(fact_4703_case__prodD,axiom,
    ! [F: fun_fu5117548769634986908t_bool,A: fun_in6429348457993719614e_term,B: product_prod_int_int] :
      ( pp(aa_Pro7591400126937138665t_bool(produc6627716702706253960t_bool(F),aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,A),B)))
     => pp(aa_Pro4501881432599481050t_bool(aa_fun4446166836006121821t_bool(F,A),B)) ) ).

% case_prodD
tff(fact_4704_case__prodD,axiom,
    ! [F: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(F,A),B)) ) ).

% case_prodD
tff(fact_4705_case__prodD,axiom,
    ! [F: fun_int_fun_int_bool,A: int,B: int] :
      ( pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)))
     => pp(aa_int_bool(aa_int_fun_int_bool(F,A),B)) ) ).

% case_prodD
tff(fact_4706_case__prodE_H,axiom,
    ! [C: fun_na4786964484885125043t_bool,P2: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(C),P2),Z))
     => ~ ! [X4: nat,Y2: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y2) )
           => ~ pp(aa_Pro2232559814647311250t_bool(aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(C,X4),Y2),Z)) ) ) ).

% case_prodE'
tff(fact_4707_case__prodD_H,axiom,
    ! [R: fun_na4786964484885125043t_bool,A: nat,B: nat,C: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(R),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),C))
     => pp(aa_Pro2232559814647311250t_bool(aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(R,A),B),C)) ) ).

% case_prodD'
tff(fact_4708_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),zero_n3304061248610475627l_real(P))) ).

% zero_less_eq_of_bool
tff(fact_4709_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),zero_n2052037380579107095ol_rat(P))) ).

% zero_less_eq_of_bool
tff(fact_4710_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),zero_n2687167440665602831ol_nat(P))) ).

% zero_less_eq_of_bool
tff(fact_4711_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_bool_int(zero_n2684676970156552555ol_int,P))) ).

% zero_less_eq_of_bool
tff(fact_4712_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_n3304061248610475627l_real(P)),one_one_real)) ).

% of_bool_less_eq_one
tff(fact_4713_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_n2052037380579107095ol_rat(P)),one_one_rat)) ).

% of_bool_less_eq_one
tff(fact_4714_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_n2687167440665602831ol_nat(P)),one_one_nat)) ).

% of_bool_less_eq_one
tff(fact_4715_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),one_one_int)) ).

% of_bool_less_eq_one
tff(fact_4716_split__of__bool__asm,axiom,
    ! [P: fun_complex_bool,P2: bool] :
      ( pp(aa_complex_bool(P,zero_n1201886186963655149omplex(P2)))
    <=> ~ ( ( pp(P2)
            & ~ pp(aa_complex_bool(P,one_one_complex)) )
          | ( ~ pp(P2)
            & ~ pp(aa_complex_bool(P,zero_zero_complex)) ) ) ) ).

% split_of_bool_asm
tff(fact_4717_split__of__bool__asm,axiom,
    ! [P: fun_real_bool,P2: bool] :
      ( pp(aa_real_bool(P,zero_n3304061248610475627l_real(P2)))
    <=> ~ ( ( pp(P2)
            & ~ pp(aa_real_bool(P,one_one_real)) )
          | ( ~ pp(P2)
            & ~ pp(aa_real_bool(P,zero_zero_real)) ) ) ) ).

% split_of_bool_asm
tff(fact_4718_split__of__bool__asm,axiom,
    ! [P: fun_rat_bool,P2: bool] :
      ( pp(aa_rat_bool(P,zero_n2052037380579107095ol_rat(P2)))
    <=> ~ ( ( pp(P2)
            & ~ pp(aa_rat_bool(P,one_one_rat)) )
          | ( ~ pp(P2)
            & ~ pp(aa_rat_bool(P,zero_zero_rat)) ) ) ) ).

% split_of_bool_asm
tff(fact_4719_split__of__bool__asm,axiom,
    ! [P: fun_nat_bool,P2: bool] :
      ( pp(aa_nat_bool(P,zero_n2687167440665602831ol_nat(P2)))
    <=> ~ ( ( pp(P2)
            & ~ pp(aa_nat_bool(P,one_one_nat)) )
          | ( ~ pp(P2)
            & ~ pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).

% split_of_bool_asm
tff(fact_4720_split__of__bool__asm,axiom,
    ! [P: fun_int_bool,P2: bool] :
      ( pp(aa_int_bool(P,aa_bool_int(zero_n2684676970156552555ol_int,P2)))
    <=> ~ ( ( pp(P2)
            & ~ pp(aa_int_bool(P,one_one_int)) )
          | ( ~ pp(P2)
            & ~ pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% split_of_bool_asm
tff(fact_4721_split__of__bool,axiom,
    ! [P: fun_complex_bool,P2: bool] :
      ( pp(aa_complex_bool(P,zero_n1201886186963655149omplex(P2)))
    <=> ( ( pp(P2)
         => pp(aa_complex_bool(P,one_one_complex)) )
        & ( ~ pp(P2)
         => pp(aa_complex_bool(P,zero_zero_complex)) ) ) ) ).

% split_of_bool
tff(fact_4722_split__of__bool,axiom,
    ! [P: fun_real_bool,P2: bool] :
      ( pp(aa_real_bool(P,zero_n3304061248610475627l_real(P2)))
    <=> ( ( pp(P2)
         => pp(aa_real_bool(P,one_one_real)) )
        & ( ~ pp(P2)
         => pp(aa_real_bool(P,zero_zero_real)) ) ) ) ).

% split_of_bool
tff(fact_4723_split__of__bool,axiom,
    ! [P: fun_rat_bool,P2: bool] :
      ( pp(aa_rat_bool(P,zero_n2052037380579107095ol_rat(P2)))
    <=> ( ( pp(P2)
         => pp(aa_rat_bool(P,one_one_rat)) )
        & ( ~ pp(P2)
         => pp(aa_rat_bool(P,zero_zero_rat)) ) ) ) ).

% split_of_bool
tff(fact_4724_split__of__bool,axiom,
    ! [P: fun_nat_bool,P2: bool] :
      ( pp(aa_nat_bool(P,zero_n2687167440665602831ol_nat(P2)))
    <=> ( ( pp(P2)
         => pp(aa_nat_bool(P,one_one_nat)) )
        & ( ~ pp(P2)
         => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).

% split_of_bool
tff(fact_4725_split__of__bool,axiom,
    ! [P: fun_int_bool,P2: bool] :
      ( pp(aa_int_bool(P,aa_bool_int(zero_n2684676970156552555ol_int,P2)))
    <=> ( ( pp(P2)
         => pp(aa_int_bool(P,one_one_int)) )
        & ( ~ pp(P2)
         => pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% split_of_bool
tff(fact_4726_of__bool__def,axiom,
    ! [P2: bool] :
      ( ( pp(P2)
       => ( zero_n1201886186963655149omplex(P2) = one_one_complex ) )
      & ( ~ pp(P2)
       => ( zero_n1201886186963655149omplex(P2) = zero_zero_complex ) ) ) ).

% of_bool_def
tff(fact_4727_of__bool__def,axiom,
    ! [P2: bool] :
      ( ( pp(P2)
       => ( zero_n3304061248610475627l_real(P2) = one_one_real ) )
      & ( ~ pp(P2)
       => ( zero_n3304061248610475627l_real(P2) = zero_zero_real ) ) ) ).

% of_bool_def
tff(fact_4728_of__bool__def,axiom,
    ! [P2: bool] :
      ( ( pp(P2)
       => ( zero_n2052037380579107095ol_rat(P2) = one_one_rat ) )
      & ( ~ pp(P2)
       => ( zero_n2052037380579107095ol_rat(P2) = zero_zero_rat ) ) ) ).

% of_bool_def
tff(fact_4729_of__bool__def,axiom,
    ! [P2: bool] :
      ( ( pp(P2)
       => ( zero_n2687167440665602831ol_nat(P2) = one_one_nat ) )
      & ( ~ pp(P2)
       => ( zero_n2687167440665602831ol_nat(P2) = zero_zero_nat ) ) ) ).

% of_bool_def
tff(fact_4730_of__bool__def,axiom,
    ! [P2: bool] :
      ( ( pp(P2)
       => ( aa_bool_int(zero_n2684676970156552555ol_int,P2) = one_one_int ) )
      & ( ~ pp(P2)
       => ( aa_bool_int(zero_n2684676970156552555ol_int,P2) = zero_zero_int ) ) ) ).

% of_bool_def
tff(fact_4731_finite__set__decode,axiom,
    ! [N: nat] : pp(aa_set_nat_bool(finite_finite_nat,nat_set_decode(N))) ).

% finite_set_decode
tff(fact_4732_set__encode__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ( aa_set_nat_nat(nat_set_encode,A2) = aa_set_nat_nat(nat_set_encode,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% set_encode_eq
tff(fact_4733_set__encode__inf,axiom,
    ! [A2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_set_nat_nat(nat_set_encode,A2) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_4734_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_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_4735_divmod__int__def,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(M,N) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ).

% divmod_int_def
tff(fact_4736_divmod__def,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(M,N) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ).

% divmod_def
tff(fact_4737_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_4738_divmod__def,axiom,
    ! [M: num,N: num] : unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N)),modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N))) ).

% divmod_def
tff(fact_4739_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_4740_exp__mod__exp,axiom,
    ! [M: nat,N: nat] : modulo364778990260209775nteger(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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_4741_exp__mod__exp,axiom,
    ! [M: 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_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,zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_4742_exp__mod__exp,axiom,
    ! [M: nat,N: nat] : aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),aa_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_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_4743_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_4744_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] : divide6298287555418463151nteger(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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)),zero_z3403309356797280102nteger))),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ).

% exp_div_exp_eq
tff(fact_4745_exp__div__exp__eq,axiom,
    ! [M: nat,N: 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))),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,zero_n2687167440665602831ol_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_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))),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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ).

% exp_div_exp_eq
tff(fact_4746_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] : aa_int_int(divide_divide_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),aa_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_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),zero_zero_int))),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)))),aa_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,M),N))) ).

% exp_div_exp_eq
tff(fact_4747_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_4748_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(aa_int2565445551602696444nt_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) = aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(N),unique5052692396658037445od_int(M,aa_num_num(bit0,N))) ) ) ) ).

% divmod_divmod_step
tff(fact_4749_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_4750_listrel1p__def,axiom,
    ! [R2: fun_nat_fun_nat_bool,Xs: list_nat,Ys: list_nat] :
      ( pp(aa_list_nat_bool(listrel1p_nat(R2,Xs),Ys))
    <=> member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),listrel1_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(R2)))) ) ).

% listrel1p_def
tff(fact_4751_listrel1p__def,axiom,
    ! [R2: fun_int_fun_int_bool,Xs: list_int,Ys: list_int] :
      ( pp(aa_list_int_bool(listrel1p_int(R2,Xs),Ys))
    <=> member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),listrel1_int(collec213857154873943460nt_int(produc8560705615207772305t_bool(R2)))) ) ).

% listrel1p_def
tff(fact_4752_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,N)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_fn),unique5052692396658037445od_int(M,N)) ).

% divmod_algorithm_code(6)
tff(fact_4753_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] : unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,N)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_fo),unique5055182867167087721od_nat(M,N)) ).

% divmod_algorithm_code(6)
tff(fact_4754_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] : unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,N)) = produc6916734918728496179nteger(aTP_Lamm_fp,unique3479559517661332726nteger(M,N)) ).

% divmod_algorithm_code(6)
tff(fact_4755_Divides_Oadjust__div__eq,axiom,
    ! [Q4: int,R2: int] : adjust_div(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q4),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_4756_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_fq(D))) ).

% int_ge_less_than2_def
tff(fact_4757_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_fr(D))) ).

% int_ge_less_than_def
tff(fact_4758_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(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( aa_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(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_4759_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_4760_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(aa_int2565445551602696444nt_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)) = aa_Pro6970504127699319548nt_int(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_4761_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_4762_add_Oinverse__inverse,axiom,
    ! [A: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,A)) = A ).

% add.inverse_inverse
tff(fact_4763_add_Oinverse__inverse,axiom,
    ! [A: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,A)) = A ).

% add.inverse_inverse
tff(fact_4764_add_Oinverse__inverse,axiom,
    ! [A: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,A)) = A ).

% add.inverse_inverse
tff(fact_4765_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).

% add.inverse_inverse
tff(fact_4766_neg__equal__iff__equal,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4767_neg__equal__iff__equal,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4768_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4769_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4770_compl__le__compl__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(X)),uminus1532241313380277803et_int(Y)))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(Y),X)) ) ).

% compl_le_compl_iff
tff(fact_4771_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% neg_le_iff_le
tff(fact_4772_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).

% neg_le_iff_le
tff(fact_4773_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% neg_le_iff_le
tff(fact_4774_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% neg_le_iff_le
tff(fact_4775_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_4776_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_4777_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_4778_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_4779_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_4780_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_4781_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_4782_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_4783_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
tff(fact_4784_neg__equal__0__iff__equal,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
tff(fact_4785_neg__equal__0__iff__equal,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
tff(fact_4786_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
tff(fact_4787_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
tff(fact_4788_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex = aa_complex_complex(uminus1482373934393186551omplex,A) )
    <=> ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4789_neg__0__equal__iff__equal,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(uminus_uminus_int,A) )
    <=> ( zero_zero_int = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4790_neg__0__equal__iff__equal,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(uminus_uminus_real,A) )
    <=> ( zero_zero_real = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4791_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4792_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4793_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_4794_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_4795_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_4796_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_4797_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_4798_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% neg_less_iff_less
tff(fact_4799_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% neg_less_iff_less
tff(fact_4800_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% neg_less_iff_less
tff(fact_4801_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).

% neg_less_iff_less
tff(fact_4802_add__minus__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B)) = B ).

% add_minus_cancel
tff(fact_4803_add__minus__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B)) = B ).

% add_minus_cancel
tff(fact_4804_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) = B ).

% add_minus_cancel
tff(fact_4805_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(A,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B)) = B ).

% add_minus_cancel
tff(fact_4806_minus__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = B ).

% minus_add_cancel
tff(fact_4807_minus__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = B ).

% minus_add_cancel
tff(fact_4808_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = B ).

% minus_add_cancel
tff(fact_4809_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),plus_p5714425477246183910nteger(A,B)) = B ).

% minus_add_cancel
tff(fact_4810_minus__add__distrib,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) ).

% minus_add_distrib
tff(fact_4811_minus__add__distrib,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) ).

% minus_add_distrib
tff(fact_4812_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_add_distrib
tff(fact_4813_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_add_distrib
tff(fact_4814_minus__diff__eq,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ).

% minus_diff_eq
tff(fact_4815_minus__diff__eq,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A) ).

% minus_diff_eq
tff(fact_4816_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(minus_minus_rat(B),A) ).

% minus_diff_eq
tff(fact_4817_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A,B)) = minus_8373710615458151222nteger(B,A) ).

% minus_diff_eq
tff(fact_4818_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_4819_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% neg_0_le_iff_le
tff(fact_4820_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).

% neg_0_le_iff_le
tff(fact_4821_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% neg_0_le_iff_le
tff(fact_4822_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% neg_0_le_iff_le
tff(fact_4823_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% neg_le_0_iff_le
tff(fact_4824_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_le_0_iff_le
tff(fact_4825_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% neg_le_0_iff_le
tff(fact_4826_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% neg_le_0_iff_le
tff(fact_4827_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% less_eq_neg_nonpos
tff(fact_4828_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).

% less_eq_neg_nonpos
tff(fact_4829_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% less_eq_neg_nonpos
tff(fact_4830_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% less_eq_neg_nonpos
tff(fact_4831_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% neg_less_eq_nonneg
tff(fact_4832_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_eq_nonneg
tff(fact_4833_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% neg_less_eq_nonneg
tff(fact_4834_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% neg_less_eq_nonneg
tff(fact_4835_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% neg_less_0_iff_less
tff(fact_4836_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% neg_less_0_iff_less
tff(fact_4837_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% neg_less_0_iff_less
tff(fact_4838_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_0_iff_less
tff(fact_4839_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% neg_0_less_iff_less
tff(fact_4840_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% neg_0_less_iff_less
tff(fact_4841_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% neg_0_less_iff_less
tff(fact_4842_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% neg_0_less_iff_less
tff(fact_4843_neg__less__pos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% neg_less_pos
tff(fact_4844_neg__less__pos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% neg_less_pos
tff(fact_4845_neg__less__pos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% neg_less_pos
tff(fact_4846_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_pos
tff(fact_4847_less__neg__neg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% less_neg_neg
tff(fact_4848_less__neg__neg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% less_neg_neg
tff(fact_4849_less__neg__neg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% less_neg_neg
tff(fact_4850_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% less_neg_neg
tff(fact_4851_add_Oright__inverse,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,A)) = zero_zero_complex ).

% add.right_inverse
tff(fact_4852_add_Oright__inverse,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,A)) = zero_zero_int ).

% add.right_inverse
tff(fact_4853_add_Oright__inverse,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,A)) = zero_zero_real ).

% add.right_inverse
tff(fact_4854_add_Oright__inverse,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_4855_add_Oright__inverse,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_4856_ab__left__minus,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).

% ab_left_minus
tff(fact_4857_ab__left__minus,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_left_minus
tff(fact_4858_ab__left__minus,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_left_minus
tff(fact_4859_ab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_4860_ab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_4861_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% verit_minus_simplify(3)
tff(fact_4862_verit__minus__simplify_I3_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),B) = aa_int_int(uminus_uminus_int,B) ).

% verit_minus_simplify(3)
tff(fact_4863_verit__minus__simplify_I3_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),B) = aa_real_real(uminus_uminus_real,B) ).

% verit_minus_simplify(3)
tff(fact_4864_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% verit_minus_simplify(3)
tff(fact_4865_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% verit_minus_simplify(3)
tff(fact_4866_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_4867_diff__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),A) = aa_int_int(uminus_uminus_int,A) ).

% diff_0
tff(fact_4868_diff__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),A) = aa_real_real(uminus_uminus_real,A) ).

% diff_0
tff(fact_4869_diff__0,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),A) = aa_rat_rat(uminus_uminus_rat,A) ).

% diff_0
tff(fact_4870_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% diff_0
tff(fact_4871_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ).

% uminus_add_conv_diff
tff(fact_4872_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A) ).

% uminus_add_conv_diff
tff(fact_4873_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(minus_minus_rat(B),A) ).

% uminus_add_conv_diff
tff(fact_4874_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = minus_8373710615458151222nteger(B,A) ).

% uminus_add_conv_diff
tff(fact_4875_diff__minus__eq__add,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) ).

% diff_minus_eq_add
tff(fact_4876_diff__minus__eq__add,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) ).

% diff_minus_eq_add
tff(fact_4877_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) ).

% diff_minus_eq_add
tff(fact_4878_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = plus_p5714425477246183910nteger(A,B) ).

% diff_minus_eq_add
tff(fact_4879_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_4880_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_4881_add__neg__numeral__special_I7_J,axiom,
    aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% add_neg_numeral_special(7)
tff(fact_4882_add__neg__numeral__special_I7_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% add_neg_numeral_special(7)
tff(fact_4883_add__neg__numeral__special_I7_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% add_neg_numeral_special(7)
tff(fact_4884_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_4885_add__neg__numeral__special_I7_J,axiom,
    plus_p5714425477246183910nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(7)
tff(fact_4886_add__neg__numeral__special_I8_J,axiom,
    aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = zero_zero_complex ).

% add_neg_numeral_special(8)
tff(fact_4887_add__neg__numeral__special_I8_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = zero_zero_int ).

% add_neg_numeral_special(8)
tff(fact_4888_add__neg__numeral__special_I8_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = zero_zero_real ).

% add_neg_numeral_special(8)
tff(fact_4889_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_4890_add__neg__numeral__special_I8_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(8)
tff(fact_4891_diff__numeral__special_I12_J,axiom,
    aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% diff_numeral_special(12)
tff(fact_4892_diff__numeral__special_I12_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% diff_numeral_special(12)
tff(fact_4893_diff__numeral__special_I12_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% diff_numeral_special(12)
tff(fact_4894_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_4895_diff__numeral__special_I12_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_4896_mod__minus1__right,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_4897_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_4898_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U)) ) ) ) ).

% max_number_of(4)
tff(fact_4899_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)) ) ) ) ).

% max_number_of(4)
tff(fact_4900_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))))
       => ( aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))))
       => ( aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U)) ) ) ) ).

% max_number_of(4)
tff(fact_4901_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
       => ( aa_int_int(ord_max_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
       => ( aa_int_int(ord_max_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U)) ) ) ) ).

% max_number_of(4)
tff(fact_4902_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_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_real_real(uminus_uminus_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_real_real(uminus_uminus_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_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U)) ) ) ) ).

% max_number_of(3)
tff(fact_4903_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = numera6620942414471956472nteger(V) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)) ) ) ) ).

% max_number_of(3)
tff(fact_4904_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)))
       => ( aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_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_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)))
       => ( aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U)) ) ) ) ).

% max_number_of(3)
tff(fact_4905_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)))
       => ( aa_int_int(ord_max_int(aa_int_int(uminus_uminus_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_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)))
       => ( aa_int_int(ord_max_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U)) ) ) ) ).

% max_number_of(3)
tff(fact_4906_max__number__of_I2_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_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_real_real(uminus_uminus_real,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_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_num_real(numeral_numeral_real,U) ) ) ) ).

% max_number_of(2)
tff(fact_4907_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = numera6620942414471956472nteger(U) ) ) ) ).

% max_number_of(2)
tff(fact_4908_max__number__of_I2_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_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))))
       => ( aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_rat_rat(uminus_uminus_rat,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_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))))
       => ( aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_num_rat(numeral_numeral_rat,U) ) ) ) ).

% max_number_of(2)
tff(fact_4909_max__number__of_I2_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_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
       => ( aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_int_int(uminus_uminus_int,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_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
       => ( aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_num_int(numeral_numeral_int,U) ) ) ) ).

% max_number_of(2)
tff(fact_4910_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,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))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).

% neg_numeral_le_iff
tff(fact_4911_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).

% neg_numeral_le_iff
tff(fact_4912_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,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))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).

% neg_numeral_le_iff
tff(fact_4913_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_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))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).

% neg_numeral_le_iff
tff(fact_4914_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_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))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).

% neg_numeral_less_iff
tff(fact_4915_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,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))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).

% neg_numeral_less_iff
tff(fact_4916_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,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))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).

% neg_numeral_less_iff
tff(fact_4917_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).

% neg_numeral_less_iff
tff(fact_4918_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_4919_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_4920_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4921_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4922_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4923_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4924_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),B)) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4925_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),B)) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4926_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4927_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4928_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W2: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) = A )
    <=> ( ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) ) )
        & ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4929_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) = A )
    <=> ( ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) ) )
        & ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4930_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) = A )
    <=> ( ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) != zero_zero_rat )
         => ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) ) )
        & ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4931_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W2: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) )
    <=> ( ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) = B ) )
        & ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4932_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) )
    <=> ( ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) = B ) )
        & ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4933_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) )
    <=> ( ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) != zero_zero_rat )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) = B ) )
        & ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4934_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4935_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,one_one_real)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4936_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4937_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4938_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_4939_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_4940_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),B)) ) ).

% divide_less_eq_numeral1(2)
tff(fact_4941_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),B)) ) ).

% divide_less_eq_numeral1(2)
tff(fact_4942_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,N)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_4943_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_4944_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_4945_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_4946_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(aa_int2565445551602696444nt_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)) = aa_Pro6970504127699319548nt_int(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_4947_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_4948_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% equation_minus_iff
tff(fact_4949_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% equation_minus_iff
tff(fact_4950_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% equation_minus_iff
tff(fact_4951_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% equation_minus_iff
tff(fact_4952_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(uminus_uminus_int,B) = A ) ) ).

% minus_equation_iff
tff(fact_4953_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(uminus_uminus_real,B) = A ) ) ).

% minus_equation_iff
tff(fact_4954_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(uminus_uminus_rat,B) = A ) ) ).

% minus_equation_iff
tff(fact_4955_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) = A ) ) ).

% minus_equation_iff
tff(fact_4956_compl__le__swap2,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(Y)),X))
     => pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(X)),Y)) ) ).

% compl_le_swap2
tff(fact_4957_compl__le__swap1,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),uminus1532241313380277803et_int(X)))
     => pp(aa_set_int_bool(ord_less_eq_set_int(X),uminus1532241313380277803et_int(Y))) ) ).

% compl_le_swap1
tff(fact_4958_compl__mono,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(Y)),uminus1532241313380277803et_int(X))) ) ).

% compl_mono
tff(fact_4959_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))) ) ).

% le_imp_neg_le
tff(fact_4960_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).

% le_imp_neg_le
tff(fact_4961_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% le_imp_neg_le
tff(fact_4962_le__imp__neg__le,axiom,
    ! [A: int,B: 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_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))) ) ).

% le_imp_neg_le
tff(fact_4963_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),A)) ) ).

% minus_le_iff
tff(fact_4964_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A)) ) ).

% minus_le_iff
tff(fact_4965_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),A)) ) ).

% minus_le_iff
tff(fact_4966_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A)) ) ).

% minus_le_iff
tff(fact_4967_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(uminus_uminus_real,A))) ) ).

% le_minus_iff
tff(fact_4968_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).

% le_minus_iff
tff(fact_4969_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% le_minus_iff
tff(fact_4970_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A))) ) ).

% le_minus_iff
tff(fact_4971_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),A)) ) ).

% minus_less_iff
tff(fact_4972_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),A)) ) ).

% minus_less_iff
tff(fact_4973_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),A)) ) ).

% minus_less_iff
tff(fact_4974_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A)) ) ).

% minus_less_iff
tff(fact_4975_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(uminus_uminus_int,B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(uminus_uminus_int,A))) ) ).

% less_minus_iff
tff(fact_4976_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(uminus_uminus_real,A))) ) ).

% less_minus_iff
tff(fact_4977_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% less_minus_iff
tff(fact_4978_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).

% less_minus_iff
tff(fact_4979_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: 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,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_4980_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_4981_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_4982_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_4983_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(uminus_uminus_int,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_4984_group__cancel_Oneg1,axiom,
    ! [A2: real,K: real,A: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(uminus_uminus_real,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_4985_group__cancel_Oneg1,axiom,
    ! [A2: rat,K: rat,A: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(uminus_uminus_rat,A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).

% group_cancel.neg1
tff(fact_4986_group__cancel_Oneg1,axiom,
    ! [A2: code_integer,K: code_integer,A: code_integer] :
      ( ( A2 = plus_p5714425477246183910nteger(K,A) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).

% group_cancel.neg1
tff(fact_4987_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ).

% add.inverse_distrib_swap
tff(fact_4988_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% add.inverse_distrib_swap
tff(fact_4989_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% add.inverse_distrib_swap
tff(fact_4990_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% add.inverse_distrib_swap
tff(fact_4991_minus__diff__commute,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,B)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),B) ).

% minus_diff_commute
tff(fact_4992_minus__diff__commute,axiom,
    ! [B: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,B)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),B) ).

% minus_diff_commute
tff(fact_4993_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,B)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_diff_commute
tff(fact_4994_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),A) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% minus_diff_commute
tff(fact_4995_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_4996_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)))) ).

% not_numeral_le_neg_numeral
tff(fact_4997_not__numeral__le__neg__numeral,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_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_4998_not__numeral__le__neg__numeral,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_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_4999_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,N))) ).

% neg_numeral_le_numeral
tff(fact_5000_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),numera6620942414471956472nteger(N))) ).

% neg_numeral_le_numeral
tff(fact_5001_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N))) ).

% neg_numeral_le_numeral
tff(fact_5002_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N))) ).

% neg_numeral_le_numeral
tff(fact_5003_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% zero_neq_neg_numeral
tff(fact_5004_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% zero_neq_neg_numeral
tff(fact_5005_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) ).

% zero_neq_neg_numeral
tff(fact_5006_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) ).

% zero_neq_neg_numeral
tff(fact_5007_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) ).

% zero_neq_neg_numeral
tff(fact_5008_not__numeral__less__neg__numeral,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_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_5009_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_5010_not__numeral__less__neg__numeral,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_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_5011_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)))) ).

% not_numeral_less_neg_numeral
tff(fact_5012_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N))) ).

% neg_numeral_less_numeral
tff(fact_5013_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,N))) ).

% neg_numeral_less_numeral
tff(fact_5014_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N))) ).

% neg_numeral_less_numeral
tff(fact_5015_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),numera6620942414471956472nteger(N))) ).

% neg_numeral_less_numeral
tff(fact_5016_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X != product_Pair_num_num(one,one) )
     => ( ! [N2: num] : X != product_Pair_num_num(one,aa_num_num(bit0,N2))
       => ( ! [N2: num] : X != product_Pair_num_num(one,aa_num_num(bit1,N2))
         => ( ! [M5: num] : X != product_Pair_num_num(aa_num_num(bit0,M5),one)
           => ( ! [M5: num,N2: num] : X != product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit0,N2))
             => ( ! [M5: num,N2: num] : X != product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit1,N2))
               => ( ! [M5: num] : X != product_Pair_num_num(aa_num_num(bit1,M5),one)
                 => ( ! [M5: num,N2: num] : X != product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit0,N2))
                   => ~ ! [M5: num,N2: num] : X != product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit1,N2)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_5017_le__minus__one__simps_I2_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real)) ).

% le_minus_one_simps(2)
tff(fact_5018_le__minus__one__simps_I2_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer)) ).

% le_minus_one_simps(2)
tff(fact_5019_le__minus__one__simps_I2_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat)) ).

% le_minus_one_simps(2)
tff(fact_5020_le__minus__one__simps_I2_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int)) ).

% le_minus_one_simps(2)
tff(fact_5021_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% le_minus_one_simps(4)
tff(fact_5022_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% le_minus_one_simps(4)
tff(fact_5023_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% le_minus_one_simps(4)
tff(fact_5024_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% le_minus_one_simps(4)
tff(fact_5025_add__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex )
    <=> ( B = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% add_eq_0_iff
tff(fact_5026_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% add_eq_0_iff
tff(fact_5027_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% add_eq_0_iff
tff(fact_5028_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% add_eq_0_iff
tff(fact_5029_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% add_eq_0_iff
tff(fact_5030_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).

% ab_group_add_class.ab_left_minus
tff(fact_5031_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_group_add_class.ab_left_minus
tff(fact_5032_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_group_add_class.ab_left_minus
tff(fact_5033_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_5034_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_group_add_class.ab_left_minus
tff(fact_5035_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A) = B ) ) ).

% add.inverse_unique
tff(fact_5036_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,A) = B ) ) ).

% add.inverse_unique
tff(fact_5037_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,A) = B ) ) ).

% add.inverse_unique
tff(fact_5038_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A) = B ) ) ).

% add.inverse_unique
tff(fact_5039_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B ) ) ).

% add.inverse_unique
tff(fact_5040_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5041_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5042_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5043_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5044_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5045_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B )
    <=> ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5046_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5047_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5048_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5049_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5050_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_5051_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_5052_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_5053_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_5054_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_5055_less__minus__one__simps_I2_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int)) ).

% less_minus_one_simps(2)
tff(fact_5056_less__minus__one__simps_I2_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real)) ).

% less_minus_one_simps(2)
tff(fact_5057_less__minus__one__simps_I2_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat)) ).

% less_minus_one_simps(2)
tff(fact_5058_less__minus__one__simps_I2_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer)) ).

% less_minus_one_simps(2)
tff(fact_5059_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% less_minus_one_simps(4)
tff(fact_5060_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% less_minus_one_simps(4)
tff(fact_5061_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% less_minus_one_simps(4)
tff(fact_5062_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% less_minus_one_simps(4)
tff(fact_5063_nonzero__minus__divide__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5064_nonzero__minus__divide__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5065_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5066_nonzero__minus__divide__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5067_nonzero__minus__divide__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5068_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5069_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5070_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5071_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5072_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5073_diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ).

% diff_conv_add_uminus
tff(fact_5074_diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ).

% diff_conv_add_uminus
tff(fact_5075_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).

% diff_conv_add_uminus
tff(fact_5076_diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% diff_conv_add_uminus
tff(fact_5077_group__cancel_Osub2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5078_group__cancel_Osub2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5079_group__cancel_Osub2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B) )
     => ( aa_rat_rat(minus_minus_rat(A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5080_group__cancel_Osub2,axiom,
    ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B2 = plus_p5714425477246183910nteger(K,B) )
     => ( minus_8373710615458151222nteger(A,B2) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),minus_8373710615458151222nteger(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_5081_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))),zero_zero_real)) ).

% neg_numeral_le_zero
tff(fact_5082_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))),zero_z3403309356797280102nteger)) ).

% neg_numeral_le_zero
tff(fact_5083_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))),zero_zero_rat)) ).

% neg_numeral_le_zero
tff(fact_5084_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),zero_zero_int)) ).

% neg_numeral_le_zero
tff(fact_5085_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_zero_le_neg_numeral
tff(fact_5086_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)))) ).

% not_zero_le_neg_numeral
tff(fact_5087_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_zero_le_neg_numeral
tff(fact_5088_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_zero_le_neg_numeral
tff(fact_5089_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_zero_less_neg_numeral
tff(fact_5090_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_zero_less_neg_numeral
tff(fact_5091_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_zero_less_neg_numeral
tff(fact_5092_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)))) ).

% not_zero_less_neg_numeral
tff(fact_5093_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),zero_zero_int)) ).

% neg_numeral_less_zero
tff(fact_5094_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))),zero_zero_real)) ).

% neg_numeral_less_zero
tff(fact_5095_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))),zero_zero_rat)) ).

% neg_numeral_less_zero
tff(fact_5096_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))),zero_z3403309356797280102nteger)) ).

% neg_numeral_less_zero
tff(fact_5097_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% le_minus_one_simps(3)
tff(fact_5098_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% le_minus_one_simps(3)
tff(fact_5099_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% le_minus_one_simps(3)
tff(fact_5100_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% le_minus_one_simps(3)
tff(fact_5101_le__minus__one__simps_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real)) ).

% le_minus_one_simps(1)
tff(fact_5102_le__minus__one__simps_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger)) ).

% le_minus_one_simps(1)
tff(fact_5103_le__minus__one__simps_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat)) ).

% le_minus_one_simps(1)
tff(fact_5104_le__minus__one__simps_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int)) ).

% le_minus_one_simps(1)
tff(fact_5105_less__minus__one__simps_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int)) ).

% less_minus_one_simps(1)
tff(fact_5106_less__minus__one__simps_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real)) ).

% less_minus_one_simps(1)
tff(fact_5107_less__minus__one__simps_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat)) ).

% less_minus_one_simps(1)
tff(fact_5108_less__minus__one__simps_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger)) ).

% less_minus_one_simps(1)
tff(fact_5109_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% less_minus_one_simps(3)
tff(fact_5110_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% less_minus_one_simps(3)
tff(fact_5111_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% less_minus_one_simps(3)
tff(fact_5112_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% less_minus_one_simps(3)
tff(fact_5113_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real)) ).

% neg_numeral_le_one
tff(fact_5114_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),one_one_Code_integer)) ).

% neg_numeral_le_one
tff(fact_5115_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat)) ).

% neg_numeral_le_one
tff(fact_5116_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int)) ).

% neg_numeral_le_one
tff(fact_5117_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,M))) ).

% neg_one_le_numeral
tff(fact_5118_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(M))) ).

% neg_one_le_numeral
tff(fact_5119_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,M))) ).

% neg_one_le_numeral
tff(fact_5120_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M))) ).

% neg_one_le_numeral
tff(fact_5121_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,one_one_real))) ).

% neg_numeral_le_neg_one
tff(fact_5122_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% neg_numeral_le_neg_one
tff(fact_5123_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% neg_numeral_le_neg_one
tff(fact_5124_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int))) ).

% neg_numeral_le_neg_one
tff(fact_5125_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,one_one_real))) ).

% not_numeral_le_neg_one
tff(fact_5126_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% not_numeral_le_neg_one
tff(fact_5127_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% not_numeral_le_neg_one
tff(fact_5128_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int))) ).

% not_numeral_le_neg_one
tff(fact_5129_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)))) ).

% not_one_le_neg_numeral
tff(fact_5130_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)))) ).

% not_one_le_neg_numeral
tff(fact_5131_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)))) ).

% not_one_le_neg_numeral
tff(fact_5132_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))) ).

% not_one_le_neg_numeral
tff(fact_5133_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5134_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5135_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5136_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5137_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))) ).

% not_one_less_neg_numeral
tff(fact_5138_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)))) ).

% not_one_less_neg_numeral
tff(fact_5139_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)))) ).

% not_one_less_neg_numeral
tff(fact_5140_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)))) ).

% not_one_less_neg_numeral
tff(fact_5141_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int))) ).

% not_numeral_less_neg_one
tff(fact_5142_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,one_one_real))) ).

% not_numeral_less_neg_one
tff(fact_5143_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% not_numeral_less_neg_one
tff(fact_5144_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% not_numeral_less_neg_one
tff(fact_5145_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M))) ).

% neg_one_less_numeral
tff(fact_5146_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,M))) ).

% neg_one_less_numeral
tff(fact_5147_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,M))) ).

% neg_one_less_numeral
tff(fact_5148_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(M))) ).

% neg_one_less_numeral
tff(fact_5149_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int)) ).

% neg_numeral_less_one
tff(fact_5150_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real)) ).

% neg_numeral_less_one
tff(fact_5151_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat)) ).

% neg_numeral_less_one
tff(fact_5152_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),one_one_Code_integer)) ).

% neg_numeral_less_one
tff(fact_5153_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(uminus1482373934393186551omplex,B) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_5154_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(uminus_uminus_real,B) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_5155_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(uminus_uminus_rat,B) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_5156_minus__divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) = A )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,B) = aa_complex_complex(times_times_complex(A),C) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_5157_minus__divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = A )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_5158_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) = A )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(uminus_uminus_rat,B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_5159_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = C )
      <=> ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(times_times_complex(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5160_nonzero__neg__divide__eq__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = C )
      <=> ( aa_real_real(uminus_uminus_real,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5161_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = C )
      <=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5162_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) )
      <=> ( aa_complex_complex(times_times_complex(C),B) = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5163_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: real,C: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( C = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) = aa_real_real(uminus_uminus_real,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5164_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5165_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( B != zero_zero_complex )
        & ( A = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_5166_divide__eq__minus__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( ( B != zero_zero_real )
        & ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_5167_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( ( B != zero_zero_rat )
        & ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_5168_pos__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( 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,B),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% pos_minus_divide_less_eq
tff(fact_5169_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) ) ) ).

% pos_minus_divide_less_eq
tff(fact_5170_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% pos_less_minus_divide_eq
tff(fact_5171_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% pos_less_minus_divide_eq
tff(fact_5172_neg__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_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,B),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% neg_minus_divide_less_eq
tff(fact_5173_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% neg_minus_divide_less_eq
tff(fact_5174_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% neg_less_minus_divide_eq
tff(fact_5175_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) ) ) ).

% neg_less_minus_divide_eq
tff(fact_5176_minus__divide__less__eq,axiom,
    ! [B: real,C: real,A: 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,B),C))),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_5177_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_5178_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_5179_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_5180_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W2: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) )
    <=> ( ( ( C != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),C) ) )
        & ( ( C = zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5181_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) )
    <=> ( ( ( C != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C) ) )
        & ( ( C = zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5182_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) )
    <=> ( ( ( C != zero_zero_rat )
         => ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C) ) )
        & ( ( C = zero_zero_rat )
         => ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5183_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: complex,C: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),C) = B ) )
        & ( ( C = zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5184_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C) = B ) )
        & ( ( C = zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5185_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C) = B ) )
        & ( ( C = zero_zero_rat )
         => ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5186_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q4: 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,Q4))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_5187_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q4: num] : aa_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,Q4))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_5188_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q4: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_5189_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = B ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_5190_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = B ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_5191_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = B ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_5192_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5193_minus__divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z))),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5194_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(X),Z))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5195_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_5196_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = aa_real_real(uminus_uminus_real,B) ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_5197_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = aa_rat_rat(uminus_uminus_rat,B) ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_5198_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),aa_complex_complex(times_times_complex(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_5199_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = aa_real_real(uminus_uminus_real,B) ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_5200_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = aa_rat_rat(uminus_uminus_rat,B) ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_5201_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5202_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z))),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5203_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(X),Z))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5204_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_5205_verit__less__mono__div__int2,axiom,
    ! [A2: int,B2: int,N: 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,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(divide_divide_int(B2),N)),aa_int_int(divide_divide_int(A2),N))) ) ) ).

% verit_less_mono_div_int2
tff(fact_5206_div__eq__minus1,axiom,
    ! [B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_5207_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : aa_num_nat(size_num,aa_num_num(bit1,X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(3)
tff(fact_5208_num_Osize_I6_J,axiom,
    ! [X32: num] : aa_num_nat(size_size_num,aa_num_num(bit1,X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(6)
tff(fact_5209_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_5210_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_5211_minus__divide__le__eq,axiom,
    ! [B: real,C: real,A: 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,B),C))),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_5212_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_5213_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% neg_le_minus_divide_eq
tff(fact_5214_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) ) ) ).

% neg_le_minus_divide_eq
tff(fact_5215_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_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,B),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5216_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5217_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5218_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5219_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( 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,B),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5220_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5221_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5222_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5223_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5224_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5225_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: 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,Q4))) = 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,Q4))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q4)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5226_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,N: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) = aa_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,Q4))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q4)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5227_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,N: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(Q4)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5228_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: 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,Q4))) = 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,Q4))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q4)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5229_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: num] :
      ( ( aa_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,Q4))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q4)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5230_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) = modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q4)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5231_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5232_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5233_neg__one__power__add__eq__neg__one__power__diff,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(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = 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(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5234_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5235_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5236_zmod__minus1,axiom,
    ! [B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),one_one_int) ) ) ).

% zmod_minus1
tff(fact_5237_zminus1__lemma,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R2)))
     => ( ( B != zero_zero_int )
       => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(uminus_uminus_int,A),B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int),aa_int_int(uminus_uminus_int,Q4)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q4)),one_one_int))),aa_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,B),R2))))) ) ) ).

% zminus1_lemma
tff(fact_5238_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5239_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5240_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5241_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5242_square__le__1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% square_le_1
tff(fact_5243_square__le__1,axiom,
    ! [X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),X))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)) ) ) ).

% square_le_1
tff(fact_5244_square__le__1,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)) ) ) ).

% square_le_1
tff(fact_5245_square__le__1,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)) ) ) ).

% square_le_1
tff(fact_5246_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(divide_divide_int(K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_5247_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_5248_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_5249_signed__take__bit__rec,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( N = zero_zero_nat )
       => ( bit_ri6519982836138164636nteger(N,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) )
      & ( ( N != zero_zero_nat )
       => ( bit_ri6519982836138164636nteger(N,A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ) ) ) ).

% signed_take_bit_rec
tff(fact_5250_signed__take__bit__rec,axiom,
    ! [N: nat,A: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_int_int(bit_ri631733984087533419it_int(N),A) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) )
      & ( ( N != zero_zero_nat )
       => ( aa_int_int(bit_ri631733984087533419it_int(N),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),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(bit_ri631733984087533419it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% signed_take_bit_rec
tff(fact_5251_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique5052692396658037445od_int(aa_num_num(bitM,M),aa_num_num(bit0,one)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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_5252_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_complex(ring_17405671764205052669omplex,K) = zero_zero_complex ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_complex(ring_17405671764205052669omplex,K) = aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_complex(ring_17405671764205052669omplex,K) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_complex)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_5253_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_int(ring_1_of_int_int,K) = zero_zero_int ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_int(ring_1_of_int_int,K) = aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_int(ring_1_of_int_int,K) = aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_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(ring_1_of_int_int,aa_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(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(ring_1_of_int_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_int)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_5254_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_real(ring_1_of_int_real,K) = zero_zero_real ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_real(ring_1_of_int_real,K) = aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_real(ring_1_of_int_real,K) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),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_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_real)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_5255_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_rat(ring_1_of_int_rat,K) = zero_zero_rat ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_rat(ring_1_of_int_rat,K) = aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_rat(ring_1_of_int_rat,K) = aa_rat_rat(if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_rat)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_5256_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = zero_z3403309356797280102nteger ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = if_Code_integer(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_Code_integer)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_5257_signed__take__bit__0,axiom,
    ! [A: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_5258_signed__take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_ri631733984087533419it_int(zero_zero_nat),A) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_5259_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5260_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5261_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5262_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5263_Compl__anti__mono,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(uminus1532241313380277803et_int(B2)),uminus1532241313380277803et_int(A2))) ) ).

% Compl_anti_mono
tff(fact_5264_Compl__subset__Compl__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(A2)),uminus1532241313380277803et_int(B2)))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2)) ) ).

% Compl_subset_Compl_iff
tff(fact_5265_signed__take__bit__of__0,axiom,
    ! [N: nat] : aa_int_int(bit_ri631733984087533419it_int(N),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_5266_of__int__0,axiom,
    aa_int_complex(ring_17405671764205052669omplex,zero_zero_int) = zero_zero_complex ).

% of_int_0
tff(fact_5267_of__int__0,axiom,
    aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_5268_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_5269_of__int__0,axiom,
    aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).

% of_int_0
tff(fact_5270_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_complex = aa_int_complex(ring_17405671764205052669omplex,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_5271_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_int = aa_int_int(ring_1_of_int_int,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_5272_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_real = aa_int_real(ring_1_of_int_real,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_5273_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_rat = aa_int_rat(ring_1_of_int_rat,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_5274_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = zero_zero_complex )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_5275_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = zero_zero_int )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_5276_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = zero_zero_real )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_5277_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = zero_zero_rat )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_5278_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z)) ) ).

% of_int_le_iff
tff(fact_5279_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z)) ) ).

% of_int_le_iff
tff(fact_5280_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z)) ) ).

% of_int_le_iff
tff(fact_5281_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ).

% of_int_less_iff
tff(fact_5282_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ).

% of_int_less_iff
tff(fact_5283_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ).

% of_int_less_iff
tff(fact_5284_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_5285_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_5286_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_5287_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_5288_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_5289_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_5290_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_5291_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_5292_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_5293_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_5294_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_5295_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_5296_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_5297_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_5298_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_5299_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_5300_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_5301_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_5302_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_5303_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_5304_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_5305_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_5306_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_5307_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_5308_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_5309_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_5310_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_5311_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_5312_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_5313_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_5314_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_5315_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_5316_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_5317_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_5318_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_5319_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_5320_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: 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,B)),W2)),aa_int_real(ring_1_of_int_real,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W2)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5321_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: 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,B)),W2)),aa_int_rat(ring_1_of_int_rat,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W2)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5322_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2)),aa_int_int(ring_1_of_int_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W2)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5323_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5324_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5325_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5326_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: 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,B)),W2)),aa_int_real(ring_1_of_int_real,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W2)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5327_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: 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,B)),W2)),aa_int_rat(ring_1_of_int_rat,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W2)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5328_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2)),aa_int_int(ring_1_of_int_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W2)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5329_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5330_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5331_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5332_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: 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,X)),N)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5333_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: 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,X)),N)),aa_int_rat(ring_1_of_int_rat,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5334_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5335_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5336_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5337_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5338_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: 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,X)),N)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5339_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: 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,X)),N)),aa_int_rat(ring_1_of_int_rat,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5340_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5341_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5342_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5343_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5344_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( 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(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5345_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5346_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N)),aa_int_rat(ring_1_of_int_rat,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5347_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5348_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5349_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5350_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5351_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5352_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5353_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5354_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N)),aa_int_rat(ring_1_of_int_rat,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5355_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5356_subset__Compl__self__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),uminus6524753893492686040at_nat(A2)))
    <=> ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% subset_Compl_self_eq
tff(fact_5357_subset__Compl__self__eq,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),uminus612125837232591019t_real(A2)))
    <=> ( A2 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
tff(fact_5358_subset__Compl__self__eq,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),uminus5710092332889474511et_nat(A2)))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_5359_subset__Compl__self__eq,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),uminus1532241313380277803et_int(A2)))
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_5360_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(uminus_uminus_real,X))) ) ).

% real_add_less_0_iff
tff(fact_5361_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,X)),Y)) ) ).

% real_0_less_add_iff
tff(fact_5362_signed__take__bit__take__bit,axiom,
    ! [M: nat,N: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(M),aa_int_int(bit_se2923211474154528505it_int(N),A)) = aa_int_int(if_fun_int_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M),bit_se2923211474154528505it_int(N),bit_ri631733984087533419it_int(M)),A) ).

% signed_take_bit_take_bit
tff(fact_5363_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_5364_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_5365_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_5366_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_5367_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_5368_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_5369_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_5370_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_5371_take__bit__signed__take__bit,axiom,
    ! [M: nat,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_int_int(bit_se2923211474154528505it_int(M),aa_int_int(bit_ri631733984087533419it_int(N),A)) = aa_int_int(bit_se2923211474154528505it_int(M),A) ) ) ).

% take_bit_signed_take_bit
tff(fact_5372_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(bit_ri631733984087533419it_int(N),K)),aa_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_5373_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(bit_ri631733984087533419it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_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_5374_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(bit_ri631733984087533419it_int(N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_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_5375_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(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(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_5376_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_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(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(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_5377_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(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(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( aa_int_int(bit_ri631733984087533419it_int(N),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_5378_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(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(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N)))),aa_int_int(bit_ri631733984087533419it_int(N),K))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_5379_floor__exists1,axiom,
    ! [X: real] :
    ? [X4: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X4)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),one_one_int))))
      & ! [Y3: int] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y3)),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y3),one_one_int)))) )
         => ( Y3 = X4 ) ) ) ).

% floor_exists1
tff(fact_5380_floor__exists1,axiom,
    ! [X: 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)),X))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),one_one_int))))
      & ! [Y3: int] :
          ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y3)),X))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y3),one_one_int)))) )
         => ( Y3 = X4 ) ) ) ).

% floor_exists1
tff(fact_5381_floor__exists,axiom,
    ! [X: real] :
    ? [Z3: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z3)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z3),one_one_int)))) ) ).

% floor_exists
tff(fact_5382_floor__exists,axiom,
    ! [X: rat] :
    ? [Z3: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z3)),X))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z3),one_one_int)))) ) ).

% floor_exists
tff(fact_5383_round__unique,axiom,
    ! [X: real,Y: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),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,Y)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),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,X) = Y ) ) ) ).

% round_unique
tff(fact_5384_round__unique,axiom,
    ! [X: rat,Y: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(X),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,Y)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),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,X) = Y ) ) ) ).

% round_unique
tff(fact_5385_set__encode__insert,axiom,
    ! [A2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(N),A2))
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(N,A2)) = 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,A2)) ) ) ) ).

% set_encode_insert
tff(fact_5386_arith__series__nat,axiom,
    ! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_fs(A),D)),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))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_5387_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),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)))),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_5388_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),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)))),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_5389_insertCI,axiom,
    ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),B2))
       => ( A = B ) )
     => pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(B,B2))) ) ).

% insertCI
tff(fact_5390_insertCI,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),B2))
       => ( A = B ) )
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(B,B2))) ) ).

% insertCI
tff(fact_5391_insertCI,axiom,
    ! [A: real,B2: set_real,B: real] :
      ( ( ~ pp(aa_set_real_bool(member_real(A),B2))
       => ( A = B ) )
     => pp(aa_set_real_bool(member_real(A),insert_real(B,B2))) ) ).

% insertCI
tff(fact_5392_insertCI,axiom,
    ! [A: set_nat,B2: set_set_nat,B: set_nat] :
      ( ( ~ pp(aa_set_set_nat_bool(member_set_nat(A),B2))
       => ( A = B ) )
     => pp(aa_set_set_nat_bool(member_set_nat(A),insert_set_nat(B,B2))) ) ).

% insertCI
tff(fact_5393_insertCI,axiom,
    ! [A: nat,B2: set_nat,B: nat] :
      ( ( ~ pp(aa_set_nat_bool(member_nat(A),B2))
       => ( A = B ) )
     => pp(aa_set_nat_bool(member_nat(A),insert_nat(B,B2))) ) ).

% insertCI
tff(fact_5394_insertCI,axiom,
    ! [A: int,B2: set_int,B: int] :
      ( ( ~ pp(aa_set_int_bool(member_int(A),B2))
       => ( A = B ) )
     => pp(aa_set_int_bool(member_int(A),insert_int(B,B2))) ) ).

% insertCI
tff(fact_5395_insert__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(B,A2)))
    <=> ( ( A = B )
        | pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),A2)) ) ) ).

% insert_iff
tff(fact_5396_insert__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(B,A2)))
    <=> ( ( A = B )
        | pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),A2)) ) ) ).

% insert_iff
tff(fact_5397_insert__iff,axiom,
    ! [A: real,B: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(A),insert_real(B,A2)))
    <=> ( ( A = B )
        | pp(aa_set_real_bool(member_real(A),A2)) ) ) ).

% insert_iff
tff(fact_5398_insert__iff,axiom,
    ! [A: set_nat,B: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),insert_set_nat(B,A2)))
    <=> ( ( A = B )
        | pp(aa_set_set_nat_bool(member_set_nat(A),A2)) ) ) ).

% insert_iff
tff(fact_5399_insert__iff,axiom,
    ! [A: nat,B: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),insert_nat(B,A2)))
    <=> ( ( A = B )
        | pp(aa_set_nat_bool(member_nat(A),A2)) ) ) ).

% insert_iff
tff(fact_5400_insert__iff,axiom,
    ! [A: int,B: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(A),insert_int(B,A2)))
    <=> ( ( A = B )
        | pp(aa_set_int_bool(member_int(A),A2)) ) ) ).

% insert_iff
tff(fact_5401_insert__absorb2,axiom,
    ! [X: nat,A2: set_nat] : insert_nat(X,insert_nat(X,A2)) = insert_nat(X,A2) ).

% insert_absorb2
tff(fact_5402_insert__absorb2,axiom,
    ! [X: int,A2: set_int] : insert_int(X,insert_int(X,A2)) = insert_int(X,A2) ).

% insert_absorb2
tff(fact_5403_insert__absorb2,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(X,insert8211810215607154385at_nat(X,A2)) = insert8211810215607154385at_nat(X,A2) ).

% insert_absorb2
tff(fact_5404_insert__absorb2,axiom,
    ! [X: real,A2: set_real] : insert_real(X,insert_real(X,A2)) = insert_real(X,A2) ).

% insert_absorb2
tff(fact_5405_insert__absorb2,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(X,insert9069300056098147895at_nat(X,A2)) = insert9069300056098147895at_nat(X,A2) ).

% insert_absorb2
tff(fact_5406_ComplI,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A2))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),uminus6524753893492686040at_nat(A2))) ) ).

% ComplI
tff(fact_5407_ComplI,axiom,
    ! [C: real,A2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(C),A2))
     => pp(aa_set_real_bool(member_real(C),uminus612125837232591019t_real(A2))) ) ).

% ComplI
tff(fact_5408_ComplI,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(C),A2))
     => pp(aa_set_set_nat_bool(member_set_nat(C),uminus613421341184616069et_nat(A2))) ) ).

% ComplI
tff(fact_5409_ComplI,axiom,
    ! [C: nat,A2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(C),A2))
     => pp(aa_set_nat_bool(member_nat(C),uminus5710092332889474511et_nat(A2))) ) ).

% ComplI
tff(fact_5410_ComplI,axiom,
    ! [C: int,A2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(C),A2))
     => pp(aa_set_int_bool(member_int(C),uminus1532241313380277803et_int(A2))) ) ).

% ComplI
tff(fact_5411_Compl__iff,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),uminus6524753893492686040at_nat(A2)))
    <=> ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A2)) ) ).

% Compl_iff
tff(fact_5412_Compl__iff,axiom,
    ! [C: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),uminus612125837232591019t_real(A2)))
    <=> ~ pp(aa_set_real_bool(member_real(C),A2)) ) ).

% Compl_iff
tff(fact_5413_Compl__iff,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),uminus613421341184616069et_nat(A2)))
    <=> ~ pp(aa_set_set_nat_bool(member_set_nat(C),A2)) ) ).

% Compl_iff
tff(fact_5414_Compl__iff,axiom,
    ! [C: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),uminus5710092332889474511et_nat(A2)))
    <=> ~ pp(aa_set_nat_bool(member_nat(C),A2)) ) ).

% Compl_iff
tff(fact_5415_Compl__iff,axiom,
    ! [C: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),uminus1532241313380277803et_int(A2)))
    <=> ~ pp(aa_set_int_bool(member_int(C),A2)) ) ).

% Compl_iff
tff(fact_5416_singletonI,axiom,
    ! [A: produc3843707927480180839at_nat] : pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) ).

% singletonI
tff(fact_5417_singletonI,axiom,
    ! [A: set_nat] : pp(aa_set_set_nat_bool(member_set_nat(A),insert_set_nat(A,bot_bot_set_set_nat))) ).

% singletonI
tff(fact_5418_singletonI,axiom,
    ! [A: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) ).

% singletonI
tff(fact_5419_singletonI,axiom,
    ! [A: real] : pp(aa_set_real_bool(member_real(A),insert_real(A,bot_bot_set_real))) ).

% singletonI
tff(fact_5420_singletonI,axiom,
    ! [A: nat] : pp(aa_set_nat_bool(member_nat(A),insert_nat(A,bot_bot_set_nat))) ).

% singletonI
tff(fact_5421_singletonI,axiom,
    ! [A: int] : pp(aa_set_int_bool(member_int(A),insert_int(A,bot_bot_set_int))) ).

% singletonI
tff(fact_5422_finite__insert,axiom,
    ! [A: real,A2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,insert_real(A,A2)))
    <=> pp(aa_set_real_bool(finite_finite_real,A2)) ) ).

% finite_insert
tff(fact_5423_finite__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,insert9069300056098147895at_nat(A,A2)))
    <=> pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A2)) ) ).

% finite_insert
tff(fact_5424_finite__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,insert_nat(A,A2)))
    <=> pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ).

% finite_insert
tff(fact_5425_finite__insert,axiom,
    ! [A: int,A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,insert_int(A,A2)))
    <=> pp(aa_set_int_bool(finite_finite_int,A2)) ) ).

% finite_insert
tff(fact_5426_finite__insert,axiom,
    ! [A: complex,A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,insert_complex(A,A2)))
    <=> pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ).

% finite_insert
tff(fact_5427_finite__insert,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,insert8211810215607154385at_nat(A,A2)))
    <=> pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2)) ) ).

% finite_insert
tff(fact_5428_insert__subset,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(insert9069300056098147895at_nat(X,A2)),B2))
    <=> ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),B2))
        & pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(A2),B2)) ) ) ).

% insert_subset
tff(fact_5429_insert__subset,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(insert8211810215607154385at_nat(X,A2)),B2))
    <=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B2))
        & pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B2)) ) ) ).

% insert_subset
tff(fact_5430_insert__subset,axiom,
    ! [X: real,A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(insert_real(X,A2)),B2))
    <=> ( pp(aa_set_real_bool(member_real(X),B2))
        & pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2)) ) ) ).

% insert_subset
tff(fact_5431_insert__subset,axiom,
    ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(insert_set_nat(X,A2)),B2))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(X),B2))
        & pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B2)) ) ) ).

% insert_subset
tff(fact_5432_insert__subset,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,insert_nat(X,A2)),B2))
    <=> ( pp(aa_set_nat_bool(member_nat(X),B2))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2)) ) ) ).

% insert_subset
tff(fact_5433_insert__subset,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(insert_int(X,A2)),B2))
    <=> ( pp(aa_set_int_bool(member_int(X),B2))
        & pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ) ).

% insert_subset
tff(fact_5434_Diff__insert0,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),A2))
     => ( minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(X,B2)) = minus_3314409938677909166at_nat(A2,B2) ) ) ).

% Diff_insert0
tff(fact_5435_Diff__insert0,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2))
     => ( minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(X,B2)) = minus_1356011639430497352at_nat(A2,B2) ) ) ).

% Diff_insert0
tff(fact_5436_Diff__insert0,axiom,
    ! [X: real,A2: set_real,B2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X),A2))
     => ( minus_minus_set_real(A2,insert_real(X,B2)) = minus_minus_set_real(A2,B2) ) ) ).

% Diff_insert0
tff(fact_5437_Diff__insert0,axiom,
    ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A2))
     => ( minus_2163939370556025621et_nat(A2,insert_set_nat(X,B2)) = minus_2163939370556025621et_nat(A2,B2) ) ) ).

% Diff_insert0
tff(fact_5438_Diff__insert0,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X),A2))
     => ( minus_minus_set_int(A2,insert_int(X,B2)) = minus_minus_set_int(A2,B2) ) ) ).

% Diff_insert0
tff(fact_5439_Diff__insert0,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
     => ( minus_minus_set_nat(A2,insert_nat(X,B2)) = minus_minus_set_nat(A2,B2) ) ) ).

% Diff_insert0
tff(fact_5440_insert__Diff1,axiom,
    ! [X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),B2))
     => ( minus_3314409938677909166at_nat(insert9069300056098147895at_nat(X,A2),B2) = minus_3314409938677909166at_nat(A2,B2) ) ) ).

% insert_Diff1
tff(fact_5441_insert__Diff1,axiom,
    ! [X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B2))
     => ( minus_1356011639430497352at_nat(insert8211810215607154385at_nat(X,A2),B2) = minus_1356011639430497352at_nat(A2,B2) ) ) ).

% insert_Diff1
tff(fact_5442_insert__Diff1,axiom,
    ! [X: real,B2: set_real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(X),B2))
     => ( minus_minus_set_real(insert_real(X,A2),B2) = minus_minus_set_real(A2,B2) ) ) ).

% insert_Diff1
tff(fact_5443_insert__Diff1,axiom,
    ! [X: set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),B2))
     => ( minus_2163939370556025621et_nat(insert_set_nat(X,A2),B2) = minus_2163939370556025621et_nat(A2,B2) ) ) ).

% insert_Diff1
tff(fact_5444_insert__Diff1,axiom,
    ! [X: int,B2: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(X),B2))
     => ( minus_minus_set_int(insert_int(X,A2),B2) = minus_minus_set_int(A2,B2) ) ) ).

% insert_Diff1
tff(fact_5445_insert__Diff1,axiom,
    ! [X: nat,B2: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),B2))
     => ( minus_minus_set_nat(insert_nat(X,A2),B2) = minus_minus_set_nat(A2,B2) ) ) ).

% insert_Diff1
tff(fact_5446_xor_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),zero_zero_nat) = A ).

% xor.right_neutral
tff(fact_5447_xor_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(A),zero_zero_int) = A ).

% xor.right_neutral
tff(fact_5448_xor_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,zero_zero_nat),A) = A ).

% xor.left_neutral
tff(fact_5449_xor_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(zero_zero_int),A) = A ).

% xor.left_neutral
tff(fact_5450_xor__self__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),A) = zero_zero_nat ).

% xor_self_eq
tff(fact_5451_xor__self__eq,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(A),A) = zero_zero_int ).

% xor_self_eq
tff(fact_5452_bit_Oxor__self,axiom,
    ! [X: int] : aa_int_int(bit_se6526347334894502574or_int(X),X) = zero_zero_int ).

% bit.xor_self
tff(fact_5453_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ck),A2) = zero_zero_nat ).

% sum.neutral_const
tff(fact_5454_sum_Oneutral__const,axiom,
    ! [A2: set_complex] : groups7754918857620584856omplex(aTP_Lamm_ch,A2) = zero_zero_complex ).

% sum.neutral_const
tff(fact_5455_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : groups6591440286371151544t_real(aTP_Lamm_ft,A2) = zero_zero_real ).

% sum.neutral_const
tff(fact_5456_singleton__conv,axiom,
    ! [A: produc3843707927480180839at_nat] : collec6321179662152712658at_nat(aTP_Lamm_fu(A)) = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ).

% singleton_conv
tff(fact_5457_singleton__conv,axiom,
    ! [A: list_nat] : collect_list_nat(aTP_Lamm_fv(A)) = insert_list_nat(A,bot_bot_set_list_nat) ).

% singleton_conv
tff(fact_5458_singleton__conv,axiom,
    ! [A: set_nat] : collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_fw,A)) = insert_set_nat(A,bot_bot_set_set_nat) ).

% singleton_conv
tff(fact_5459_singleton__conv,axiom,
    ! [A: product_prod_nat_nat] : collec3392354462482085612at_nat(aa_Pro595088089689268830t_bool(aTP_Lamm_fx,A)) = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ).

% singleton_conv
tff(fact_5460_singleton__conv,axiom,
    ! [A: real] : collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_fy,A)) = insert_real(A,bot_bot_set_real) ).

% singleton_conv
tff(fact_5461_singleton__conv,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_fz,A)) = insert_nat(A,bot_bot_set_nat) ).

% singleton_conv
tff(fact_5462_singleton__conv,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_bool(aTP_Lamm_ga,A)) = insert_int(A,bot_bot_set_int) ).

% singleton_conv
tff(fact_5463_singleton__conv2,axiom,
    ! [A: produc3843707927480180839at_nat] : collec6321179662152712658at_nat(fequal7949547063978738364at_nat(A)) = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ).

% singleton_conv2
tff(fact_5464_singleton__conv2,axiom,
    ! [A: list_nat] : collect_list_nat(fequal_list_nat(A)) = insert_list_nat(A,bot_bot_set_list_nat) ).

% singleton_conv2
tff(fact_5465_singleton__conv2,axiom,
    ! [A: set_nat] : collect_set_nat(aa_set5116768664143533222t_bool(fequal_set_nat,A)) = insert_set_nat(A,bot_bot_set_set_nat) ).

% singleton_conv2
tff(fact_5466_singleton__conv2,axiom,
    ! [A: product_prod_nat_nat] : collec3392354462482085612at_nat(aa_Pro595088089689268830t_bool(fequal9188845342289841878at_nat,A)) = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ).

% singleton_conv2
tff(fact_5467_singleton__conv2,axiom,
    ! [A: real] : collect_real(aa_rea7809428808904475482l_bool(fequal_real,A)) = insert_real(A,bot_bot_set_real) ).

% singleton_conv2
tff(fact_5468_singleton__conv2,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_bool(fequal_nat,A)) = insert_nat(A,bot_bot_set_nat) ).

% singleton_conv2
tff(fact_5469_singleton__conv2,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_bool(fequal_int,A)) = insert_int(A,bot_bot_set_int) ).

% singleton_conv2
tff(fact_5470_sum_Oempty,axiom,
    ! [G: fun_real_complex] : groups5754745047067104278omplex(G,bot_bot_set_real) = zero_zero_complex ).

% sum.empty
tff(fact_5471_sum_Oempty,axiom,
    ! [G: fun_real_real] : groups8097168146408367636l_real(G,bot_bot_set_real) = zero_zero_real ).

% sum.empty
tff(fact_5472_sum_Oempty,axiom,
    ! [G: fun_real_rat] : groups1300246762558778688al_rat(G,bot_bot_set_real) = zero_zero_rat ).

% sum.empty
tff(fact_5473_sum_Oempty,axiom,
    ! [G: fun_real_nat] : groups1935376822645274424al_nat(G,bot_bot_set_real) = zero_zero_nat ).

% sum.empty
tff(fact_5474_sum_Oempty,axiom,
    ! [G: fun_real_int] : groups1932886352136224148al_int(G,bot_bot_set_real) = zero_zero_int ).

% sum.empty
tff(fact_5475_sum_Oempty,axiom,
    ! [G: fun_nat_complex] : groups2073611262835488442omplex(G,bot_bot_set_nat) = zero_zero_complex ).

% sum.empty
tff(fact_5476_sum_Oempty,axiom,
    ! [G: fun_nat_rat] : groups2906978787729119204at_rat(G,bot_bot_set_nat) = zero_zero_rat ).

% sum.empty
tff(fact_5477_sum_Oempty,axiom,
    ! [G: fun_nat_int] : groups3539618377306564664at_int(G,bot_bot_set_nat) = zero_zero_int ).

% sum.empty
tff(fact_5478_sum_Oempty,axiom,
    ! [G: fun_int_complex] : groups3049146728041665814omplex(G,bot_bot_set_int) = zero_zero_complex ).

% sum.empty
tff(fact_5479_sum_Oempty,axiom,
    ! [G: fun_int_real] : groups8778361861064173332t_real(G,bot_bot_set_int) = zero_zero_real ).

% sum.empty
tff(fact_5480_sum__eq__0__iff,axiom,
    ! [F3: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( ( groups4541462559716669496nt_nat(F,F3) = zero_zero_nat )
      <=> ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),F3))
           => ( aa_int_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5481_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( ( groups5693394587270226106ex_nat(F,F3) = zero_zero_nat )
      <=> ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),F3))
           => ( aa_complex_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5482_sum__eq__0__iff,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F3))
     => ( ( groups977919841031483927at_nat(F,F3) = zero_zero_nat )
      <=> ! [X3: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),F3))
           => ( aa_Pro4243173770044397973at_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5483_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),F3) = zero_zero_nat )
      <=> ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),F3))
           => ( aa_nat_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5484_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_complex] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups2073611262835488442omplex(G,A2) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_5485_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3049146728041665814omplex(G,A2) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_5486_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups8778361861064173332t_real(G,A2) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_5487_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5808333547571424918x_real(G,A2) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_5488_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_rat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups2906978787729119204at_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_5489_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3906332499630173760nt_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_5490_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5058264527183730370ex_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_5491_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups4541462559716669496nt_nat(G,A2) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_5492_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_nat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5693394587270226106ex_nat(G,A2) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_5493_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_int] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups3539618377306564664at_int(G,A2) = zero_zero_int ) ) ).

% sum.infinite
tff(fact_5494_singleton__insert__inj__eq,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat) = insert9069300056098147895at_nat(A,A2) )
    <=> ( ( A = B )
        & pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(A2),insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat))) ) ) ).

% singleton_insert_inj_eq
tff(fact_5495_singleton__insert__inj__eq,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat) = insert8211810215607154385at_nat(A,A2) )
    <=> ( ( A = B )
        & pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat))) ) ) ).

% singleton_insert_inj_eq
tff(fact_5496_singleton__insert__inj__eq,axiom,
    ! [B: real,A: real,A2: set_real] :
      ( ( insert_real(B,bot_bot_set_real) = insert_real(A,A2) )
    <=> ( ( A = B )
        & pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(B,bot_bot_set_real))) ) ) ).

% singleton_insert_inj_eq
tff(fact_5497_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A2: set_nat] :
      ( ( insert_nat(B,bot_bot_set_nat) = insert_nat(A,A2) )
    <=> ( ( A = B )
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(B,bot_bot_set_nat))) ) ) ).

% singleton_insert_inj_eq
tff(fact_5498_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A2: set_int] :
      ( ( insert_int(B,bot_bot_set_int) = insert_int(A,A2) )
    <=> ( ( A = B )
        & pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(B,bot_bot_set_int))) ) ) ).

% singleton_insert_inj_eq
tff(fact_5499_singleton__insert__inj__eq_H,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( insert9069300056098147895at_nat(A,A2) = insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat) )
    <=> ( ( A = B )
        & pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(A2),insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_5500_singleton__insert__inj__eq_H,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A,A2) = insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat) )
    <=> ( ( A = B )
        & pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_5501_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A2: set_real,B: real] :
      ( ( insert_real(A,A2) = insert_real(B,bot_bot_set_real) )
    <=> ( ( A = B )
        & pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(B,bot_bot_set_real))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_5502_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B: nat] :
      ( ( insert_nat(A,A2) = insert_nat(B,bot_bot_set_nat) )
    <=> ( ( A = B )
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(B,bot_bot_set_nat))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_5503_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A2: set_int,B: int] :
      ( ( insert_int(A,A2) = insert_int(B,bot_bot_set_int) )
    <=> ( ( A = B )
        & pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(B,bot_bot_set_int))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_5504_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( set_or1269000886237332187st_nat(A,B) = insert_nat(C,bot_bot_set_nat) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_5505_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B) = insert_int(C,bot_bot_set_int) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_5506_atLeastAtMost__singleton__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( set_or1222579329274155063t_real(A,B) = insert_real(C,bot_bot_set_real) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_5507_atLeastAtMost__singleton,axiom,
    ! [A: nat] : set_or1269000886237332187st_nat(A,A) = insert_nat(A,bot_bot_set_nat) ).

% atLeastAtMost_singleton
tff(fact_5508_atLeastAtMost__singleton,axiom,
    ! [A: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),A) = insert_int(A,bot_bot_set_int) ).

% atLeastAtMost_singleton
tff(fact_5509_atLeastAtMost__singleton,axiom,
    ! [A: real] : set_or1222579329274155063t_real(A,A) = insert_real(A,bot_bot_set_real) ).

% atLeastAtMost_singleton
tff(fact_5510_insert__Diff__single,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A,minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) = insert9069300056098147895at_nat(A,A2) ).

% insert_Diff_single
tff(fact_5511_insert__Diff__single,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A,minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = insert8211810215607154385at_nat(A,A2) ).

% insert_Diff_single
tff(fact_5512_insert__Diff__single,axiom,
    ! [A: real,A2: set_real] : insert_real(A,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = insert_real(A,A2) ).

% insert_Diff_single
tff(fact_5513_insert__Diff__single,axiom,
    ! [A: int,A2: set_int] : insert_int(A,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = insert_int(A,A2) ).

% insert_Diff_single
tff(fact_5514_insert__Diff__single,axiom,
    ! [A: nat,A2: set_nat] : insert_nat(A,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = insert_nat(A,A2) ).

% insert_Diff_single
tff(fact_5515_finite__Diff__insert,axiom,
    ! [A2: set_real,A: real,B2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(A2,insert_real(A,B2))))
    <=> pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(A2,B2))) ) ).

% finite_Diff_insert
tff(fact_5516_finite__Diff__insert,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(A,B2))))
    <=> pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,minus_3314409938677909166at_nat(A2,B2))) ) ).

% finite_Diff_insert
tff(fact_5517_finite__Diff__insert,axiom,
    ! [A2: set_int,A: int,B2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A2,insert_int(A,B2))))
    <=> pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A2,B2))) ) ).

% finite_Diff_insert
tff(fact_5518_finite__Diff__insert,axiom,
    ! [A2: set_complex,A: complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,insert_complex(A,B2))))
    <=> pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B2))) ) ).

% finite_Diff_insert
tff(fact_5519_finite__Diff__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,B2))))
    <=> pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A2,B2))) ) ).

% finite_Diff_insert
tff(fact_5520_finite__Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A2,insert_nat(A,B2))))
    <=> pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A2,B2))) ) ).

% finite_Diff_insert
tff(fact_5521_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_5522_round__0,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,zero_zero_rat) = zero_zero_int ).

% round_0
tff(fact_5523_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_gb(A),B),S2) = aa_real_complex(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_gb(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_5524_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_gc(A),B),S2) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_gc(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_5525_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_gd(A),B),S2) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_gd(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_5526_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ge(A),B),S2) = aa_real_real(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ge(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_5527_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_gf(A),B),S2) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_gf(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_5528_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_gg(A),B),S2) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_gg(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_5529_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_gh(A),B),S2) = aa_real_rat(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_gh(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_5530_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gi(A),B),S2) = aa_nat_rat(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gi(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_5531_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_gj(A),B),S2) = aa_int_rat(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_gj(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_5532_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_gk(A),B),S2) = aa_complex_rat(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_gk(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_5533_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_gl(A),B),S2) = aa_real_complex(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_gl(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_5534_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_gm(A),B),S2) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_gm(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_5535_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_gn(A),B),S2) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_gn(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_5536_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_go(A),B),S2) = aa_real_real(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_go(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_5537_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_gp(A),B),S2) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_gp(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_5538_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_gq(A),B),S2) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_gq(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_5539_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_gr(A),B),S2) = aa_real_rat(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_gr(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_5540_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gs(A),B),S2) = aa_nat_rat(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gs(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_5541_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_gt(A),B),S2) = aa_int_rat(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_gt(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_5542_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_gu(A),B),S2) = aa_complex_rat(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_gu(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_5543_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X),A2))
       => ( groups8097168146408367636l_real(G,insert_real(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X)),groups8097168146408367636l_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5544_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ~ pp(aa_set_int_bool(member_int(X),A2))
       => ( groups8778361861064173332t_real(G,insert_int(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X)),groups8778361861064173332t_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5545_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups5808333547571424918x_real(G,insert_complex(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X)),groups5808333547571424918x_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5546_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X),A2))
       => ( groups1300246762558778688al_rat(G,insert_real(X,A2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(G,X)),groups1300246762558778688al_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5547_sum_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
       => ( groups2906978787729119204at_rat(G,insert_nat(X,A2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,X)),groups2906978787729119204at_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5548_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ~ pp(aa_set_int_bool(member_int(X),A2))
       => ( groups3906332499630173760nt_rat(G,insert_int(X,A2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(G,X)),groups3906332499630173760nt_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5549_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups5058264527183730370ex_rat(G,insert_complex(X,A2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_complex_rat(G,X)),groups5058264527183730370ex_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5550_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X),A2))
       => ( groups1935376822645274424al_nat(G,insert_real(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X)),groups1935376822645274424al_nat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5551_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ~ pp(aa_set_int_bool(member_int(X),A2))
       => ( groups4541462559716669496nt_nat(G,insert_int(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(G,X)),groups4541462559716669496nt_nat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5552_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups5693394587270226106ex_nat(G,insert_complex(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X)),groups5693394587270226106ex_nat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5553_subset__Compl__singleton,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(A2),uminus935396558254630718at_nat(insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat))))
    <=> ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(B),A2)) ) ).

% subset_Compl_singleton
tff(fact_5554_subset__Compl__singleton,axiom,
    ! [A2: set_set_nat,B: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),uminus613421341184616069et_nat(insert_set_nat(B,bot_bot_set_set_nat))))
    <=> ~ pp(aa_set_set_nat_bool(member_set_nat(B),A2)) ) ).

% subset_Compl_singleton
tff(fact_5555_subset__Compl__singleton,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),uminus6524753893492686040at_nat(insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat))))
    <=> ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B),A2)) ) ).

% subset_Compl_singleton
tff(fact_5556_subset__Compl__singleton,axiom,
    ! [A2: set_real,B: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),uminus612125837232591019t_real(insert_real(B,bot_bot_set_real))))
    <=> ~ pp(aa_set_real_bool(member_real(B),A2)) ) ).

% subset_Compl_singleton
tff(fact_5557_subset__Compl__singleton,axiom,
    ! [A2: set_nat,B: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),uminus5710092332889474511et_nat(insert_nat(B,bot_bot_set_nat))))
    <=> ~ pp(aa_set_nat_bool(member_nat(B),A2)) ) ).

% subset_Compl_singleton
tff(fact_5558_subset__Compl__singleton,axiom,
    ! [A2: set_int,B: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),uminus1532241313380277803et_int(insert_int(B,bot_bot_set_int))))
    <=> ~ pp(aa_set_int_bool(member_int(B),A2)) ) ).

% subset_Compl_singleton
tff(fact_5559_set__replicate,axiom,
    ! [N: nat,X: produc3843707927480180839at_nat] :
      ( ( N != zero_zero_nat )
     => ( set_Pr3765526544606949372at_nat(replic2264142908078655527at_nat(N,X)) = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ).

% set_replicate
tff(fact_5560_set__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( N != zero_zero_nat )
     => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ) ) ).

% set_replicate
tff(fact_5561_set__replicate,axiom,
    ! [N: nat,X: product_prod_nat_nat] :
      ( ( N != zero_zero_nat )
     => ( set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(N,X)) = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ).

% set_replicate
tff(fact_5562_set__replicate,axiom,
    ! [N: nat,X: real] :
      ( ( N != zero_zero_nat )
     => ( set_real2(replicate_real(N,X)) = insert_real(X,bot_bot_set_real) ) ) ).

% set_replicate
tff(fact_5563_set__replicate,axiom,
    ! [N: nat,X: nat] :
      ( ( N != zero_zero_nat )
     => ( set_nat2(replicate_nat(N,X)) = insert_nat(X,bot_bot_set_nat) ) ) ).

% set_replicate
tff(fact_5564_set__replicate,axiom,
    ! [N: nat,X: int] :
      ( ( N != zero_zero_nat )
     => ( set_int2(replicate_int(N,X)) = insert_int(X,bot_bot_set_int) ) ) ).

% set_replicate
tff(fact_5565_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(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_5566_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(aa_rat_fun_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_5567_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_5568_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_5569_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_5570_xor__nat__numerals_I1_J,axiom,
    ! [Y: 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,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_5571_xor__nat__numerals_I2_J,axiom,
    ! [Y: 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,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y)) ).

% xor_nat_numerals(2)
tff(fact_5572_xor__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% xor_nat_numerals(3)
tff(fact_5573_xor__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X)) ).

% xor_nat_numerals(4)
tff(fact_5574_sum__zero__power,axiom,
    ! [A2: set_nat,C: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aTP_Lamm_gv(C),A2) = aa_nat_complex(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aTP_Lamm_gv(C),A2) = zero_zero_complex ) ) ) ).

% sum_zero_power
tff(fact_5575_sum__zero__power,axiom,
    ! [A2: set_nat,C: fun_nat_rat] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2906978787729119204at_rat(aTP_Lamm_gw(C),A2) = aa_nat_rat(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2906978787729119204at_rat(aTP_Lamm_gw(C),A2) = zero_zero_rat ) ) ) ).

% sum_zero_power
tff(fact_5576_sum__zero__power,axiom,
    ! [A2: set_nat,C: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups6591440286371151544t_real(aTP_Lamm_gx(C),A2) = aa_nat_real(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups6591440286371151544t_real(aTP_Lamm_gx(C),A2) = zero_zero_real ) ) ) ).

% sum_zero_power
tff(fact_5577_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: fun_nat_complex,D: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_gy(C),D),A2) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(C,zero_zero_nat)),aa_nat_complex(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_gy(C),D),A2) = zero_zero_complex ) ) ) ).

% sum_zero_power'
tff(fact_5578_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: fun_nat_rat,D: fun_nat_rat] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gz(C),D),A2) = aa_rat_rat(divide_divide_rat(aa_nat_rat(C,zero_zero_nat)),aa_nat_rat(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gz(C),D),A2) = zero_zero_rat ) ) ) ).

% sum_zero_power'
tff(fact_5579_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: fun_nat_real,D: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ha(C),D),A2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(C,zero_zero_nat)),aa_nat_real(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ha(C),D),A2) = zero_zero_real ) ) ) ).

% sum_zero_power'
tff(fact_5580_uminus__set__def,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(A2) = collec3392354462482085612at_nat(uminus6270006169978340066t_bool(aTP_Lamm_a(A2))) ).

% uminus_set_def
tff(fact_5581_uminus__set__def,axiom,
    ! [A2: set_real] : uminus612125837232591019t_real(A2) = collect_real(uminus4638318004487240379l_bool(aTP_Lamm_aa(A2))) ).

% uminus_set_def
tff(fact_5582_uminus__set__def,axiom,
    ! [A2: set_list_nat] : uminus3195874150345416415st_nat(A2) = collect_list_nat(uminus5119773305880778631t_bool(aTP_Lamm_ab(A2))) ).

% uminus_set_def
tff(fact_5583_uminus__set__def,axiom,
    ! [A2: set_set_nat] : uminus613421341184616069et_nat(A2) = collect_set_nat(uminus6226608070181861729t_bool(aTP_Lamm_ac(A2))) ).

% uminus_set_def
tff(fact_5584_uminus__set__def,axiom,
    ! [A2: set_nat] : uminus5710092332889474511et_nat(A2) = collect_nat(uminus1995910650241132439t_bool(aa_set3744802623701456604t_bool(aTP_Lamm_ad,A2))) ).

% uminus_set_def
tff(fact_5585_uminus__set__def,axiom,
    ! [A2: set_int] : uminus1532241313380277803et_int(A2) = collect_int(uminus8094991720627284027t_bool(aTP_Lamm_ae(A2))) ).

% uminus_set_def
tff(fact_5586_Collect__neg__eq,axiom,
    ! [P: fun_real_bool] : collect_real(aTP_Lamm_hb(P)) = uminus612125837232591019t_real(collect_real(P)) ).

% Collect_neg_eq
tff(fact_5587_Collect__neg__eq,axiom,
    ! [P: fun_list_nat_bool] : collect_list_nat(aTP_Lamm_hc(P)) = uminus3195874150345416415st_nat(collect_list_nat(P)) ).

% Collect_neg_eq
tff(fact_5588_Collect__neg__eq,axiom,
    ! [P: fun_set_nat_bool] : collect_set_nat(aTP_Lamm_hd(P)) = uminus613421341184616069et_nat(collect_set_nat(P)) ).

% Collect_neg_eq
tff(fact_5589_Collect__neg__eq,axiom,
    ! [P: fun_nat_bool] : collect_nat(aTP_Lamm_he(P)) = uminus5710092332889474511et_nat(collect_nat(P)) ).

% Collect_neg_eq
tff(fact_5590_Collect__neg__eq,axiom,
    ! [P: fun_int_bool] : collect_int(aTP_Lamm_hf(P)) = uminus1532241313380277803et_int(collect_int(P)) ).

% Collect_neg_eq
tff(fact_5591_Compl__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(A2) = collec3392354462482085612at_nat(aTP_Lamm_hg(A2)) ).

% Compl_eq
tff(fact_5592_Compl__eq,axiom,
    ! [A2: set_real] : uminus612125837232591019t_real(A2) = collect_real(aTP_Lamm_hh(A2)) ).

% Compl_eq
tff(fact_5593_Compl__eq,axiom,
    ! [A2: set_list_nat] : uminus3195874150345416415st_nat(A2) = collect_list_nat(aTP_Lamm_hi(A2)) ).

% Compl_eq
tff(fact_5594_Compl__eq,axiom,
    ! [A2: set_set_nat] : uminus613421341184616069et_nat(A2) = collect_set_nat(aTP_Lamm_hj(A2)) ).

% Compl_eq
tff(fact_5595_Compl__eq,axiom,
    ! [A2: set_nat] : uminus5710092332889474511et_nat(A2) = collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_hk,A2)) ).

% Compl_eq
tff(fact_5596_Compl__eq,axiom,
    ! [A2: set_int] : uminus1532241313380277803et_int(A2) = collect_int(aTP_Lamm_hl(A2)) ).

% Compl_eq
tff(fact_5597_ComplD,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),uminus6524753893492686040at_nat(A2)))
     => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A2)) ) ).

% ComplD
tff(fact_5598_ComplD,axiom,
    ! [C: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),uminus612125837232591019t_real(A2)))
     => ~ pp(aa_set_real_bool(member_real(C),A2)) ) ).

% ComplD
tff(fact_5599_ComplD,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),uminus613421341184616069et_nat(A2)))
     => ~ pp(aa_set_set_nat_bool(member_set_nat(C),A2)) ) ).

% ComplD
tff(fact_5600_ComplD,axiom,
    ! [C: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),uminus5710092332889474511et_nat(A2)))
     => ~ pp(aa_set_nat_bool(member_nat(C),A2)) ) ).

% ComplD
tff(fact_5601_ComplD,axiom,
    ! [C: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),uminus1532241313380277803et_int(A2)))
     => ~ pp(aa_set_int_bool(member_int(C),A2)) ) ).

% ComplD
tff(fact_5602_insert__Collect,axiom,
    ! [A: product_prod_nat_nat,P: fun_Pr7757163693017731481t_bool] : insert8211810215607154385at_nat(A,collec3392354462482085612at_nat(P)) = collec3392354462482085612at_nat(aa_fun830836278467416944t_bool(aTP_Lamm_hm(A),P)) ).

% insert_Collect
tff(fact_5603_insert__Collect,axiom,
    ! [A: produc3843707927480180839at_nat,P: fun_Pr1607443439640623155t_bool] : insert9069300056098147895at_nat(A,collec6321179662152712658at_nat(P)) = collec6321179662152712658at_nat(aa_fun5022828243006652836t_bool(aTP_Lamm_hn(A),P)) ).

% insert_Collect
tff(fact_5604_insert__Collect,axiom,
    ! [A: real,P: fun_real_bool] : insert_real(A,collect_real(P)) = collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_ho(A),P)) ).

% insert_Collect
tff(fact_5605_insert__Collect,axiom,
    ! [A: list_nat,P: fun_list_nat_bool] : insert_list_nat(A,collect_list_nat(P)) = collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_hp(A),P)) ).

% insert_Collect
tff(fact_5606_insert__Collect,axiom,
    ! [A: set_nat,P: fun_set_nat_bool] : insert_set_nat(A,collect_set_nat(P)) = collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_hq(A),P)) ).

% insert_Collect
tff(fact_5607_insert__Collect,axiom,
    ! [A: nat,P: fun_nat_bool] : insert_nat(A,collect_nat(P)) = collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_hr(A),P)) ).

% insert_Collect
tff(fact_5608_insert__Collect,axiom,
    ! [A: int,P: fun_int_bool] : insert_int(A,collect_int(P)) = collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_hs(A),P)) ).

% insert_Collect
tff(fact_5609_insert__compr,axiom,
    ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A,B2) = collec6321179662152712658at_nat(aa_set377272637606474814t_bool(aTP_Lamm_ht(A),B2)) ).

% insert_compr
tff(fact_5610_insert__compr,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A,B2) = collec3392354462482085612at_nat(aa_set5223363028666124222t_bool(aTP_Lamm_hu(A),B2)) ).

% insert_compr
tff(fact_5611_insert__compr,axiom,
    ! [A: real,B2: set_real] : insert_real(A,B2) = collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_hv(A),B2)) ).

% insert_compr
tff(fact_5612_insert__compr,axiom,
    ! [A: list_nat,B2: set_list_nat] : insert_list_nat(A,B2) = collect_list_nat(aa_set257547205302447420t_bool(aTP_Lamm_hw(A),B2)) ).

% insert_compr
tff(fact_5613_insert__compr,axiom,
    ! [A: set_nat,B2: set_set_nat] : insert_set_nat(A,B2) = collect_set_nat(aa_set9121396911425987952t_bool(aTP_Lamm_hx(A),B2)) ).

% insert_compr
tff(fact_5614_insert__compr,axiom,
    ! [A: nat,B2: set_nat] : insert_nat(A,B2) = collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_hy(A),B2)) ).

% insert_compr
tff(fact_5615_insert__compr,axiom,
    ! [A: int,B2: set_int] : insert_int(A,B2) = collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_hz(A),B2)) ).

% insert_compr
tff(fact_5616_insertE,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(B,A2)))
     => ( ( A != B )
       => pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),A2)) ) ) ).

% insertE
tff(fact_5617_insertE,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(B,A2)))
     => ( ( A != B )
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),A2)) ) ) ).

% insertE
tff(fact_5618_insertE,axiom,
    ! [A: real,B: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(A),insert_real(B,A2)))
     => ( ( A != B )
       => pp(aa_set_real_bool(member_real(A),A2)) ) ) ).

% insertE
tff(fact_5619_insertE,axiom,
    ! [A: set_nat,B: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),insert_set_nat(B,A2)))
     => ( ( A != B )
       => pp(aa_set_set_nat_bool(member_set_nat(A),A2)) ) ) ).

% insertE
tff(fact_5620_insertE,axiom,
    ! [A: nat,B: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),insert_nat(B,A2)))
     => ( ( A != B )
       => pp(aa_set_nat_bool(member_nat(A),A2)) ) ) ).

% insertE
tff(fact_5621_insertE,axiom,
    ! [A: int,B: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(A),insert_int(B,A2)))
     => ( ( A != B )
       => pp(aa_set_int_bool(member_int(A),A2)) ) ) ).

% insertE
tff(fact_5622_insertI1,axiom,
    ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(A,B2))) ).

% insertI1
tff(fact_5623_insertI1,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(A,B2))) ).

% insertI1
tff(fact_5624_insertI1,axiom,
    ! [A: real,B2: set_real] : pp(aa_set_real_bool(member_real(A),insert_real(A,B2))) ).

% insertI1
tff(fact_5625_insertI1,axiom,
    ! [A: set_nat,B2: set_set_nat] : pp(aa_set_set_nat_bool(member_set_nat(A),insert_set_nat(A,B2))) ).

% insertI1
tff(fact_5626_insertI1,axiom,
    ! [A: nat,B2: set_nat] : pp(aa_set_nat_bool(member_nat(A),insert_nat(A,B2))) ).

% insertI1
tff(fact_5627_insertI1,axiom,
    ! [A: int,B2: set_int] : pp(aa_set_int_bool(member_int(A),insert_int(A,B2))) ).

% insertI1
tff(fact_5628_insertI2,axiom,
    ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),B2))
     => pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(B,B2))) ) ).

% insertI2
tff(fact_5629_insertI2,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),B2))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(B,B2))) ) ).

% insertI2
tff(fact_5630_insertI2,axiom,
    ! [A: real,B2: set_real,B: real] :
      ( pp(aa_set_real_bool(member_real(A),B2))
     => pp(aa_set_real_bool(member_real(A),insert_real(B,B2))) ) ).

% insertI2
tff(fact_5631_insertI2,axiom,
    ! [A: set_nat,B2: set_set_nat,B: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),B2))
     => pp(aa_set_set_nat_bool(member_set_nat(A),insert_set_nat(B,B2))) ) ).

% insertI2
tff(fact_5632_insertI2,axiom,
    ! [A: nat,B2: set_nat,B: nat] :
      ( pp(aa_set_nat_bool(member_nat(A),B2))
     => pp(aa_set_nat_bool(member_nat(A),insert_nat(B,B2))) ) ).

% insertI2
tff(fact_5633_insertI2,axiom,
    ! [A: int,B2: set_int,B: int] :
      ( pp(aa_set_int_bool(member_int(A),B2))
     => pp(aa_set_int_bool(member_int(A),insert_int(B,B2))) ) ).

% insertI2
tff(fact_5634_Set_Oset__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),A2))
     => ~ ! [B7: set_Pr4329608150637261639at_nat] :
            ( ( A2 = insert9069300056098147895at_nat(X,B7) )
           => pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),B7)) ) ) ).

% Set.set_insert
tff(fact_5635_Set_Oset__insert,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2))
     => ~ ! [B7: set_Pr1261947904930325089at_nat] :
            ( ( A2 = insert8211810215607154385at_nat(X,B7) )
           => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B7)) ) ) ).

% Set.set_insert
tff(fact_5636_Set_Oset__insert,axiom,
    ! [X: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(X),A2))
     => ~ ! [B7: set_real] :
            ( ( A2 = insert_real(X,B7) )
           => pp(aa_set_real_bool(member_real(X),B7)) ) ) ).

% Set.set_insert
tff(fact_5637_Set_Oset__insert,axiom,
    ! [X: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
     => ~ ! [B7: set_set_nat] :
            ( ( A2 = insert_set_nat(X,B7) )
           => pp(aa_set_set_nat_bool(member_set_nat(X),B7)) ) ) ).

% Set.set_insert
tff(fact_5638_Set_Oset__insert,axiom,
    ! [X: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),A2))
     => ~ ! [B7: set_nat] :
            ( ( A2 = insert_nat(X,B7) )
           => pp(aa_set_nat_bool(member_nat(X),B7)) ) ) ).

% Set.set_insert
tff(fact_5639_Set_Oset__insert,axiom,
    ! [X: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(X),A2))
     => ~ ! [B7: set_int] :
            ( ( A2 = insert_int(X,B7) )
           => pp(aa_set_int_bool(member_int(X),B7)) ) ) ).

% Set.set_insert
tff(fact_5640_insert__ident,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),A2))
     => ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),B2))
       => ( ( insert9069300056098147895at_nat(X,A2) = insert9069300056098147895at_nat(X,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_5641_insert__ident,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2))
     => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B2))
       => ( ( insert8211810215607154385at_nat(X,A2) = insert8211810215607154385at_nat(X,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_5642_insert__ident,axiom,
    ! [X: real,A2: set_real,B2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X),A2))
     => ( ~ pp(aa_set_real_bool(member_real(X),B2))
       => ( ( insert_real(X,A2) = insert_real(X,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_5643_insert__ident,axiom,
    ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A2))
     => ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),B2))
       => ( ( insert_set_nat(X,A2) = insert_set_nat(X,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_5644_insert__ident,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(X),B2))
       => ( ( insert_nat(X,A2) = insert_nat(X,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_5645_insert__ident,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X),A2))
     => ( ~ pp(aa_set_int_bool(member_int(X),B2))
       => ( ( insert_int(X,A2) = insert_int(X,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_5646_insert__absorb,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),A2))
     => ( insert9069300056098147895at_nat(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_5647_insert__absorb,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),A2))
     => ( insert8211810215607154385at_nat(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_5648_insert__absorb,axiom,
    ! [A: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(A),A2))
     => ( insert_real(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_5649_insert__absorb,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
     => ( insert_set_nat(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_5650_insert__absorb,axiom,
    ! [A: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),A2))
     => ( insert_nat(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_5651_insert__absorb,axiom,
    ! [A: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(A),A2))
     => ( insert_int(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_5652_insert__eq__iff,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),A2))
     => ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(B),B2))
       => ( ( insert9069300056098147895at_nat(A,A2) = insert9069300056098147895at_nat(B,B2) )
        <=> ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C5: set_Pr4329608150637261639at_nat] :
                  ( ( A2 = insert9069300056098147895at_nat(B,C5) )
                  & ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(B),C5))
                  & ( B2 = insert9069300056098147895at_nat(A,C5) )
                  & ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),C5)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_5653_insert__eq__iff,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),A2))
     => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B),B2))
       => ( ( insert8211810215607154385at_nat(A,A2) = insert8211810215607154385at_nat(B,B2) )
        <=> ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C5: set_Pr1261947904930325089at_nat] :
                  ( ( A2 = insert8211810215607154385at_nat(B,C5) )
                  & ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B),C5))
                  & ( B2 = insert8211810215607154385at_nat(A,C5) )
                  & ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),C5)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_5654_insert__eq__iff,axiom,
    ! [A: real,A2: set_real,B: real,B2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(A),A2))
     => ( ~ pp(aa_set_real_bool(member_real(B),B2))
       => ( ( insert_real(A,A2) = insert_real(B,B2) )
        <=> ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C5: set_real] :
                  ( ( A2 = insert_real(B,C5) )
                  & ~ pp(aa_set_real_bool(member_real(B),C5))
                  & ( B2 = insert_real(A,C5) )
                  & ~ pp(aa_set_real_bool(member_real(A),C5)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_5655_insert__eq__iff,axiom,
    ! [A: set_nat,A2: set_set_nat,B: set_nat,B2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(A),A2))
     => ( ~ pp(aa_set_set_nat_bool(member_set_nat(B),B2))
       => ( ( insert_set_nat(A,A2) = insert_set_nat(B,B2) )
        <=> ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C5: set_set_nat] :
                  ( ( A2 = insert_set_nat(B,C5) )
                  & ~ pp(aa_set_set_nat_bool(member_set_nat(B),C5))
                  & ( B2 = insert_set_nat(A,C5) )
                  & ~ pp(aa_set_set_nat_bool(member_set_nat(A),C5)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_5656_insert__eq__iff,axiom,
    ! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(A),A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(B),B2))
       => ( ( insert_nat(A,A2) = insert_nat(B,B2) )
        <=> ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C5: set_nat] :
                  ( ( A2 = insert_nat(B,C5) )
                  & ~ pp(aa_set_nat_bool(member_nat(B),C5))
                  & ( B2 = insert_nat(A,C5) )
                  & ~ pp(aa_set_nat_bool(member_nat(A),C5)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_5657_insert__eq__iff,axiom,
    ! [A: int,A2: set_int,B: int,B2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(A),A2))
     => ( ~ pp(aa_set_int_bool(member_int(B),B2))
       => ( ( insert_int(A,A2) = insert_int(B,B2) )
        <=> ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C5: set_int] :
                  ( ( A2 = insert_int(B,C5) )
                  & ~ pp(aa_set_int_bool(member_int(B),C5))
                  & ( B2 = insert_int(A,C5) )
                  & ~ pp(aa_set_int_bool(member_int(A),C5)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_5658_insert__commute,axiom,
    ! [X: nat,Y: nat,A2: set_nat] : insert_nat(X,insert_nat(Y,A2)) = insert_nat(Y,insert_nat(X,A2)) ).

% insert_commute
tff(fact_5659_insert__commute,axiom,
    ! [X: int,Y: int,A2: set_int] : insert_int(X,insert_int(Y,A2)) = insert_int(Y,insert_int(X,A2)) ).

% insert_commute
tff(fact_5660_insert__commute,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(X,insert8211810215607154385at_nat(Y,A2)) = insert8211810215607154385at_nat(Y,insert8211810215607154385at_nat(X,A2)) ).

% insert_commute
tff(fact_5661_insert__commute,axiom,
    ! [X: real,Y: real,A2: set_real] : insert_real(X,insert_real(Y,A2)) = insert_real(Y,insert_real(X,A2)) ).

% insert_commute
tff(fact_5662_insert__commute,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(X,insert9069300056098147895at_nat(Y,A2)) = insert9069300056098147895at_nat(Y,insert9069300056098147895at_nat(X,A2)) ).

% insert_commute
tff(fact_5663_mk__disjoint__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),A2))
     => ? [B7: set_Pr4329608150637261639at_nat] :
          ( ( A2 = insert9069300056098147895at_nat(A,B7) )
          & ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),B7)) ) ) ).

% mk_disjoint_insert
tff(fact_5664_mk__disjoint__insert,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),A2))
     => ? [B7: set_Pr1261947904930325089at_nat] :
          ( ( A2 = insert8211810215607154385at_nat(A,B7) )
          & ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),B7)) ) ) ).

% mk_disjoint_insert
tff(fact_5665_mk__disjoint__insert,axiom,
    ! [A: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(A),A2))
     => ? [B7: set_real] :
          ( ( A2 = insert_real(A,B7) )
          & ~ pp(aa_set_real_bool(member_real(A),B7)) ) ) ).

% mk_disjoint_insert
tff(fact_5666_mk__disjoint__insert,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
     => ? [B7: set_set_nat] :
          ( ( A2 = insert_set_nat(A,B7) )
          & ~ pp(aa_set_set_nat_bool(member_set_nat(A),B7)) ) ) ).

% mk_disjoint_insert
tff(fact_5667_mk__disjoint__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),A2))
     => ? [B7: set_nat] :
          ( ( A2 = insert_nat(A,B7) )
          & ~ pp(aa_set_nat_bool(member_nat(A),B7)) ) ) ).

% mk_disjoint_insert
tff(fact_5668_mk__disjoint__insert,axiom,
    ! [A: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(A),A2))
     => ? [B7: set_int] :
          ( ( A2 = insert_int(A,B7) )
          & ~ pp(aa_set_int_bool(member_int(A),B7)) ) ) ).

% mk_disjoint_insert
tff(fact_5669_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(X),A2))
         => ( groups8097168146408367636l_real(G,insert_real(X,A2)) = groups8097168146408367636l_real(G,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A2))
         => ( groups8097168146408367636l_real(G,insert_real(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X)),groups8097168146408367636l_real(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_5670_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( pp(aa_set_int_bool(member_int(X),A2))
         => ( groups8778361861064173332t_real(G,insert_int(X,A2)) = groups8778361861064173332t_real(G,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A2))
         => ( groups8778361861064173332t_real(G,insert_int(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X)),groups8778361861064173332t_real(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_5671_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups5808333547571424918x_real(G,insert_complex(X,A2)) = groups5808333547571424918x_real(G,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups5808333547571424918x_real(G,insert_complex(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X)),groups5808333547571424918x_real(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_5672_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(X),A2))
         => ( groups1300246762558778688al_rat(G,insert_real(X,A2)) = groups1300246762558778688al_rat(G,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A2))
         => ( groups1300246762558778688al_rat(G,insert_real(X,A2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(G,X)),groups1300246762558778688al_rat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_5673_sum_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( pp(aa_set_nat_bool(member_nat(X),A2))
         => ( groups2906978787729119204at_rat(G,insert_nat(X,A2)) = groups2906978787729119204at_rat(G,A2) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
         => ( groups2906978787729119204at_rat(G,insert_nat(X,A2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,X)),groups2906978787729119204at_rat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_5674_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( pp(aa_set_int_bool(member_int(X),A2))
         => ( groups3906332499630173760nt_rat(G,insert_int(X,A2)) = groups3906332499630173760nt_rat(G,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A2))
         => ( groups3906332499630173760nt_rat(G,insert_int(X,A2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(G,X)),groups3906332499630173760nt_rat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_5675_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups5058264527183730370ex_rat(G,insert_complex(X,A2)) = groups5058264527183730370ex_rat(G,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups5058264527183730370ex_rat(G,insert_complex(X,A2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_complex_rat(G,X)),groups5058264527183730370ex_rat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_5676_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(X),A2))
         => ( groups1935376822645274424al_nat(G,insert_real(X,A2)) = groups1935376822645274424al_nat(G,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A2))
         => ( groups1935376822645274424al_nat(G,insert_real(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X)),groups1935376822645274424al_nat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_5677_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( pp(aa_set_int_bool(member_int(X),A2))
         => ( groups4541462559716669496nt_nat(G,insert_int(X,A2)) = groups4541462559716669496nt_nat(G,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A2))
         => ( groups4541462559716669496nt_nat(G,insert_int(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(G,X)),groups4541462559716669496nt_nat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_5678_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups5693394587270226106ex_nat(G,insert_complex(X,A2)) = groups5693394587270226106ex_nat(G,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups5693394587270226106ex_nat(G,insert_complex(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X)),groups5693394587270226106ex_nat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_5679_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_complex,A2: set_real] :
      ( ( groups5754745047067104278omplex(G,A2) != zero_zero_complex )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A2))
           => ( aa_real_complex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5680_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_complex,A2: set_nat] :
      ( ( groups2073611262835488442omplex(G,A2) != zero_zero_complex )
     => ~ ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A2))
           => ( aa_nat_complex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5681_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_complex,A2: set_int] :
      ( ( groups3049146728041665814omplex(G,A2) != zero_zero_complex )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A2))
           => ( aa_int_complex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5682_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A2: set_real] :
      ( ( groups8097168146408367636l_real(G,A2) != zero_zero_real )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A2))
           => ( aa_real_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5683_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A2: set_int] :
      ( ( groups8778361861064173332t_real(G,A2) != zero_zero_real )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A2))
           => ( aa_int_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5684_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A2: set_real] :
      ( ( groups1300246762558778688al_rat(G,A2) != zero_zero_rat )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A2))
           => ( aa_real_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5685_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A2: set_nat] :
      ( ( groups2906978787729119204at_rat(G,A2) != zero_zero_rat )
     => ~ ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A2))
           => ( aa_nat_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5686_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_rat,A2: set_int] :
      ( ( groups3906332499630173760nt_rat(G,A2) != zero_zero_rat )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A2))
           => ( aa_int_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5687_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_nat,A2: set_real] :
      ( ( groups1935376822645274424al_nat(G,A2) != zero_zero_nat )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A2))
           => ( aa_real_nat(G,A3) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5688_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_nat,A2: set_int] :
      ( ( groups4541462559716669496nt_nat(G,A2) != zero_zero_nat )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A2))
           => ( aa_int_nat(G,A3) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5689_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( aa_nat_nat(G,X4) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_5690_sum_Oneutral,axiom,
    ! [A2: set_complex,G: fun_complex_complex] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => ( aa_complex_complex(G,X4) = zero_zero_complex ) )
     => ( groups7754918857620584856omplex(G,A2) = zero_zero_complex ) ) ).

% sum.neutral
tff(fact_5691_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( aa_nat_real(G,X4) = zero_zero_real ) )
     => ( groups6591440286371151544t_real(G,A2) = zero_zero_real ) ) ).

% sum.neutral
tff(fact_5692_singleton__inject,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) = insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_5693_singleton__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) = insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_5694_singleton__inject,axiom,
    ! [A: real,B: real] :
      ( ( insert_real(A,bot_bot_set_real) = insert_real(B,bot_bot_set_real) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_5695_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( insert_nat(A,bot_bot_set_nat) = insert_nat(B,bot_bot_set_nat) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_5696_singleton__inject,axiom,
    ! [A: int,B: int] :
      ( ( insert_int(A,bot_bot_set_int) = insert_int(B,bot_bot_set_int) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_5697_insert__not__empty,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A,A2) != bot_bo228742789529271731at_nat ).

% insert_not_empty
tff(fact_5698_insert__not__empty,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A,A2) != bot_bo2099793752762293965at_nat ).

% insert_not_empty
tff(fact_5699_insert__not__empty,axiom,
    ! [A: real,A2: set_real] : insert_real(A,A2) != bot_bot_set_real ).

% insert_not_empty
tff(fact_5700_insert__not__empty,axiom,
    ! [A: nat,A2: set_nat] : insert_nat(A,A2) != bot_bot_set_nat ).

% insert_not_empty
tff(fact_5701_insert__not__empty,axiom,
    ! [A: int,A2: set_int] : insert_int(A,A2) != bot_bot_set_int ).

% insert_not_empty
tff(fact_5702_doubleton__eq__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,C: produc3843707927480180839at_nat,D: produc3843707927480180839at_nat] :
      ( ( insert9069300056098147895at_nat(A,insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat)) = insert9069300056098147895at_nat(C,insert9069300056098147895at_nat(D,bot_bo228742789529271731at_nat)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_5703_doubleton__eq__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat,D: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A,insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat)) = insert8211810215607154385at_nat(C,insert8211810215607154385at_nat(D,bot_bo2099793752762293965at_nat)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_5704_doubleton__eq__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( insert_real(A,insert_real(B,bot_bot_set_real)) = insert_real(C,insert_real(D,bot_bot_set_real)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_5705_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( insert_nat(A,insert_nat(B,bot_bot_set_nat)) = insert_nat(C,insert_nat(D,bot_bot_set_nat)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_5706_doubleton__eq__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( insert_int(A,insert_int(B,bot_bot_set_int)) = insert_int(C,insert_int(D,bot_bot_set_int)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_5707_singleton__iff,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(B),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_5708_singleton__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(B),insert_set_nat(A,bot_bot_set_set_nat)))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_5709_singleton__iff,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_5710_singleton__iff,axiom,
    ! [B: real,A: real] :
      ( pp(aa_set_real_bool(member_real(B),insert_real(A,bot_bot_set_real)))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_5711_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_set_nat_bool(member_nat(B),insert_nat(A,bot_bot_set_nat)))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_5712_singleton__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_set_int_bool(member_int(B),insert_int(A,bot_bot_set_int)))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_5713_singletonD,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(B),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)))
     => ( B = A ) ) ).

% singletonD
tff(fact_5714_singletonD,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(B),insert_set_nat(A,bot_bot_set_set_nat)))
     => ( B = A ) ) ).

% singletonD
tff(fact_5715_singletonD,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)))
     => ( B = A ) ) ).

% singletonD
tff(fact_5716_singletonD,axiom,
    ! [B: real,A: real] :
      ( pp(aa_set_real_bool(member_real(B),insert_real(A,bot_bot_set_real)))
     => ( B = A ) ) ).

% singletonD
tff(fact_5717_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_set_nat_bool(member_nat(B),insert_nat(A,bot_bot_set_nat)))
     => ( B = A ) ) ).

% singletonD
tff(fact_5718_singletonD,axiom,
    ! [B: int,A: int] :
      ( pp(aa_set_int_bool(member_int(B),insert_int(A,bot_bot_set_int)))
     => ( B = A ) ) ).

% singletonD
tff(fact_5719_finite_OinsertI,axiom,
    ! [A2: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => pp(aa_set_real_bool(finite_finite_real,insert_real(A,A2))) ) ).

% finite.insertI
tff(fact_5720_finite_OinsertI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A2))
     => pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,insert9069300056098147895at_nat(A,A2))) ) ).

% finite.insertI
tff(fact_5721_finite_OinsertI,axiom,
    ! [A2: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_set_nat_bool(finite_finite_nat,insert_nat(A,A2))) ) ).

% finite.insertI
tff(fact_5722_finite_OinsertI,axiom,
    ! [A2: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => pp(aa_set_int_bool(finite_finite_int,insert_int(A,A2))) ) ).

% finite.insertI
tff(fact_5723_finite_OinsertI,axiom,
    ! [A2: set_complex,A: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,insert_complex(A,A2))) ) ).

% finite.insertI
tff(fact_5724_finite_OinsertI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
     => pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,insert8211810215607154385at_nat(A,A2))) ) ).

% finite.insertI
tff(fact_5725_subset__insertI2,axiom,
    ! [A2: set_nat,B2: set_nat,B: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(B,B2))) ) ).

% subset_insertI2
tff(fact_5726_subset__insertI2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B2))
     => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),insert8211810215607154385at_nat(B,B2))) ) ).

% subset_insertI2
tff(fact_5727_subset__insertI2,axiom,
    ! [A2: set_real,B2: set_real,B: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
     => pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(B,B2))) ) ).

% subset_insertI2
tff(fact_5728_subset__insertI2,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(A2),B2))
     => pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(A2),insert9069300056098147895at_nat(B,B2))) ) ).

% subset_insertI2
tff(fact_5729_subset__insertI2,axiom,
    ! [A2: set_int,B2: set_int,B: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(B,B2))) ) ).

% subset_insertI2
tff(fact_5730_subset__insertI,axiom,
    ! [B2: set_nat,A: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),insert_nat(A,B2))) ).

% subset_insertI
tff(fact_5731_subset__insertI,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(B2),insert8211810215607154385at_nat(A,B2))) ).

% subset_insertI
tff(fact_5732_subset__insertI,axiom,
    ! [B2: set_real,A: real] : pp(aa_set_real_bool(ord_less_eq_set_real(B2),insert_real(A,B2))) ).

% subset_insertI
tff(fact_5733_subset__insertI,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] : pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(B2),insert9069300056098147895at_nat(A,B2))) ).

% subset_insertI
tff(fact_5734_subset__insertI,axiom,
    ! [B2: set_int,A: int] : pp(aa_set_int_bool(ord_less_eq_set_int(B2),insert_int(A,B2))) ).

% subset_insertI
tff(fact_5735_subset__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),A2))
     => ( pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(A2),insert9069300056098147895at_nat(X,B2)))
      <=> pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(A2),B2)) ) ) ).

% subset_insert
tff(fact_5736_subset__insert,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),insert8211810215607154385at_nat(X,B2)))
      <=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B2)) ) ) ).

% subset_insert
tff(fact_5737_subset__insert,axiom,
    ! [X: real,A2: set_real,B2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X),A2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(X,B2)))
      <=> pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2)) ) ) ).

% subset_insert
tff(fact_5738_subset__insert,axiom,
    ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A2))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),insert_set_nat(X,B2)))
      <=> pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B2)) ) ) ).

% subset_insert
tff(fact_5739_subset__insert,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(X,B2)))
      <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2)) ) ) ).

% subset_insert
tff(fact_5740_subset__insert,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(X,B2)))
      <=> pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ) ).

% subset_insert
tff(fact_5741_insert__mono,axiom,
    ! [C2: set_nat,D6: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C2),D6))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,insert_nat(A,C2)),insert_nat(A,D6))) ) ).

% insert_mono
tff(fact_5742_insert__mono,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,D6: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(C2),D6))
     => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(insert8211810215607154385at_nat(A,C2)),insert8211810215607154385at_nat(A,D6))) ) ).

% insert_mono
tff(fact_5743_insert__mono,axiom,
    ! [C2: set_real,D6: set_real,A: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(C2),D6))
     => pp(aa_set_real_bool(ord_less_eq_set_real(insert_real(A,C2)),insert_real(A,D6))) ) ).

% insert_mono
tff(fact_5744_insert__mono,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,D6: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(C2),D6))
     => pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(insert9069300056098147895at_nat(A,C2)),insert9069300056098147895at_nat(A,D6))) ) ).

% insert_mono
tff(fact_5745_insert__mono,axiom,
    ! [C2: set_int,D6: set_int,A: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(C2),D6))
     => pp(aa_set_int_bool(ord_less_eq_set_int(insert_int(A,C2)),insert_int(A,D6))) ) ).

% insert_mono
tff(fact_5746_insert__subsetI,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,X6: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),A2))
     => ( pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(X6),A2))
       => pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(insert9069300056098147895at_nat(X,X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_5747_insert__subsetI,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,X6: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X6),A2))
       => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(insert8211810215607154385at_nat(X,X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_5748_insert__subsetI,axiom,
    ! [X: real,A2: set_real,X6: set_real] :
      ( pp(aa_set_real_bool(member_real(X),A2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(X6),A2))
       => pp(aa_set_real_bool(ord_less_eq_set_real(insert_real(X,X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_5749_insert__subsetI,axiom,
    ! [X: set_nat,A2: set_set_nat,X6: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(X6),A2))
       => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(insert_set_nat(X,X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_5750_insert__subsetI,axiom,
    ! [X: nat,A2: set_nat,X6: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X6),A2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,insert_nat(X,X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_5751_insert__subsetI,axiom,
    ! [X: int,A2: set_int,X6: set_int] :
      ( pp(aa_set_int_bool(member_int(X),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X6),A2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(insert_int(X,X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_5752_insert__Diff__if,axiom,
    ! [X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),B2))
       => ( minus_3314409938677909166at_nat(insert9069300056098147895at_nat(X,A2),B2) = minus_3314409938677909166at_nat(A2,B2) ) )
      & ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),B2))
       => ( minus_3314409938677909166at_nat(insert9069300056098147895at_nat(X,A2),B2) = insert9069300056098147895at_nat(X,minus_3314409938677909166at_nat(A2,B2)) ) ) ) ).

% insert_Diff_if
tff(fact_5753_insert__Diff__if,axiom,
    ! [X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B2))
       => ( minus_1356011639430497352at_nat(insert8211810215607154385at_nat(X,A2),B2) = minus_1356011639430497352at_nat(A2,B2) ) )
      & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B2))
       => ( minus_1356011639430497352at_nat(insert8211810215607154385at_nat(X,A2),B2) = insert8211810215607154385at_nat(X,minus_1356011639430497352at_nat(A2,B2)) ) ) ) ).

% insert_Diff_if
tff(fact_5754_insert__Diff__if,axiom,
    ! [X: real,B2: set_real,A2: set_real] :
      ( ( pp(aa_set_real_bool(member_real(X),B2))
       => ( minus_minus_set_real(insert_real(X,A2),B2) = minus_minus_set_real(A2,B2) ) )
      & ( ~ pp(aa_set_real_bool(member_real(X),B2))
       => ( minus_minus_set_real(insert_real(X,A2),B2) = insert_real(X,minus_minus_set_real(A2,B2)) ) ) ) ).

% insert_Diff_if
tff(fact_5755_insert__Diff__if,axiom,
    ! [X: set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( ( pp(aa_set_set_nat_bool(member_set_nat(X),B2))
       => ( minus_2163939370556025621et_nat(insert_set_nat(X,A2),B2) = minus_2163939370556025621et_nat(A2,B2) ) )
      & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),B2))
       => ( minus_2163939370556025621et_nat(insert_set_nat(X,A2),B2) = insert_set_nat(X,minus_2163939370556025621et_nat(A2,B2)) ) ) ) ).

% insert_Diff_if
tff(fact_5756_insert__Diff__if,axiom,
    ! [X: int,B2: set_int,A2: set_int] :
      ( ( pp(aa_set_int_bool(member_int(X),B2))
       => ( minus_minus_set_int(insert_int(X,A2),B2) = minus_minus_set_int(A2,B2) ) )
      & ( ~ pp(aa_set_int_bool(member_int(X),B2))
       => ( minus_minus_set_int(insert_int(X,A2),B2) = insert_int(X,minus_minus_set_int(A2,B2)) ) ) ) ).

% insert_Diff_if
tff(fact_5757_insert__Diff__if,axiom,
    ! [X: nat,B2: set_nat,A2: set_nat] :
      ( ( pp(aa_set_nat_bool(member_nat(X),B2))
       => ( minus_minus_set_nat(insert_nat(X,A2),B2) = minus_minus_set_nat(A2,B2) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(X),B2))
       => ( minus_minus_set_nat(insert_nat(X,A2),B2) = insert_nat(X,minus_minus_set_nat(A2,B2)) ) ) ) ).

% insert_Diff_if
tff(fact_5758_sum__mono,axiom,
    ! [K4: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),K4))
         => 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,K4)),groups1300246762558778688al_rat(G,K4))) ) ).

% sum_mono
tff(fact_5759_sum__mono,axiom,
    ! [K4: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),K4))
         => 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,K4)),groups2906978787729119204at_rat(G,K4))) ) ).

% sum_mono
tff(fact_5760_sum__mono,axiom,
    ! [K4: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),K4))
         => 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,K4)),groups3906332499630173760nt_rat(G,K4))) ) ).

% sum_mono
tff(fact_5761_sum__mono,axiom,
    ! [K4: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),K4))
         => 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,K4)),groups1935376822645274424al_nat(G,K4))) ) ).

% sum_mono
tff(fact_5762_sum__mono,axiom,
    ! [K4: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),K4))
         => 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,K4)),groups4541462559716669496nt_nat(G,K4))) ) ).

% sum_mono
tff(fact_5763_sum__mono,axiom,
    ! [K4: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),K4))
         => 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,K4)),groups1932886352136224148al_int(G,K4))) ) ).

% sum_mono
tff(fact_5764_sum__mono,axiom,
    ! [K4: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),K4))
         => 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,K4)),groups3539618377306564664at_int(G,K4))) ) ).

% sum_mono
tff(fact_5765_sum__mono,axiom,
    ! [K4: set_int,F: fun_int_int,G: fun_int_int] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),K4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,I2)),aa_int_int(G,I2))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups4538972089207619220nt_int(F,K4)),groups4538972089207619220nt_int(G,K4))) ) ).

% sum_mono
tff(fact_5766_sum__mono,axiom,
    ! [K4: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),K4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,I2)),aa_nat_nat(G,I2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),K4)),aa_set_nat_nat(groups3542108847815614940at_nat(G),K4))) ) ).

% sum_mono
tff(fact_5767_sum__mono,axiom,
    ! [K4: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),K4))
         => 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,groups6591440286371151544t_real(F,K4)),groups6591440286371151544t_real(G,K4))) ) ).

% sum_mono
tff(fact_5768_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_nat,G: fun_real_fun_nat_nat,R: fun_re9121486324101951821t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups1935376822645274424al_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_ib(B2),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_id(A2),G),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_5769_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_nat,G: fun_int_fun_nat_nat,R: fun_int_fun_nat_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups4541462559716669496nt_nat(aa_fun5132377469993263754nt_nat(aa_fun348247319442030677nt_nat(aTP_Lamm_if(B2),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun4133023758092209198at_nat(aa_fun7384697781603784697at_nat(aTP_Lamm_ih(A2),G),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_5770_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B2: set_nat,G: fun_co636942507509095880at_nat,R: fun_co2841405903858895691t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups5693394587270226106ex_nat(aa_fun9050545535516344074ex_nat(aa_fun2757644259618170323ex_nat(aTP_Lamm_ij(B2),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun996456248398975532at_nat(aa_fun8754082201773474805at_nat(aTP_Lamm_il(A2),G),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_5771_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_complex,G: fun_re7208826350674946562omplex,R: fun_re3010880487650781423x_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( groups5754745047067104278omplex(aa_fun4873726854752153866omplex(aa_fun883667864959905433omplex(aTP_Lamm_in(B2),G),R),A2) = groups7754918857620584856omplex(aa_fun8128191506076533516omplex(aa_fun840110154952133019omplex(aTP_Lamm_ip(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_5772_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_complex,G: fun_na1591161823130732966omplex,R: fun_na5941084329111710283x_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( groups2073611262835488442omplex(aa_fun2700696402887535882omplex(aa_fun8019143150332516725omplex(aTP_Lamm_ir(B2),G),R),A2) = groups7754918857620584856omplex(aa_fun1613615970879741416omplex(aa_fun7586814990830484307omplex(aTP_Lamm_it(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_5773_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_complex,G: fun_in1307110748760623106omplex,R: fun_in5750378413879446255x_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( groups3049146728041665814omplex(aa_fun751309451324587018omplex(aa_fun3438240386181387033omplex(aTP_Lamm_iv(B2),G),R),A2) = groups7754918857620584856omplex(aa_fun2686559102040213132omplex(aa_fun5422379709423809691omplex(aTP_Lamm_ix(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_5774_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_nat,G: fun_re4238438383274954274t_real,R: fun_re9121486324101951821t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups8097168146408367636l_real(aa_fun7981600115099051494l_real(aa_fun2847282595593119957l_real(aTP_Lamm_iy(B2),G),R),A2) = groups6591440286371151544t_real(aa_fun555827300007793674t_real(aa_fun8353879187736340217t_real(aTP_Lamm_ja(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_5775_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_nat,G: fun_int_fun_nat_real,R: fun_int_fun_nat_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups8778361861064173332t_real(aa_fun8542488008249130982t_real(aa_fun1029605435529198677t_real(aTP_Lamm_jb(B2),G),R),A2) = groups6591440286371151544t_real(aa_fun6355566433556109194t_real(aa_fun7034273878219604473t_real(aTP_Lamm_jd(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_5776_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B2: set_nat,G: fun_co8378349014452122532t_real,R: fun_co2841405903858895691t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups5808333547571424918x_real(aa_fun7842767745293442150x_real(aa_fun7038367274271708115x_real(aTP_Lamm_je(B2),G),R),A2) = groups6591440286371151544t_real(aa_fun4093520059284520584t_real(aa_fun3817074898130815221t_real(aTP_Lamm_jg(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_5777_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_real,G: fun_nat_fun_real_nat,R: fun_na6801867027684862029l_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_real_bool(finite_finite_real,B2))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8064793419647352750at_nat(aa_fun581391774837177593at_nat(aTP_Lamm_ji(B2),G),R)),A2) = groups1935376822645274424al_nat(aa_fun389422839623581578al_nat(aa_fun9122955408058973909al_nat(aTP_Lamm_jl(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_5778_sum__diff1__nat,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,F: fun_Pr1253225528153422666at_nat] :
      ( ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),A2))
       => ( groups3860910324918113789at_nat(F,minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups3860910324918113789at_nat(F,A2)),aa_Pro6313114768868464763at_nat(F,A)) ) )
      & ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),A2))
       => ( groups3860910324918113789at_nat(F,minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) = groups3860910324918113789at_nat(F,A2) ) ) ) ).

% sum_diff1_nat
tff(fact_5779_sum__diff1__nat,axiom,
    ! [A: set_nat,A2: set_set_nat,F: fun_set_nat_nat] :
      ( ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
       => ( groups8294997508430121362at_nat(F,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A2)),aa_set_nat_nat(F,A)) ) )
      & ( ~ pp(aa_set_set_nat_bool(member_set_nat(A),A2))
       => ( groups8294997508430121362at_nat(F,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = groups8294997508430121362at_nat(F,A2) ) ) ) ).

% sum_diff1_nat
tff(fact_5780_sum__diff1__nat,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),A2))
       => ( groups977919841031483927at_nat(F,minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A2)),aa_Pro4243173770044397973at_nat(F,A)) ) )
      & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),A2))
       => ( groups977919841031483927at_nat(F,minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = groups977919841031483927at_nat(F,A2) ) ) ) ).

% sum_diff1_nat
tff(fact_5781_sum__diff1__nat,axiom,
    ! [A: real,A2: set_real,F: fun_real_nat] :
      ( ( pp(aa_set_real_bool(member_real(A),A2))
       => ( groups1935376822645274424al_nat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A2)),aa_real_nat(F,A)) ) )
      & ( ~ pp(aa_set_real_bool(member_real(A),A2))
       => ( groups1935376822645274424al_nat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = groups1935376822645274424al_nat(F,A2) ) ) ) ).

% sum_diff1_nat
tff(fact_5782_sum__diff1__nat,axiom,
    ! [A: int,A2: set_int,F: fun_int_nat] :
      ( ( pp(aa_set_int_bool(member_int(A),A2))
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),aa_int_nat(F,A)) ) )
      & ( ~ pp(aa_set_int_bool(member_int(A),A2))
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = groups4541462559716669496nt_nat(F,A2) ) ) ) ).

% sum_diff1_nat
tff(fact_5783_sum__diff1__nat,axiom,
    ! [A: nat,A2: set_nat,F: fun_nat_nat] :
      ( ( pp(aa_set_nat_bool(member_nat(A),A2))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_nat_nat(F,A)) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(A),A2))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) ) ) ) ).

% sum_diff1_nat
tff(fact_5784_Collect__conv__if,axiom,
    ! [P: fun_Pr1607443439640623155t_bool,A: produc3843707927480180839at_nat] :
      ( ( pp(aa_Pro1809015098595223084t_bool(P,A))
       => ( collec6321179662152712658at_nat(aa_Pro2859564254541897054t_bool(aTP_Lamm_jm(P),A)) = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ) )
      & ( ~ pp(aa_Pro1809015098595223084t_bool(P,A))
       => ( collec6321179662152712658at_nat(aa_Pro2859564254541897054t_bool(aTP_Lamm_jm(P),A)) = bot_bo228742789529271731at_nat ) ) ) ).

% Collect_conv_if
tff(fact_5785_Collect__conv__if,axiom,
    ! [P: fun_list_nat_bool,A: list_nat] :
      ( ( pp(aa_list_nat_bool(P,A))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_jn(P),A)) = insert_list_nat(A,bot_bot_set_list_nat) ) )
      & ( ~ pp(aa_list_nat_bool(P,A))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_jn(P),A)) = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if
tff(fact_5786_Collect__conv__if,axiom,
    ! [P: fun_set_nat_bool,A: set_nat] :
      ( ( pp(aa_set_nat_bool(P,A))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_jo(P),A)) = insert_set_nat(A,bot_bot_set_set_nat) ) )
      & ( ~ pp(aa_set_nat_bool(P,A))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_jo(P),A)) = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if
tff(fact_5787_Collect__conv__if,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,A: product_prod_nat_nat] :
      ( ( pp(aa_Pro2232559814647311250t_bool(P,A))
       => ( collec3392354462482085612at_nat(aa_Pro595088089689268830t_bool(aTP_Lamm_jp(P),A)) = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ) )
      & ( ~ pp(aa_Pro2232559814647311250t_bool(P,A))
       => ( collec3392354462482085612at_nat(aa_Pro595088089689268830t_bool(aTP_Lamm_jp(P),A)) = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if
tff(fact_5788_Collect__conv__if,axiom,
    ! [P: fun_real_bool,A: real] :
      ( ( pp(aa_real_bool(P,A))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_jq(P),A)) = insert_real(A,bot_bot_set_real) ) )
      & ( ~ pp(aa_real_bool(P,A))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_jq(P),A)) = bot_bot_set_real ) ) ) ).

% Collect_conv_if
tff(fact_5789_Collect__conv__if,axiom,
    ! [P: fun_nat_bool,A: nat] :
      ( ( pp(aa_nat_bool(P,A))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_jr(P),A)) = insert_nat(A,bot_bot_set_nat) ) )
      & ( ~ pp(aa_nat_bool(P,A))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_jr(P),A)) = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
tff(fact_5790_Collect__conv__if,axiom,
    ! [P: fun_int_bool,A: int] :
      ( ( pp(aa_int_bool(P,A))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_js(P),A)) = insert_int(A,bot_bot_set_int) ) )
      & ( ~ pp(aa_int_bool(P,A))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_js(P),A)) = bot_bot_set_int ) ) ) ).

% Collect_conv_if
tff(fact_5791_Collect__conv__if2,axiom,
    ! [P: fun_Pr1607443439640623155t_bool,A: produc3843707927480180839at_nat] :
      ( ( pp(aa_Pro1809015098595223084t_bool(P,A))
       => ( collec6321179662152712658at_nat(aa_Pro2859564254541897054t_bool(aTP_Lamm_jt(P),A)) = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ) )
      & ( ~ pp(aa_Pro1809015098595223084t_bool(P,A))
       => ( collec6321179662152712658at_nat(aa_Pro2859564254541897054t_bool(aTP_Lamm_jt(P),A)) = bot_bo228742789529271731at_nat ) ) ) ).

% Collect_conv_if2
tff(fact_5792_Collect__conv__if2,axiom,
    ! [P: fun_list_nat_bool,A: list_nat] :
      ( ( pp(aa_list_nat_bool(P,A))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_ju(P),A)) = insert_list_nat(A,bot_bot_set_list_nat) ) )
      & ( ~ pp(aa_list_nat_bool(P,A))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_ju(P),A)) = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if2
tff(fact_5793_Collect__conv__if2,axiom,
    ! [P: fun_set_nat_bool,A: set_nat] :
      ( ( pp(aa_set_nat_bool(P,A))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_jv(P),A)) = insert_set_nat(A,bot_bot_set_set_nat) ) )
      & ( ~ pp(aa_set_nat_bool(P,A))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_jv(P),A)) = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if2
tff(fact_5794_Collect__conv__if2,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,A: product_prod_nat_nat] :
      ( ( pp(aa_Pro2232559814647311250t_bool(P,A))
       => ( collec3392354462482085612at_nat(aa_Pro595088089689268830t_bool(aTP_Lamm_jw(P),A)) = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ) )
      & ( ~ pp(aa_Pro2232559814647311250t_bool(P,A))
       => ( collec3392354462482085612at_nat(aa_Pro595088089689268830t_bool(aTP_Lamm_jw(P),A)) = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if2
tff(fact_5795_Collect__conv__if2,axiom,
    ! [P: fun_real_bool,A: real] :
      ( ( pp(aa_real_bool(P,A))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_jx(P),A)) = insert_real(A,bot_bot_set_real) ) )
      & ( ~ pp(aa_real_bool(P,A))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_jx(P),A)) = bot_bot_set_real ) ) ) ).

% Collect_conv_if2
tff(fact_5796_Collect__conv__if2,axiom,
    ! [P: fun_nat_bool,A: nat] :
      ( ( pp(aa_nat_bool(P,A))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_jy(P),A)) = insert_nat(A,bot_bot_set_nat) ) )
      & ( ~ pp(aa_nat_bool(P,A))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_jy(P),A)) = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
tff(fact_5797_Collect__conv__if2,axiom,
    ! [P: fun_int_bool,A: int] :
      ( ( pp(aa_int_bool(P,A))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_jz(P),A)) = insert_int(A,bot_bot_set_int) ) )
      & ( ~ pp(aa_int_bool(P,A))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_jz(P),A)) = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
tff(fact_5798_round__mono,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,X)),aa_rat_int(archim7778729529865785530nd_rat,Y))) ) ).

% round_mono
tff(fact_5799_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_real,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5808333547571424918x_real(G,insert_complex(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_5800_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_rat,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5058264527183730370ex_rat(G,insert_complex(X,A2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_complex_rat(G,X)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_5801_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_nat,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5693394587270226106ex_nat(G,insert_complex(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_5802_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_int,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5690904116761175830ex_int(G,insert_complex(X,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,X)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_5803_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_real,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups8097168146408367636l_real(G,insert_real(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X)),groups8097168146408367636l_real(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_5804_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_rat,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1300246762558778688al_rat(G,insert_real(X,A2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(G,X)),groups1300246762558778688al_rat(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_5805_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_nat,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1935376822645274424al_nat(G,insert_real(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X)),groups1935376822645274424al_nat(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_5806_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_int,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1932886352136224148al_int(G,insert_real(X,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(G,X)),groups1932886352136224148al_int(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_5807_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: fun_int_real,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups8778361861064173332t_real(G,insert_int(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X)),groups8778361861064173332t_real(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ).

% sum.insert_remove
tff(fact_5808_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: fun_int_rat,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3906332499630173760nt_rat(G,insert_int(X,A2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(G,X)),groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ).

% sum.insert_remove
tff(fact_5809_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups5808333547571424918x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_5810_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups5058264527183730370ex_rat(G,A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_complex_rat(G,X)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_5811_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups5693394587270226106ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_5812_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups5690904116761175830ex_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,X)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_5813_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(X),A2))
       => ( groups8097168146408367636l_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X)),groups8097168146408367636l_real(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_5814_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(X),A2))
       => ( groups1300246762558778688al_rat(G,A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(G,X)),groups1300246762558778688al_rat(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_5815_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(X),A2))
       => ( groups1935376822645274424al_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X)),groups1935376822645274424al_nat(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_5816_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(X),A2))
       => ( groups1932886352136224148al_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(G,X)),groups1932886352136224148al_int(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_5817_sum_Oremove,axiom,
    ! [A2: set_int,X: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(X),A2))
       => ( groups8778361861064173332t_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X)),groups8778361861064173332t_real(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% sum.remove
tff(fact_5818_sum_Oremove,axiom,
    ! [A2: set_int,X: int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(X),A2))
       => ( groups3906332499630173760nt_rat(G,A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(G,X)),groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% sum.remove
tff(fact_5819_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),A2))
         => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(F,A2)),aa_complex_real(F,A)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),A2))
         => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = groups5808333547571424918x_real(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_5820_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(A),A2))
         => ( groups8097168146408367636l_real(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8097168146408367636l_real(F,A2)),aa_real_real(F,A)) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),A2))
         => ( groups8097168146408367636l_real(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = groups8097168146408367636l_real(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_5821_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( pp(aa_set_int_bool(member_int(A),A2))
         => ( groups8778361861064173332t_real(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(F,A2)),aa_int_real(F,A)) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),A2))
         => ( groups8778361861064173332t_real(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = groups8778361861064173332t_real(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_5822_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),A2))
         => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(F,A2)),aa_complex_rat(F,A)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),A2))
         => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = groups5058264527183730370ex_rat(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_5823_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(A),A2))
         => ( groups1300246762558778688al_rat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = aa_rat_rat(minus_minus_rat(groups1300246762558778688al_rat(F,A2)),aa_real_rat(F,A)) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),A2))
         => ( groups1300246762558778688al_rat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = groups1300246762558778688al_rat(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_5824_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( pp(aa_set_int_bool(member_int(A),A2))
         => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = aa_rat_rat(minus_minus_rat(groups3906332499630173760nt_rat(F,A2)),aa_int_rat(F,A)) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),A2))
         => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = groups3906332499630173760nt_rat(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_5825_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),A2))
         => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A2)),aa_complex_int(F,A)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),A2))
         => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = groups5690904116761175830ex_int(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_5826_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(A),A2))
         => ( groups1932886352136224148al_int(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups1932886352136224148al_int(F,A2)),aa_real_int(F,A)) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),A2))
         => ( groups1932886352136224148al_int(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = groups1932886352136224148al_int(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_5827_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( pp(aa_set_int_bool(member_int(A),A2))
         => ( groups4538972089207619220nt_int(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A2)),aa_int_int(F,A)) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),A2))
         => ( groups4538972089207619220nt_int(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = groups4538972089207619220nt_int(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_5828_sum__diff1,axiom,
    ! [A2: set_nat,A: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),A2))
         => ( groups2906978787729119204at_rat(F,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(F,A2)),aa_nat_rat(F,A)) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),A2))
         => ( groups2906978787729119204at_rat(F,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = groups2906978787729119204at_rat(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_5829_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => 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,A2))) ) ).

% sum_nonneg
tff(fact_5830_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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,A2))) ) ).

% sum_nonneg
tff(fact_5831_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => 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,A2))) ) ).

% sum_nonneg
tff(fact_5832_sum__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => 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,A2))) ) ).

% sum_nonneg
tff(fact_5833_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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,A2))) ) ).

% sum_nonneg
tff(fact_5834_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => 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,A2))) ) ).

% sum_nonneg
tff(fact_5835_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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,A2))) ) ).

% sum_nonneg
tff(fact_5836_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_int] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups1932886352136224148al_int(F,A2))) ) ).

% sum_nonneg
tff(fact_5837_sum__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => 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),groups3539618377306564664at_int(F,A2))) ) ).

% sum_nonneg
tff(fact_5838_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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),groups4538972089207619220nt_int(F,A2))) ) ).

% sum_nonneg
tff(fact_5839_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => 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,A2)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_5840_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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,A2)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_5841_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => 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,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_5842_sum__nonpos,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => 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,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_5843_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_5844_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => 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,A2)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_5845_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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,A2)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_5846_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_int] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,X4)),zero_zero_int)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups1932886352136224148al_int(F,A2)),zero_zero_int)) ) ).

% sum_nonpos
tff(fact_5847_sum__nonpos,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,X4)),zero_zero_int)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,A2)),zero_zero_int)) ) ).

% sum_nonpos
tff(fact_5848_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,X4)),zero_zero_int)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups4538972089207619220nt_int(F,A2)),zero_zero_int)) ) ).

% sum_nonpos
tff(fact_5849_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_5850_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_5851_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_5852_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_5853_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_5854_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_5855_sum__mono__inv,axiom,
    ! [F: fun_complex_nat,I6: set_complex,G: fun_complex_nat,I: complex] :
      ( ( groups5693394587270226106ex_nat(F,I6) = groups5693394587270226106ex_nat(G,I6) )
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),I6))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2))) )
       => ( pp(aa_set_complex_bool(member_complex(I),I6))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
           => ( aa_complex_nat(F,I) = aa_complex_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5856_sum__mono__inv,axiom,
    ! [F: fun_real_int,I6: set_real,G: fun_real_int,I: real] :
      ( ( groups1932886352136224148al_int(F,I6) = groups1932886352136224148al_int(G,I6) )
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),I6))
           => 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_set_real_bool(member_real(I),I6))
         => ( pp(aa_set_real_bool(finite_finite_real,I6))
           => ( aa_real_int(F,I) = aa_real_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5857_sum__mono__inv,axiom,
    ! [F: fun_nat_int,I6: set_nat,G: fun_nat_int,I: nat] :
      ( ( groups3539618377306564664at_int(F,I6) = groups3539618377306564664at_int(G,I6) )
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),I6))
           => 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_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_set_nat_bool(finite_finite_nat,I6))
           => ( aa_nat_int(F,I) = aa_nat_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5858_sum__mono__inv,axiom,
    ! [F: fun_int_int,I6: set_int,G: fun_int_int,I: int] :
      ( ( groups4538972089207619220nt_int(F,I6) = groups4538972089207619220nt_int(G,I6) )
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),I6))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,I2)),aa_int_int(G,I2))) )
       => ( pp(aa_set_int_bool(member_int(I),I6))
         => ( pp(aa_set_int_bool(finite_finite_int,I6))
           => ( aa_int_int(F,I) = aa_int_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5859_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X4)),A2))
           => ( 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),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) ) ) ) ).

% sum_cong_Suc
tff(fact_5860_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X4)),A2))
           => ( aa_nat_real(F,aa_nat_nat(suc,X4)) = aa_nat_real(G,aa_nat_nat(suc,X4)) ) )
       => ( groups6591440286371151544t_real(F,A2) = groups6591440286371151544t_real(G,A2) ) ) ) ).

% sum_cong_Suc
tff(fact_5861_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real,C: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_ka(A),B),C),S2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(B,A)),groups5808333547571424918x_real(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_ka(A),B),C),S2) = groups5808333547571424918x_real(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_5862_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat,C: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_kb(A),B),C),S2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_complex_rat(B,A)),groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_kb(A),B),C),S2) = groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_5863_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_nat,C: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_kc(A),B),C),S2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(B,A)),groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_kc(A),B),C),S2) = groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_5864_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_int,C: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_kd(A),B),C),S2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(B,A)),groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_kd(A),B),C),S2) = groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_5865_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_real,C: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_ke(A),B),C),S2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(B,A)),groups8097168146408367636l_real(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_ke(A),B),C),S2) = groups8097168146408367636l_real(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% sum.delta_remove
tff(fact_5866_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat,C: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_kf(A),B),C),S2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(B,A)),groups1300246762558778688al_rat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_kf(A),B),C),S2) = groups1300246762558778688al_rat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% sum.delta_remove
tff(fact_5867_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_nat,C: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_kg(A),B),C),S2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(B,A)),groups1935376822645274424al_nat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_kg(A),B),C),S2) = groups1935376822645274424al_nat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% sum.delta_remove
tff(fact_5868_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_int,C: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_kh(A),B),C),S2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(B,A)),groups1932886352136224148al_int(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_kh(A),B),C),S2) = groups1932886352136224148al_int(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% sum.delta_remove
tff(fact_5869_sum_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B: fun_int_real,C: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_ki(A),B),C),S2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(B,A)),groups8778361861064173332t_real(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int)))) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_ki(A),B),C),S2) = groups8778361861064173332t_real(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int))) ) ) ) ) ).

% sum.delta_remove
tff(fact_5870_sum_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B: fun_int_rat,C: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_kj(A),B),C),S2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(B,A)),groups3906332499630173760nt_rat(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int)))) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_kj(A),B),C),S2) = groups3906332499630173760nt_rat(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int))) ) ) ) ) ).

% sum.delta_remove
tff(fact_5871_infinite__finite__induct,axiom,
    ! [P: fun_se8353504738346895699t_bool,A2: set_Pr4329608150637261639at_nat] :
      ( ! [A4: set_Pr4329608150637261639at_nat] :
          ( ~ pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A4))
         => pp(aa_set4148026973699692364t_bool(P,A4)) )
     => ( pp(aa_set4148026973699692364t_bool(P,bot_bo228742789529271731at_nat))
       => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
              ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,F4))
             => ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X4),F4))
               => ( pp(aa_set4148026973699692364t_bool(P,F4))
                 => pp(aa_set4148026973699692364t_bool(P,insert9069300056098147895at_nat(X4,F4))) ) ) )
         => pp(aa_set4148026973699692364t_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_5872_infinite__finite__induct,axiom,
    ! [P: fun_set_set_nat_bool,A2: set_set_nat] :
      ( ! [A4: set_set_nat] :
          ( ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A4))
         => pp(aa_set_set_nat_bool(P,A4)) )
     => ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
       => ( ! [X4: set_nat,F4: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
             => ( ~ pp(aa_set_set_nat_bool(member_set_nat(X4),F4))
               => ( pp(aa_set_set_nat_bool(P,F4))
                 => pp(aa_set_set_nat_bool(P,insert_set_nat(X4,F4))) ) ) )
         => pp(aa_set_set_nat_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_5873_infinite__finite__induct,axiom,
    ! [P: fun_set_complex_bool,A2: set_complex] :
      ( ! [A4: set_complex] :
          ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
         => pp(aa_set_complex_bool(P,A4)) )
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X4: complex,F4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
             => ( ~ pp(aa_set_complex_bool(member_complex(X4),F4))
               => ( pp(aa_set_complex_bool(P,F4))
                 => pp(aa_set_complex_bool(P,insert_complex(X4,F4))) ) ) )
         => pp(aa_set_complex_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_5874_infinite__finite__induct,axiom,
    ! [P: fun_se8074476271188067385t_bool,A2: set_Pr1261947904930325089at_nat] :
      ( ! [A4: set_Pr1261947904930325089at_nat] :
          ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
         => pp(aa_set3135850010578026034t_bool(P,A4)) )
     => ( pp(aa_set3135850010578026034t_bool(P,bot_bo2099793752762293965at_nat))
       => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
              ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F4))
             => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),F4))
               => ( pp(aa_set3135850010578026034t_bool(P,F4))
                 => pp(aa_set3135850010578026034t_bool(P,insert8211810215607154385at_nat(X4,F4))) ) ) )
         => pp(aa_set3135850010578026034t_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_5875_infinite__finite__induct,axiom,
    ! [P: fun_set_real_bool,A2: set_real] :
      ( ! [A4: set_real] :
          ( ~ pp(aa_set_real_bool(finite_finite_real,A4))
         => pp(aa_set_real_bool(P,A4)) )
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X4: real,F4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,F4))
             => ( ~ pp(aa_set_real_bool(member_real(X4),F4))
               => ( pp(aa_set_real_bool(P,F4))
                 => pp(aa_set_real_bool(P,insert_real(X4,F4))) ) ) )
         => pp(aa_set_real_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_5876_infinite__finite__induct,axiom,
    ! [P: fun_set_nat_bool,A2: set_nat] :
      ( ! [A4: set_nat] :
          ( ~ pp(aa_set_nat_bool(finite_finite_nat,A4))
         => pp(aa_set_nat_bool(P,A4)) )
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [X4: nat,F4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,F4))
             => ( ~ pp(aa_set_nat_bool(member_nat(X4),F4))
               => ( pp(aa_set_nat_bool(P,F4))
                 => pp(aa_set_nat_bool(P,insert_nat(X4,F4))) ) ) )
         => pp(aa_set_nat_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_5877_infinite__finite__induct,axiom,
    ! [P: fun_set_int_bool,A2: set_int] :
      ( ! [A4: set_int] :
          ( ~ pp(aa_set_int_bool(finite_finite_int,A4))
         => pp(aa_set_int_bool(P,A4)) )
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [X4: int,F4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,F4))
             => ( ~ pp(aa_set_int_bool(member_int(X4),F4))
               => ( pp(aa_set_int_bool(P,F4))
                 => pp(aa_set_int_bool(P,insert_int(X4,F4))) ) ) )
         => pp(aa_set_int_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_5878_finite__ne__induct,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,P: fun_se8353504738346895699t_bool] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,F3))
     => ( ( F3 != bot_bo228742789529271731at_nat )
       => ( ! [X4: produc3843707927480180839at_nat] : pp(aa_set4148026973699692364t_bool(P,insert9069300056098147895at_nat(X4,bot_bo228742789529271731at_nat)))
         => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
                ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,F4))
               => ( ( F4 != bot_bo228742789529271731at_nat )
                 => ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X4),F4))
                   => ( pp(aa_set4148026973699692364t_bool(P,F4))
                     => pp(aa_set4148026973699692364t_bool(P,insert9069300056098147895at_nat(X4,F4))) ) ) ) )
           => pp(aa_set4148026973699692364t_bool(P,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_5879_finite__ne__induct,axiom,
    ! [F3: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F3))
     => ( ( F3 != bot_bot_set_set_nat )
       => ( ! [X4: set_nat] : pp(aa_set_set_nat_bool(P,insert_set_nat(X4,bot_bot_set_set_nat)))
         => ( ! [X4: set_nat,F4: set_set_nat] :
                ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
               => ( ( F4 != bot_bot_set_set_nat )
                 => ( ~ pp(aa_set_set_nat_bool(member_set_nat(X4),F4))
                   => ( pp(aa_set_set_nat_bool(P,F4))
                     => pp(aa_set_set_nat_bool(P,insert_set_nat(X4,F4))) ) ) ) )
           => pp(aa_set_set_nat_bool(P,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_5880_finite__ne__induct,axiom,
    ! [F3: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( ( F3 != bot_bot_set_complex )
       => ( ! [X4: complex] : pp(aa_set_complex_bool(P,insert_complex(X4,bot_bot_set_complex)))
         => ( ! [X4: complex,F4: set_complex] :
                ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
               => ( ( F4 != bot_bot_set_complex )
                 => ( ~ pp(aa_set_complex_bool(member_complex(X4),F4))
                   => ( pp(aa_set_complex_bool(P,F4))
                     => pp(aa_set_complex_bool(P,insert_complex(X4,F4))) ) ) ) )
           => pp(aa_set_complex_bool(P,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_5881_finite__ne__induct,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,P: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F3))
     => ( ( F3 != bot_bo2099793752762293965at_nat )
       => ( ! [X4: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(P,insert8211810215607154385at_nat(X4,bot_bo2099793752762293965at_nat)))
         => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F4))
               => ( ( F4 != bot_bo2099793752762293965at_nat )
                 => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),F4))
                   => ( pp(aa_set3135850010578026034t_bool(P,F4))
                     => pp(aa_set3135850010578026034t_bool(P,insert8211810215607154385at_nat(X4,F4))) ) ) ) )
           => pp(aa_set3135850010578026034t_bool(P,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_5882_finite__ne__induct,axiom,
    ! [F3: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F3))
     => ( ( F3 != bot_bot_set_real )
       => ( ! [X4: real] : pp(aa_set_real_bool(P,insert_real(X4,bot_bot_set_real)))
         => ( ! [X4: real,F4: set_real] :
                ( pp(aa_set_real_bool(finite_finite_real,F4))
               => ( ( F4 != bot_bot_set_real )
                 => ( ~ pp(aa_set_real_bool(member_real(X4),F4))
                   => ( pp(aa_set_real_bool(P,F4))
                     => pp(aa_set_real_bool(P,insert_real(X4,F4))) ) ) ) )
           => pp(aa_set_real_bool(P,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_5883_finite__ne__induct,axiom,
    ! [F3: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( ( F3 != bot_bot_set_nat )
       => ( ! [X4: nat] : pp(aa_set_nat_bool(P,insert_nat(X4,bot_bot_set_nat)))
         => ( ! [X4: nat,F4: set_nat] :
                ( pp(aa_set_nat_bool(finite_finite_nat,F4))
               => ( ( F4 != bot_bot_set_nat )
                 => ( ~ pp(aa_set_nat_bool(member_nat(X4),F4))
                   => ( pp(aa_set_nat_bool(P,F4))
                     => pp(aa_set_nat_bool(P,insert_nat(X4,F4))) ) ) ) )
           => pp(aa_set_nat_bool(P,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_5884_finite__ne__induct,axiom,
    ! [F3: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( ( F3 != bot_bot_set_int )
       => ( ! [X4: int] : pp(aa_set_int_bool(P,insert_int(X4,bot_bot_set_int)))
         => ( ! [X4: int,F4: set_int] :
                ( pp(aa_set_int_bool(finite_finite_int,F4))
               => ( ( F4 != bot_bot_set_int )
                 => ( ~ pp(aa_set_int_bool(member_int(X4),F4))
                   => ( pp(aa_set_int_bool(P,F4))
                     => pp(aa_set_int_bool(P,insert_int(X4,F4))) ) ) ) )
           => pp(aa_set_int_bool(P,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_5885_finite__induct,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,P: fun_se8353504738346895699t_bool] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,F3))
     => ( pp(aa_set4148026973699692364t_bool(P,bot_bo228742789529271731at_nat))
       => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
              ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,F4))
             => ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X4),F4))
               => ( pp(aa_set4148026973699692364t_bool(P,F4))
                 => pp(aa_set4148026973699692364t_bool(P,insert9069300056098147895at_nat(X4,F4))) ) ) )
         => pp(aa_set4148026973699692364t_bool(P,F3)) ) ) ) ).

% finite_induct
tff(fact_5886_finite__induct,axiom,
    ! [F3: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F3))
     => ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
       => ( ! [X4: set_nat,F4: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
             => ( ~ pp(aa_set_set_nat_bool(member_set_nat(X4),F4))
               => ( pp(aa_set_set_nat_bool(P,F4))
                 => pp(aa_set_set_nat_bool(P,insert_set_nat(X4,F4))) ) ) )
         => pp(aa_set_set_nat_bool(P,F3)) ) ) ) ).

% finite_induct
tff(fact_5887_finite__induct,axiom,
    ! [F3: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X4: complex,F4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
             => ( ~ pp(aa_set_complex_bool(member_complex(X4),F4))
               => ( pp(aa_set_complex_bool(P,F4))
                 => pp(aa_set_complex_bool(P,insert_complex(X4,F4))) ) ) )
         => pp(aa_set_complex_bool(P,F3)) ) ) ) ).

% finite_induct
tff(fact_5888_finite__induct,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,P: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F3))
     => ( pp(aa_set3135850010578026034t_bool(P,bot_bo2099793752762293965at_nat))
       => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
              ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F4))
             => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),F4))
               => ( pp(aa_set3135850010578026034t_bool(P,F4))
                 => pp(aa_set3135850010578026034t_bool(P,insert8211810215607154385at_nat(X4,F4))) ) ) )
         => pp(aa_set3135850010578026034t_bool(P,F3)) ) ) ) ).

% finite_induct
tff(fact_5889_finite__induct,axiom,
    ! [F3: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F3))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X4: real,F4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,F4))
             => ( ~ pp(aa_set_real_bool(member_real(X4),F4))
               => ( pp(aa_set_real_bool(P,F4))
                 => pp(aa_set_real_bool(P,insert_real(X4,F4))) ) ) )
         => pp(aa_set_real_bool(P,F3)) ) ) ) ).

% finite_induct
tff(fact_5890_finite__induct,axiom,
    ! [F3: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [X4: nat,F4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,F4))
             => ( ~ pp(aa_set_nat_bool(member_nat(X4),F4))
               => ( pp(aa_set_nat_bool(P,F4))
                 => pp(aa_set_nat_bool(P,insert_nat(X4,F4))) ) ) )
         => pp(aa_set_nat_bool(P,F3)) ) ) ) ).

% finite_induct
tff(fact_5891_finite__induct,axiom,
    ! [F3: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [X4: int,F4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,F4))
             => ( ~ pp(aa_set_int_bool(member_int(X4),F4))
               => ( pp(aa_set_int_bool(P,F4))
                 => pp(aa_set_int_bool(P,insert_int(X4,F4))) ) ) )
         => pp(aa_set_int_bool(P,F3)) ) ) ) ).

% finite_induct
tff(fact_5892_finite_Osimps,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A))
    <=> ( ( A = bot_bo228742789529271731at_nat )
        | ? [A7: set_Pr4329608150637261639at_nat,A6: produc3843707927480180839at_nat] :
            ( ( A = insert9069300056098147895at_nat(A6,A7) )
            & pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A7)) ) ) ) ).

% finite.simps
tff(fact_5893_finite_Osimps,axiom,
    ! [A: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
    <=> ( ( A = bot_bot_set_complex )
        | ? [A7: set_complex,A6: complex] :
            ( ( A = insert_complex(A6,A7) )
            & pp(aa_set_complex_bool(finite3207457112153483333omplex,A7)) ) ) ) ).

% finite.simps
tff(fact_5894_finite_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A))
    <=> ( ( A = bot_bo2099793752762293965at_nat )
        | ? [A7: set_Pr1261947904930325089at_nat,A6: product_prod_nat_nat] :
            ( ( A = insert8211810215607154385at_nat(A6,A7) )
            & pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A7)) ) ) ) ).

% finite.simps
tff(fact_5895_finite_Osimps,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
    <=> ( ( A = bot_bot_set_real )
        | ? [A7: set_real,A6: real] :
            ( ( A = insert_real(A6,A7) )
            & pp(aa_set_real_bool(finite_finite_real,A7)) ) ) ) ).

% finite.simps
tff(fact_5896_finite_Osimps,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
    <=> ( ( A = bot_bot_set_nat )
        | ? [A7: set_nat,A6: nat] :
            ( ( A = insert_nat(A6,A7) )
            & pp(aa_set_nat_bool(finite_finite_nat,A7)) ) ) ) ).

% finite.simps
tff(fact_5897_finite_Osimps,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
    <=> ( ( A = bot_bot_set_int )
        | ? [A7: set_int,A6: int] :
            ( ( A = insert_int(A6,A7) )
            & pp(aa_set_int_bool(finite_finite_int,A7)) ) ) ) ).

% finite.simps
tff(fact_5898_finite_Ocases,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A))
     => ( ( A != bot_bo228742789529271731at_nat )
       => ~ ! [A4: set_Pr4329608150637261639at_nat] :
              ( ? [A3: produc3843707927480180839at_nat] : A = insert9069300056098147895at_nat(A3,A4)
             => ~ pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A4)) ) ) ) ).

% finite.cases
tff(fact_5899_finite_Ocases,axiom,
    ! [A: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ( A != bot_bot_set_complex )
       => ~ ! [A4: set_complex] :
              ( ? [A3: complex] : A = insert_complex(A3,A4)
             => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4)) ) ) ) ).

% finite.cases
tff(fact_5900_finite_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A))
     => ( ( A != bot_bo2099793752762293965at_nat )
       => ~ ! [A4: set_Pr1261947904930325089at_nat] :
              ( ? [A3: product_prod_nat_nat] : A = insert8211810215607154385at_nat(A3,A4)
             => ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4)) ) ) ) ).

% finite.cases
tff(fact_5901_finite_Ocases,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ( A != bot_bot_set_real )
       => ~ ! [A4: set_real] :
              ( ? [A3: real] : A = insert_real(A3,A4)
             => ~ pp(aa_set_real_bool(finite_finite_real,A4)) ) ) ) ).

% finite.cases
tff(fact_5902_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A4: set_nat] :
              ( ? [A3: nat] : A = insert_nat(A3,A4)
             => ~ pp(aa_set_nat_bool(finite_finite_nat,A4)) ) ) ) ).

% finite.cases
tff(fact_5903_finite_Ocases,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ( A != bot_bot_set_int )
       => ~ ! [A4: set_int] :
              ( ? [A3: int] : A = insert_int(A3,A4)
             => ~ pp(aa_set_int_bool(finite_finite_int,A4)) ) ) ) ).

% finite.cases
tff(fact_5904_subset__singletonD,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(A2),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)))
     => ( ( A2 = bot_bo228742789529271731at_nat )
        | ( A2 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ) ).

% subset_singletonD
tff(fact_5905_subset__singletonD,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)))
     => ( ( A2 = bot_bo2099793752762293965at_nat )
        | ( A2 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ) ).

% subset_singletonD
tff(fact_5906_subset__singletonD,axiom,
    ! [A2: set_real,X: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(X,bot_bot_set_real)))
     => ( ( A2 = bot_bot_set_real )
        | ( A2 = insert_real(X,bot_bot_set_real) ) ) ) ).

% subset_singletonD
tff(fact_5907_subset__singletonD,axiom,
    ! [A2: set_nat,X: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(X,bot_bot_set_nat)))
     => ( ( A2 = bot_bot_set_nat )
        | ( A2 = insert_nat(X,bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_5908_subset__singletonD,axiom,
    ! [A2: set_int,X: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(X,bot_bot_set_int)))
     => ( ( A2 = bot_bot_set_int )
        | ( A2 = insert_int(X,bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_5909_subset__singleton__iff,axiom,
    ! [X6: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(X6),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)))
    <=> ( ( X6 = bot_bo228742789529271731at_nat )
        | ( X6 = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_5910_subset__singleton__iff,axiom,
    ! [X6: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X6),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)))
    <=> ( ( X6 = bot_bo2099793752762293965at_nat )
        | ( X6 = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_5911_subset__singleton__iff,axiom,
    ! [X6: set_real,A: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(X6),insert_real(A,bot_bot_set_real)))
    <=> ( ( X6 = bot_bot_set_real )
        | ( X6 = insert_real(A,bot_bot_set_real) ) ) ) ).

% subset_singleton_iff
tff(fact_5912_subset__singleton__iff,axiom,
    ! [X6: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X6),insert_nat(A,bot_bot_set_nat)))
    <=> ( ( X6 = bot_bot_set_nat )
        | ( X6 = insert_nat(A,bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_5913_subset__singleton__iff,axiom,
    ! [X6: set_int,A: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X6),insert_int(A,bot_bot_set_int)))
    <=> ( ( X6 = bot_bot_set_int )
        | ( X6 = insert_int(A,bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_5914_atLeastAtMost__singleton_H,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( set_or1269000886237332187st_nat(A,B) = insert_nat(A,bot_bot_set_nat) ) ) ).

% atLeastAtMost_singleton'
tff(fact_5915_atLeastAtMost__singleton_H,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B) = insert_int(A,bot_bot_set_int) ) ) ).

% atLeastAtMost_singleton'
tff(fact_5916_atLeastAtMost__singleton_H,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( set_or1222579329274155063t_real(A,B) = insert_real(A,bot_bot_set_real) ) ) ).

% atLeastAtMost_singleton'
tff(fact_5917_Diff__insert__absorb,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),A2))
     => ( minus_3314409938677909166at_nat(insert9069300056098147895at_nat(X,A2),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5918_Diff__insert__absorb,axiom,
    ! [X: set_nat,A2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A2))
     => ( minus_2163939370556025621et_nat(insert_set_nat(X,A2),insert_set_nat(X,bot_bot_set_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5919_Diff__insert__absorb,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2))
     => ( minus_1356011639430497352at_nat(insert8211810215607154385at_nat(X,A2),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5920_Diff__insert__absorb,axiom,
    ! [X: real,A2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X),A2))
     => ( minus_minus_set_real(insert_real(X,A2),insert_real(X,bot_bot_set_real)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5921_Diff__insert__absorb,axiom,
    ! [X: int,A2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X),A2))
     => ( minus_minus_set_int(insert_int(X,A2),insert_int(X,bot_bot_set_int)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5922_Diff__insert__absorb,axiom,
    ! [X: nat,A2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
     => ( minus_minus_set_nat(insert_nat(X,A2),insert_nat(X,bot_bot_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5923_Diff__insert2,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(A,B2)) = minus_3314409938677909166at_nat(minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)),B2) ).

% Diff_insert2
tff(fact_5924_Diff__insert2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,B2)) = minus_1356011639430497352at_nat(minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)),B2) ).

% Diff_insert2
tff(fact_5925_Diff__insert2,axiom,
    ! [A2: set_real,A: real,B2: set_real] : minus_minus_set_real(A2,insert_real(A,B2)) = minus_minus_set_real(minus_minus_set_real(A2,insert_real(A,bot_bot_set_real)),B2) ).

% Diff_insert2
tff(fact_5926_Diff__insert2,axiom,
    ! [A2: set_int,A: int,B2: set_int] : minus_minus_set_int(A2,insert_int(A,B2)) = minus_minus_set_int(minus_minus_set_int(A2,insert_int(A,bot_bot_set_int)),B2) ).

% Diff_insert2
tff(fact_5927_Diff__insert2,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] : minus_minus_set_nat(A2,insert_nat(A,B2)) = minus_minus_set_nat(minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat)),B2) ).

% Diff_insert2
tff(fact_5928_insert__Diff,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),A2))
     => ( insert9069300056098147895at_nat(A,minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_5929_insert__Diff,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
     => ( insert_set_nat(A,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_5930_insert__Diff,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),A2))
     => ( insert8211810215607154385at_nat(A,minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_5931_insert__Diff,axiom,
    ! [A: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(A),A2))
     => ( insert_real(A,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = A2 ) ) ).

% insert_Diff
tff(fact_5932_insert__Diff,axiom,
    ! [A: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(A),A2))
     => ( insert_int(A,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = A2 ) ) ).

% insert_Diff
tff(fact_5933_insert__Diff,axiom,
    ! [A: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),A2))
     => ( insert_nat(A,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_5934_Diff__insert,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(A,B2)) = minus_3314409938677909166at_nat(minus_3314409938677909166at_nat(A2,B2),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)) ).

% Diff_insert
tff(fact_5935_Diff__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,B2)) = minus_1356011639430497352at_nat(minus_1356011639430497352at_nat(A2,B2),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)) ).

% Diff_insert
tff(fact_5936_Diff__insert,axiom,
    ! [A2: set_real,A: real,B2: set_real] : minus_minus_set_real(A2,insert_real(A,B2)) = minus_minus_set_real(minus_minus_set_real(A2,B2),insert_real(A,bot_bot_set_real)) ).

% Diff_insert
tff(fact_5937_Diff__insert,axiom,
    ! [A2: set_int,A: int,B2: set_int] : minus_minus_set_int(A2,insert_int(A,B2)) = minus_minus_set_int(minus_minus_set_int(A2,B2),insert_int(A,bot_bot_set_int)) ).

% Diff_insert
tff(fact_5938_Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] : minus_minus_set_nat(A2,insert_nat(A,B2)) = minus_minus_set_nat(minus_minus_set_nat(A2,B2),insert_nat(A,bot_bot_set_nat)) ).

% Diff_insert
tff(fact_5939_Compl__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] : uminus935396558254630718at_nat(insert9069300056098147895at_nat(X,A2)) = minus_3314409938677909166at_nat(uminus935396558254630718at_nat(A2),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) ).

% Compl_insert
tff(fact_5940_Compl__insert,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(insert8211810215607154385at_nat(X,A2)) = minus_1356011639430497352at_nat(uminus6524753893492686040at_nat(A2),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) ).

% Compl_insert
tff(fact_5941_Compl__insert,axiom,
    ! [X: real,A2: set_real] : uminus612125837232591019t_real(insert_real(X,A2)) = minus_minus_set_real(uminus612125837232591019t_real(A2),insert_real(X,bot_bot_set_real)) ).

% Compl_insert
tff(fact_5942_Compl__insert,axiom,
    ! [X: int,A2: set_int] : uminus1532241313380277803et_int(insert_int(X,A2)) = minus_minus_set_int(uminus1532241313380277803et_int(A2),insert_int(X,bot_bot_set_int)) ).

% Compl_insert
tff(fact_5943_Compl__insert,axiom,
    ! [X: nat,A2: set_nat] : uminus5710092332889474511et_nat(insert_nat(X,A2)) = minus_minus_set_nat(uminus5710092332889474511et_nat(A2),insert_nat(X,bot_bot_set_nat)) ).

% Compl_insert
tff(fact_5944_subset__Diff__insert,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(A2),minus_3314409938677909166at_nat(B2,insert9069300056098147895at_nat(X,C2))))
    <=> ( pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(A2),minus_3314409938677909166at_nat(B2,C2)))
        & ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),A2)) ) ) ).

% subset_Diff_insert
tff(fact_5945_subset__Diff__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),minus_1356011639430497352at_nat(B2,insert8211810215607154385at_nat(X,C2))))
    <=> ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),minus_1356011639430497352at_nat(B2,C2)))
        & ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2)) ) ) ).

% subset_Diff_insert
tff(fact_5946_subset__Diff__insert,axiom,
    ! [A2: set_real,B2: set_real,X: real,C2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),minus_minus_set_real(B2,insert_real(X,C2))))
    <=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),minus_minus_set_real(B2,C2)))
        & ~ pp(aa_set_real_bool(member_real(X),A2)) ) ) ).

% subset_Diff_insert
tff(fact_5947_subset__Diff__insert,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,X: set_nat,C2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),minus_2163939370556025621et_nat(B2,insert_set_nat(X,C2))))
    <=> ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),minus_2163939370556025621et_nat(B2,C2)))
        & ~ pp(aa_set_set_nat_bool(member_set_nat(X),A2)) ) ) ).

% subset_Diff_insert
tff(fact_5948_subset__Diff__insert,axiom,
    ! [A2: set_nat,B2: set_nat,X: nat,C2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),minus_minus_set_nat(B2,insert_nat(X,C2))))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),minus_minus_set_nat(B2,C2)))
        & ~ pp(aa_set_nat_bool(member_nat(X),A2)) ) ) ).

% subset_Diff_insert
tff(fact_5949_subset__Diff__insert,axiom,
    ! [A2: set_int,B2: set_int,X: int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),minus_minus_set_int(B2,insert_int(X,C2))))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),minus_minus_set_int(B2,C2)))
        & ~ pp(aa_set_int_bool(member_int(X),A2)) ) ) ).

% subset_Diff_insert
tff(fact_5950_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_complex,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups5754745047067104278omplex(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bm(A2),P))) = groups5754745047067104278omplex(aa_fun6813637976755515871omplex(aTP_Lamm_kk(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5951_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_complex,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups2073611262835488442omplex(G,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bp(A2),P))) = groups2073611262835488442omplex(aa_fun8588439219086927967omplex(aTP_Lamm_kl(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5952_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_complex,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3049146728041665814omplex(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bq(A2),P))) = groups3049146728041665814omplex(aa_fun3606824691168040799omplex(aTP_Lamm_km(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5953_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups8097168146408367636l_real(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bm(A2),P))) = groups8097168146408367636l_real(aa_fun6348708079608492509l_real(aTP_Lamm_kn(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5954_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_real,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups8778361861064173332t_real(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bq(A2),P))) = groups8778361861064173332t_real(aa_fun6974987029372589917t_real(aTP_Lamm_ko(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5955_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_real,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5808333547571424918x_real(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_kp(A2),P))) = groups5808333547571424918x_real(aa_fun2902006202505214429x_real(aTP_Lamm_kq(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5956_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_rat,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1300246762558778688al_rat(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bm(A2),P))) = groups1300246762558778688al_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_kr(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5957_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_rat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups2906978787729119204at_rat(G,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bp(A2),P))) = groups2906978787729119204at_rat(aa_fun5656162108953708681at_rat(aTP_Lamm_ks(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5958_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_rat,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3906332499630173760nt_rat(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bq(A2),P))) = groups3906332499630173760nt_rat(aa_fun4400602662665189257nt_rat(aTP_Lamm_kt(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5959_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_rat,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5058264527183730370ex_rat(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_kp(A2),P))) = groups5058264527183730370ex_rat(aa_fun5529999419959482889ex_rat(aTP_Lamm_ku(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5960_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(member_complex(I),A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(A2,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,A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I)),groups5808333547571424918x_real(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_5961_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(member_real(I),A2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(A2,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,A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I)),groups8097168146408367636l_real(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_5962_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(member_int(I),A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(A2,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,A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I)),groups8778361861064173332t_real(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_5963_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(member_complex(I),A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(A2,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,A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I)),groups5058264527183730370ex_rat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_5964_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(member_real(I),A2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(A2,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,A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I)),groups1300246762558778688al_rat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_5965_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(member_int(I),A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(A2,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,A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I)),groups3906332499630173760nt_rat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_5966_member__le__sum,axiom,
    ! [I: nat,A2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(member_nat(I),A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(A2,insert_nat(I,bot_bot_set_nat))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I)),groups2906978787729119204at_rat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_5967_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(member_complex(I),A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(A2,insert_complex(I,bot_bot_set_complex))))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X4))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I)),groups5693394587270226106ex_nat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_5968_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(member_real(I),A2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(A2,insert_real(I,bot_bot_set_real))))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4))) )
       => ( pp(aa_set_real_bool(finite_finite_real,A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I)),groups1935376822645274424al_nat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_5969_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(member_int(I),A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(A2,insert_int(I,bot_bot_set_int))))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4))) )
       => ( pp(aa_set_int_bool(finite_finite_int,A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I)),groups4541462559716669496nt_nat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_5970_sum__subtractf__nat,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,G: fun_Pr2602117396471556196at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(G,X4)),aa_Pro4243173770044397973at_nat(F,X4))) )
     => ( groups977919841031483927at_nat(aa_fun2351638177499875132at_nat(aTP_Lamm_kv(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A2)),groups977919841031483927at_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_5971_sum__subtractf__nat,axiom,
    ! [A2: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => 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_kw(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_5972_sum__subtractf__nat,axiom,
    ! [A2: set_set_nat,G: fun_set_nat_nat,F: fun_set_nat_nat] :
      ( ! [X4: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(G,X4)),aa_set_nat_nat(F,X4))) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_kx(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A2)),groups8294997508430121362at_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_5973_sum__subtractf__nat,axiom,
    ! [A2: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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_ky(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_5974_sum__subtractf__nat,axiom,
    ! [A2: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => 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_kz(G),F)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)) ) ) ).

% sum_subtractf_nat
tff(fact_5975_sum__le__included,axiom,
    ! [S: 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,S))
     => ( 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),S))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_int(I,Xa) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(G,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,S)),groups8778361861064173332t_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5976_sum__le__included,axiom,
    ! [S: 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,S))
     => ( 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),S))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_int(I,Xa) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_complex_real(G,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,S)),groups5808333547571424918x_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5977_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G: fun_int_real,I: fun_int_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( 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),S))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_complex(I,Xa) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),aa_int_real(G,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,S)),groups8778361861064173332t_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5978_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: fun_complex_real,I: fun_complex_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( 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),S))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_complex(I,Xa) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),aa_complex_real(G,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,S)),groups5808333547571424918x_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5979_sum__le__included,axiom,
    ! [S: 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,S))
     => ( 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),S))
               => ? [Xa: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Xa),T))
                    & ( aa_nat_nat(I,Xa) = X4 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,S)),groups2906978787729119204at_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5980_sum__le__included,axiom,
    ! [S: set_nat,T: set_int,G: fun_int_rat,I: fun_int_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X4))) )
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_nat(I,Xa) = X4 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_int_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,S)),groups3906332499630173760nt_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5981_sum__le__included,axiom,
    ! [S: set_nat,T: set_complex,G: fun_complex_rat,I: fun_complex_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X4))) )
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_nat(I,Xa) = X4 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_complex_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,S)),groups5058264527183730370ex_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5982_sum__le__included,axiom,
    ! [S: set_int,T: set_nat,G: fun_nat_rat,I: fun_nat_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( 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: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => ? [Xa: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Xa),T))
                    & ( aa_nat_int(I,Xa) = X4 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),aa_nat_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S)),groups2906978787729119204at_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5983_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: fun_int_rat,I: fun_int_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X4))) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_int(I,Xa) = X4 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),aa_int_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S)),groups3906332499630173760nt_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5984_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: fun_complex_rat,I: fun_complex_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X4))) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_int(I,Xa) = X4 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),aa_complex_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S)),groups5058264527183730370ex_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5985_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
       => ( ( groups8097168146408367636l_real(F,A2) = zero_zero_real )
        <=> ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),A2))
             => ( aa_real_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5986_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))) )
       => ( ( groups8778361861064173332t_real(F,A2) = zero_zero_real )
        <=> ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A2))
             => ( aa_int_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5987_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4))) )
       => ( ( groups5808333547571424918x_real(F,A2) = zero_zero_real )
        <=> ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),A2))
             => ( aa_complex_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5988_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4))) )
       => ( ( groups1300246762558778688al_rat(F,A2) = zero_zero_rat )
        <=> ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),A2))
             => ( aa_real_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5989_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4))) )
       => ( ( groups2906978787729119204at_rat(F,A2) = zero_zero_rat )
        <=> ! [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),A2))
             => ( aa_nat_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5990_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4))) )
       => ( ( groups3906332499630173760nt_rat(F,A2) = zero_zero_rat )
        <=> ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A2))
             => ( aa_int_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5991_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4))) )
       => ( ( groups5058264527183730370ex_rat(F,A2) = zero_zero_rat )
        <=> ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),A2))
             => ( aa_complex_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5992_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4))) )
       => ( ( groups1935376822645274424al_nat(F,A2) = zero_zero_nat )
        <=> ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),A2))
             => ( aa_real_nat(F,X3) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5993_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4))) )
       => ( ( groups4541462559716669496nt_nat(F,A2) = zero_zero_nat )
        <=> ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A2))
             => ( aa_int_nat(F,X3) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5994_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X4))) )
       => ( ( groups5693394587270226106ex_nat(F,A2) = zero_zero_nat )
        <=> ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),A2))
             => ( aa_complex_nat(F,X3) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5995_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(G,X4))) )
       => ( ? [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),A2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X2)),aa_int_real(G,X2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5996_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),aa_complex_real(G,X4))) )
       => ( ? [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),A2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X2)),aa_complex_real(G,X2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5997_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,X4))) )
       => ( ? [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),A2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(G,X2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5998_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),aa_int_rat(G,X4))) )
       => ( ? [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),A2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(G,X2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5999_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X4)),aa_complex_rat(G,X4))) )
       => ( ? [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),A2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X2)),aa_complex_rat(G,X2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6000_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X4)),aa_int_nat(G,X4))) )
       => ( ? [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),A2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,X2)),aa_int_nat(G,X2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6001_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,X4)),aa_complex_nat(G,X4))) )
       => ( ? [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),A2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,X2)),aa_complex_nat(G,X2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6002_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,X4)),aa_nat_int(G,X4))) )
       => ( ? [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),A2))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,X2)),aa_nat_int(G,X2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups3539618377306564664at_int(F,A2)),groups3539618377306564664at_int(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6003_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_int,G: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,X4)),aa_int_int(G,X4))) )
       => ( ? [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),A2))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(F,X2)),aa_int_int(G,X2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6004_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_complex_int(F,X4)),aa_complex_int(G,X4))) )
       => ( ? [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),A2))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_complex_int(F,X2)),aa_complex_int(G,X2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6005_sum_Orelated,axiom,
    ! [R: fun_co1378794544173401837x_bool,S2: 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,Y15: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X15),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y15),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(plus_plus_complex(X15),Y15)),aa_complex_complex(plus_plus_complex(X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S2))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => 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,S2)),groups2073611262835488442omplex(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_6006_sum_Orelated,axiom,
    ! [R: fun_co1378794544173401837x_bool,S2: 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,Y15: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X15),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y15),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(plus_plus_complex(X15),Y15)),aa_complex_complex(plus_plus_complex(X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => 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,S2)),groups3049146728041665814omplex(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_6007_sum_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S2: 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,Y15: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X15),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y15),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => 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,S2)),groups8778361861064173332t_real(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_6008_sum_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S2: 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,Y15: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X15),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y15),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => 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,S2)),groups5808333547571424918x_real(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_6009_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S2: 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,Y15: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X15),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y15),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X15),Y15)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S2))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => 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,S2)),groups2906978787729119204at_rat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_6010_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S2: 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,Y15: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X15),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y15),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X15),Y15)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => 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,S2)),groups3906332499630173760nt_rat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_6011_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S2: 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,Y15: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X15),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y15),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X15),Y15)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => 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,S2)),groups5058264527183730370ex_rat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_6012_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_bool,S2: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,zero_zero_nat),zero_zero_nat))
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X15),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y15),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_int_nat(H,X4)),aa_int_nat(G,X4))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,groups4541462559716669496nt_nat(H,S2)),groups4541462559716669496nt_nat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_6013_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_bool,S2: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,zero_zero_nat),zero_zero_nat))
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X15),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y15),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_complex_nat(H,X4)),aa_complex_nat(G,X4))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,groups5693394587270226106ex_nat(H,S2)),groups5693394587270226106ex_nat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_6014_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_bool,S2: set_nat,H: fun_nat_int,G: fun_nat_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(R,zero_zero_int),zero_zero_int))
     => ( ! [X15: int,Y15: int,X23: int,Y23: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(R,X15),X23))
              & pp(aa_int_bool(aa_int_fun_int_bool(R,Y15),Y23)) )
           => pp(aa_int_bool(aa_int_fun_int_bool(R,aa_int_int(aa_int_fun_int_int(plus_plus_int,X15),Y15)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S2))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => pp(aa_int_bool(aa_int_fun_int_bool(R,aa_nat_int(H,X4)),aa_nat_int(G,X4))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(R,groups3539618377306564664at_int(H,S2)),groups3539618377306564664at_int(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_6015_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A2))
             => 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,A2)),groups5808333547571424918x_real(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6016_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
             => 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,A2)),groups8097168146408367636l_real(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6017_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A2))
             => 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,A2)),groups8778361861064173332t_real(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6018_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A2))
             => 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,A2)),groups5058264527183730370ex_rat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6019_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
             => 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,A2)),groups1300246762558778688al_rat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6020_sum__strict__mono,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A2))
             => 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,A2)),groups2906978787729119204at_rat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6021_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6022_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,X4)),aa_complex_nat(G,X4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6023_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(G,X4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6024_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,X4)),aa_int_nat(G,X4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6025_sum__eq__Suc0__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups4541462559716669496nt_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
            & ( aa_int_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),A2))
               => ( ( X3 != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6026_sum__eq__Suc0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups5693394587270226106ex_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
            & ( aa_complex_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa3),A2))
               => ( ( X3 != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6027_sum__eq__Suc0__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
     => ( ( groups977919841031483927at_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),A2))
            & ( aa_Pro4243173770044397973at_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: product_prod_nat_nat] :
                ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Xa3),A2))
               => ( ( X3 != Xa3 )
                 => ( aa_Pro4243173770044397973at_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6028_sum__eq__Suc0__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
            & ( aa_nat_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa3),A2))
               => ( ( X3 != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6029_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T4: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T4))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_real_real(J,A3)),minus_minus_set_real(T2,T4))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T4)))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T4)))
                   => pp(aa_set_real_bool(member_real(aa_real_real(I,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T4))
                       => ( aa_real_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_real_complex(H,aa_real_real(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups5754745047067104278omplex(G,S2) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6030_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T4: set_int,S2: set_real,I: fun_int_real,J: fun_real_int,T2: set_int,G: fun_real_complex,H: fun_int_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T4))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_int_real(I,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_int_bool(member_int(aa_real_int(J,A3)),minus_minus_set_int(T2,T4))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T4)))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T4)))
                   => pp(aa_set_real_bool(member_real(aa_int_real(I,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T4))
                       => ( aa_int_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_int_complex(H,aa_real_int(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups5754745047067104278omplex(G,S2) = groups3049146728041665814omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6031_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T4: set_real,S2: set_int,I: fun_real_int,J: fun_int_real,T2: set_real,G: fun_int_complex,H: fun_real_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T4))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
             => ( aa_real_int(I,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_int_real(J,A3)),minus_minus_set_real(T2,T4))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T4)))
                 => ( aa_int_real(J,aa_real_int(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T4)))
                   => pp(aa_set_int_bool(member_int(aa_real_int(I,B3)),minus_minus_set_int(S2,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T4))
                       => ( aa_real_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_real_complex(H,aa_int_real(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups3049146728041665814omplex(G,S2) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6032_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T4: set_int,S2: set_int,I: fun_int_int,J: fun_int_int,T2: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T4))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
             => ( aa_int_int(I,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
               => pp(aa_set_int_bool(member_int(aa_int_int(J,A3)),minus_minus_set_int(T2,T4))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T4)))
                 => ( aa_int_int(J,aa_int_int(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T4)))
                   => pp(aa_set_int_bool(member_int(aa_int_int(I,B3)),minus_minus_set_int(S2,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T4))
                       => ( aa_int_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_int_complex(H,aa_int_int(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups3049146728041665814omplex(G,S2) = groups3049146728041665814omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6033_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T4: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T2: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T4))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_real_real(J,A3)),minus_minus_set_real(T2,T4))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T4)))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T4)))
                   => pp(aa_set_real_bool(member_real(aa_real_real(I,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T4))
                       => ( aa_real_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_real_real(H,aa_real_real(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6034_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T4: set_int,S2: set_real,I: fun_int_real,J: fun_real_int,T2: set_int,G: fun_real_real,H: fun_int_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T4))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_int_real(I,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_int_bool(member_int(aa_real_int(J,A3)),minus_minus_set_int(T2,T4))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T4)))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T4)))
                   => pp(aa_set_real_bool(member_real(aa_int_real(I,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T4))
                       => ( aa_int_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_int_real(H,aa_real_int(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups8097168146408367636l_real(G,S2) = groups8778361861064173332t_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6035_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T4: set_complex,S2: set_real,I: fun_complex_real,J: fun_real_complex,T2: set_complex,G: fun_real_real,H: fun_complex_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_complex_real(I,aa_real_complex(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_real_complex(J,A3)),minus_811609699411566653omplex(T2,T4))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T2,T4)))
                 => ( aa_real_complex(J,aa_complex_real(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T2,T4)))
                   => pp(aa_set_real_bool(member_real(aa_complex_real(I,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T4))
                       => ( aa_complex_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_complex_real(H,aa_real_complex(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups8097168146408367636l_real(G,S2) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6036_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T4: set_real,S2: set_int,I: fun_real_int,J: fun_int_real,T2: set_real,G: fun_int_real,H: fun_real_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T4))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
             => ( aa_real_int(I,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_int_real(J,A3)),minus_minus_set_real(T2,T4))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T4)))
                 => ( aa_int_real(J,aa_real_int(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T4)))
                   => pp(aa_set_int_bool(member_int(aa_real_int(I,B3)),minus_minus_set_int(S2,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T4))
                       => ( aa_real_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_real_real(H,aa_int_real(J,A3)) = aa_int_real(G,A3) ) )
                     => ( groups8778361861064173332t_real(G,S2) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6037_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T4: set_int,S2: set_int,I: fun_int_int,J: fun_int_int,T2: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T4))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
             => ( aa_int_int(I,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
               => pp(aa_set_int_bool(member_int(aa_int_int(J,A3)),minus_minus_set_int(T2,T4))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T4)))
                 => ( aa_int_int(J,aa_int_int(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T4)))
                   => pp(aa_set_int_bool(member_int(aa_int_int(I,B3)),minus_minus_set_int(S2,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T4))
                       => ( aa_int_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_int_real(H,aa_int_int(J,A3)) = aa_int_real(G,A3) ) )
                     => ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6038_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T4: set_complex,S2: set_int,I: fun_complex_int,J: fun_int_complex,T2: set_complex,G: fun_int_real,H: fun_complex_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
             => ( aa_complex_int(I,aa_int_complex(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_int_complex(J,A3)),minus_811609699411566653omplex(T2,T4))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T2,T4)))
                 => ( aa_int_complex(J,aa_complex_int(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T2,T4)))
                   => pp(aa_set_int_bool(member_int(aa_complex_int(I,B3)),minus_minus_set_int(S2,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T4))
                       => ( aa_complex_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_complex_real(H,aa_int_complex(J,A3)) = aa_int_real(G,A3) ) )
                     => ( groups8778361861064173332t_real(G,S2) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6039_sum__SucD,axiom,
    ! [F: fun_nat_nat,A2: set_nat,N: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,N) )
     => ? [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4))) ) ) ).

% sum_SucD
tff(fact_6040_sum__eq__1__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups4541462559716669496nt_nat(F,A2) = one_one_nat )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
            & ( aa_int_nat(F,X3) = one_one_nat )
            & ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),A2))
               => ( ( X3 != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6041_sum__eq__1__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups5693394587270226106ex_nat(F,A2) = one_one_nat )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
            & ( aa_complex_nat(F,X3) = one_one_nat )
            & ! [Xa3: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa3),A2))
               => ( ( X3 != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6042_sum__eq__1__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
     => ( ( groups977919841031483927at_nat(F,A2) = one_one_nat )
      <=> ? [X3: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),A2))
            & ( aa_Pro4243173770044397973at_nat(F,X3) = one_one_nat )
            & ! [Xa3: product_prod_nat_nat] :
                ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Xa3),A2))
               => ( ( X3 != Xa3 )
                 => ( aa_Pro4243173770044397973at_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6043_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = one_one_nat )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
            & ( aa_nat_nat(F,X3) = one_one_nat )
            & ! [Xa3: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa3),A2))
               => ( ( X3 != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6044_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: fun_set_complex_bool,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X4: complex,S6: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S6))
             => ( ! [Y3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y3),S6))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,Y3)),aa_complex_rat(F,X4))) )
               => ( pp(aa_set_complex_bool(P,S6))
                 => pp(aa_set_complex_bool(P,insert_complex(X4,S6))) ) ) )
         => pp(aa_set_complex_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6045_finite__ranking__induct,axiom,
    ! [S2: set_real,P: fun_set_real_bool,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X4: real,S6: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,S6))
             => ( ! [Y3: real] :
                    ( pp(aa_set_real_bool(member_real(Y3),S6))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,Y3)),aa_real_rat(F,X4))) )
               => ( pp(aa_set_real_bool(P,S6))
                 => pp(aa_set_real_bool(P,insert_real(X4,S6))) ) ) )
         => pp(aa_set_real_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6046_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: fun_set_nat_bool,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [X4: nat,S6: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,S6))
             => ( ! [Y3: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y3),S6))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,Y3)),aa_nat_rat(F,X4))) )
               => ( pp(aa_set_nat_bool(P,S6))
                 => pp(aa_set_nat_bool(P,insert_nat(X4,S6))) ) ) )
         => pp(aa_set_nat_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6047_finite__ranking__induct,axiom,
    ! [S2: set_int,P: fun_set_int_bool,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [X4: int,S6: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,S6))
             => ( ! [Y3: int] :
                    ( pp(aa_set_int_bool(member_int(Y3),S6))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,Y3)),aa_int_rat(F,X4))) )
               => ( pp(aa_set_int_bool(P,S6))
                 => pp(aa_set_int_bool(P,insert_int(X4,S6))) ) ) )
         => pp(aa_set_int_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6048_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: fun_set_complex_bool,F: fun_complex_num] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X4: complex,S6: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S6))
             => ( ! [Y3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y3),S6))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_complex_num(F,Y3)),aa_complex_num(F,X4))) )
               => ( pp(aa_set_complex_bool(P,S6))
                 => pp(aa_set_complex_bool(P,insert_complex(X4,S6))) ) ) )
         => pp(aa_set_complex_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6049_finite__ranking__induct,axiom,
    ! [S2: set_real,P: fun_set_real_bool,F: fun_real_num] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X4: real,S6: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,S6))
             => ( ! [Y3: real] :
                    ( pp(aa_set_real_bool(member_real(Y3),S6))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_real_num(F,Y3)),aa_real_num(F,X4))) )
               => ( pp(aa_set_real_bool(P,S6))
                 => pp(aa_set_real_bool(P,insert_real(X4,S6))) ) ) )
         => pp(aa_set_real_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6050_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: fun_set_nat_bool,F: fun_nat_num] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [X4: nat,S6: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,S6))
             => ( ! [Y3: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y3),S6))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,Y3)),aa_nat_num(F,X4))) )
               => ( pp(aa_set_nat_bool(P,S6))
                 => pp(aa_set_nat_bool(P,insert_nat(X4,S6))) ) ) )
         => pp(aa_set_nat_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6051_finite__ranking__induct,axiom,
    ! [S2: set_int,P: fun_set_int_bool,F: fun_int_num] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [X4: int,S6: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,S6))
             => ( ! [Y3: int] :
                    ( pp(aa_set_int_bool(member_int(Y3),S6))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F,Y3)),aa_int_num(F,X4))) )
               => ( pp(aa_set_int_bool(P,S6))
                 => pp(aa_set_int_bool(P,insert_int(X4,S6))) ) ) )
         => pp(aa_set_int_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6052_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: fun_set_complex_bool,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X4: complex,S6: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S6))
             => ( ! [Y3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y3),S6))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,Y3)),aa_complex_nat(F,X4))) )
               => ( pp(aa_set_complex_bool(P,S6))
                 => pp(aa_set_complex_bool(P,insert_complex(X4,S6))) ) ) )
         => pp(aa_set_complex_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6053_finite__ranking__induct,axiom,
    ! [S2: set_real,P: fun_set_real_bool,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X4: real,S6: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,S6))
             => ( ! [Y3: real] :
                    ( pp(aa_set_real_bool(member_real(Y3),S6))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,Y3)),aa_real_nat(F,X4))) )
               => ( pp(aa_set_real_bool(P,S6))
                 => pp(aa_set_real_bool(P,insert_real(X4,S6))) ) ) )
         => pp(aa_set_real_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6054_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [B3: real,A4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A4))
             => ( ! [X2: real] :
                    ( pp(aa_set_real_bool(member_real(X2),A4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),X2)) )
               => ( pp(aa_set_real_bool(P,A4))
                 => pp(aa_set_real_bool(P,insert_real(B3,A4))) ) ) )
         => pp(aa_set_real_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6055_finite__linorder__min__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_bool] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( pp(aa_set_rat_bool(P,bot_bot_set_rat))
       => ( ! [B3: rat,A4: set_rat] :
              ( pp(aa_set_rat_bool(finite_finite_rat,A4))
             => ( ! [X2: rat] :
                    ( pp(aa_set_rat_bool(member_rat(X2),A4))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),X2)) )
               => ( pp(aa_set_rat_bool(P,A4))
                 => pp(aa_set_rat_bool(P,insert_rat(B3,A4))) ) ) )
         => pp(aa_set_rat_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6056_finite__linorder__min__induct,axiom,
    ! [A2: set_num,P: fun_set_num_bool] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( pp(aa_set_num_bool(P,bot_bot_set_num))
       => ( ! [B3: num,A4: set_num] :
              ( pp(aa_set_num_bool(finite_finite_num,A4))
             => ( ! [X2: num] :
                    ( pp(aa_set_num_bool(member_num(X2),A4))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),X2)) )
               => ( pp(aa_set_num_bool(P,A4))
                 => pp(aa_set_num_bool(P,insert_num(B3,A4))) ) ) )
         => pp(aa_set_num_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6057_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [B3: nat,A4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A4))
             => ( ! [X2: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X2),A4))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),X2)) )
               => ( pp(aa_set_nat_bool(P,A4))
                 => pp(aa_set_nat_bool(P,insert_nat(B3,A4))) ) ) )
         => pp(aa_set_nat_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6058_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [B3: int,A4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A4))
             => ( ! [X2: int] :
                    ( pp(aa_set_int_bool(member_int(X2),A4))
                   => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),X2)) )
               => ( pp(aa_set_int_bool(P,A4))
                 => pp(aa_set_int_bool(P,insert_int(B3,A4))) ) ) )
         => pp(aa_set_int_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6059_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [B3: real,A4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A4))
             => ( ! [X2: real] :
                    ( pp(aa_set_real_bool(member_real(X2),A4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B3)) )
               => ( pp(aa_set_real_bool(P,A4))
                 => pp(aa_set_real_bool(P,insert_real(B3,A4))) ) ) )
         => pp(aa_set_real_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6060_finite__linorder__max__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_bool] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( pp(aa_set_rat_bool(P,bot_bot_set_rat))
       => ( ! [B3: rat,A4: set_rat] :
              ( pp(aa_set_rat_bool(finite_finite_rat,A4))
             => ( ! [X2: rat] :
                    ( pp(aa_set_rat_bool(member_rat(X2),A4))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),B3)) )
               => ( pp(aa_set_rat_bool(P,A4))
                 => pp(aa_set_rat_bool(P,insert_rat(B3,A4))) ) ) )
         => pp(aa_set_rat_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6061_finite__linorder__max__induct,axiom,
    ! [A2: set_num,P: fun_set_num_bool] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( pp(aa_set_num_bool(P,bot_bot_set_num))
       => ( ! [B3: num,A4: set_num] :
              ( pp(aa_set_num_bool(finite_finite_num,A4))
             => ( ! [X2: num] :
                    ( pp(aa_set_num_bool(member_num(X2),A4))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),B3)) )
               => ( pp(aa_set_num_bool(P,A4))
                 => pp(aa_set_num_bool(P,insert_num(B3,A4))) ) ) )
         => pp(aa_set_num_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6062_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [B3: nat,A4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A4))
             => ( ! [X2: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X2),A4))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),B3)) )
               => ( pp(aa_set_nat_bool(P,A4))
                 => pp(aa_set_nat_bool(P,insert_nat(B3,A4))) ) ) )
         => pp(aa_set_nat_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6063_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [B3: int,A4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A4))
             => ( ! [X2: int] :
                    ( pp(aa_set_int_bool(member_int(X2),A4))
                   => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),B3)) )
               => ( pp(aa_set_int_bool(P,A4))
                 => pp(aa_set_int_bool(P,insert_int(B3,A4))) ) ) )
         => pp(aa_set_int_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6064_finite__subset__induct,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: fun_se8353504738346895699t_bool] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,F3))
     => ( pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(F3),A2))
       => ( pp(aa_set4148026973699692364t_bool(P,bot_bo228742789529271731at_nat))
         => ( ! [A3: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
                ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,F4))
               => ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A3),A2))
                 => ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A3),F4))
                   => ( pp(aa_set4148026973699692364t_bool(P,F4))
                     => pp(aa_set4148026973699692364t_bool(P,insert9069300056098147895at_nat(A3,F4))) ) ) ) )
           => pp(aa_set4148026973699692364t_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6065_finite__subset__induct,axiom,
    ! [F3: set_set_nat,A2: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F3))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(F3),A2))
       => ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
         => ( ! [A3: set_nat,F4: set_set_nat] :
                ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
               => ( pp(aa_set_set_nat_bool(member_set_nat(A3),A2))
                 => ( ~ pp(aa_set_set_nat_bool(member_set_nat(A3),F4))
                   => ( pp(aa_set_set_nat_bool(P,F4))
                     => pp(aa_set_set_nat_bool(P,insert_set_nat(A3,F4))) ) ) ) )
           => pp(aa_set_set_nat_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6066_finite__subset__induct,axiom,
    ! [F3: set_complex,A2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F3),A2))
       => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
         => ( ! [A3: complex,F4: set_complex] :
                ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
               => ( pp(aa_set_complex_bool(member_complex(A3),A2))
                 => ( ~ pp(aa_set_complex_bool(member_complex(A3),F4))
                   => ( pp(aa_set_complex_bool(P,F4))
                     => pp(aa_set_complex_bool(P,insert_complex(A3,F4))) ) ) ) )
           => pp(aa_set_complex_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6067_finite__subset__induct,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F3))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(F3),A2))
       => ( pp(aa_set3135850010578026034t_bool(P,bot_bo2099793752762293965at_nat))
         => ( ! [A3: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F4))
               => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A3),A2))
                 => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A3),F4))
                   => ( pp(aa_set3135850010578026034t_bool(P,F4))
                     => pp(aa_set3135850010578026034t_bool(P,insert8211810215607154385at_nat(A3,F4))) ) ) ) )
           => pp(aa_set3135850010578026034t_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6068_finite__subset__induct,axiom,
    ! [F3: set_real,A2: 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),A2))
       => ( pp(aa_set_real_bool(P,bot_bot_set_real))
         => ( ! [A3: real,F4: set_real] :
                ( pp(aa_set_real_bool(finite_finite_real,F4))
               => ( pp(aa_set_real_bool(member_real(A3),A2))
                 => ( ~ pp(aa_set_real_bool(member_real(A3),F4))
                   => ( pp(aa_set_real_bool(P,F4))
                     => pp(aa_set_real_bool(P,insert_real(A3,F4))) ) ) ) )
           => pp(aa_set_real_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6069_finite__subset__induct,axiom,
    ! [F3: set_nat,A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,F3),A2))
       => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
         => ( ! [A3: nat,F4: set_nat] :
                ( pp(aa_set_nat_bool(finite_finite_nat,F4))
               => ( pp(aa_set_nat_bool(member_nat(A3),A2))
                 => ( ~ pp(aa_set_nat_bool(member_nat(A3),F4))
                   => ( pp(aa_set_nat_bool(P,F4))
                     => pp(aa_set_nat_bool(P,insert_nat(A3,F4))) ) ) ) )
           => pp(aa_set_nat_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6070_finite__subset__induct,axiom,
    ! [F3: set_int,A2: 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),A2))
       => ( pp(aa_set_int_bool(P,bot_bot_set_int))
         => ( ! [A3: int,F4: set_int] :
                ( pp(aa_set_int_bool(finite_finite_int,F4))
               => ( pp(aa_set_int_bool(member_int(A3),A2))
                 => ( ~ pp(aa_set_int_bool(member_int(A3),F4))
                   => ( pp(aa_set_int_bool(P,F4))
                     => pp(aa_set_int_bool(P,insert_int(A3,F4))) ) ) ) )
           => pp(aa_set_int_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6071_finite__subset__induct_H,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: fun_se8353504738346895699t_bool] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,F3))
     => ( pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(F3),A2))
       => ( pp(aa_set4148026973699692364t_bool(P,bot_bo228742789529271731at_nat))
         => ( ! [A3: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
                ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,F4))
               => ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A3),A2))
                 => ( pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(F4),A2))
                   => ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A3),F4))
                     => ( pp(aa_set4148026973699692364t_bool(P,F4))
                       => pp(aa_set4148026973699692364t_bool(P,insert9069300056098147895at_nat(A3,F4))) ) ) ) ) )
           => pp(aa_set4148026973699692364t_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6072_finite__subset__induct_H,axiom,
    ! [F3: set_set_nat,A2: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F3))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(F3),A2))
       => ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
         => ( ! [A3: set_nat,F4: set_set_nat] :
                ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
               => ( pp(aa_set_set_nat_bool(member_set_nat(A3),A2))
                 => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(F4),A2))
                   => ( ~ pp(aa_set_set_nat_bool(member_set_nat(A3),F4))
                     => ( pp(aa_set_set_nat_bool(P,F4))
                       => pp(aa_set_set_nat_bool(P,insert_set_nat(A3,F4))) ) ) ) ) )
           => pp(aa_set_set_nat_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6073_finite__subset__induct_H,axiom,
    ! [F3: set_complex,A2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F3),A2))
       => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
         => ( ! [A3: complex,F4: set_complex] :
                ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
               => ( pp(aa_set_complex_bool(member_complex(A3),A2))
                 => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F4),A2))
                   => ( ~ pp(aa_set_complex_bool(member_complex(A3),F4))
                     => ( pp(aa_set_complex_bool(P,F4))
                       => pp(aa_set_complex_bool(P,insert_complex(A3,F4))) ) ) ) ) )
           => pp(aa_set_complex_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6074_finite__subset__induct_H,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F3))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(F3),A2))
       => ( pp(aa_set3135850010578026034t_bool(P,bot_bo2099793752762293965at_nat))
         => ( ! [A3: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F4))
               => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A3),A2))
                 => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(F4),A2))
                   => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A3),F4))
                     => ( pp(aa_set3135850010578026034t_bool(P,F4))
                       => pp(aa_set3135850010578026034t_bool(P,insert8211810215607154385at_nat(A3,F4))) ) ) ) ) )
           => pp(aa_set3135850010578026034t_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6075_finite__subset__induct_H,axiom,
    ! [F3: set_real,A2: 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),A2))
       => ( pp(aa_set_real_bool(P,bot_bot_set_real))
         => ( ! [A3: real,F4: set_real] :
                ( pp(aa_set_real_bool(finite_finite_real,F4))
               => ( pp(aa_set_real_bool(member_real(A3),A2))
                 => ( pp(aa_set_real_bool(ord_less_eq_set_real(F4),A2))
                   => ( ~ pp(aa_set_real_bool(member_real(A3),F4))
                     => ( pp(aa_set_real_bool(P,F4))
                       => pp(aa_set_real_bool(P,insert_real(A3,F4))) ) ) ) ) )
           => pp(aa_set_real_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6076_finite__subset__induct_H,axiom,
    ! [F3: set_nat,A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,F3),A2))
       => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
         => ( ! [A3: nat,F4: set_nat] :
                ( pp(aa_set_nat_bool(finite_finite_nat,F4))
               => ( pp(aa_set_nat_bool(member_nat(A3),A2))
                 => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,F4),A2))
                   => ( ~ pp(aa_set_nat_bool(member_nat(A3),F4))
                     => ( pp(aa_set_nat_bool(P,F4))
                       => pp(aa_set_nat_bool(P,insert_nat(A3,F4))) ) ) ) ) )
           => pp(aa_set_nat_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6077_finite__subset__induct_H,axiom,
    ! [F3: set_int,A2: 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),A2))
       => ( pp(aa_set_int_bool(P,bot_bot_set_int))
         => ( ! [A3: int,F4: set_int] :
                ( pp(aa_set_int_bool(finite_finite_int,F4))
               => ( pp(aa_set_int_bool(member_int(A3),A2))
                 => ( pp(aa_set_int_bool(ord_less_eq_set_int(F4),A2))
                   => ( ~ pp(aa_set_int_bool(member_int(A3),F4))
                     => ( pp(aa_set_int_bool(P,F4))
                       => pp(aa_set_int_bool(P,insert_int(A3,F4))) ) ) ) ) )
           => pp(aa_set_int_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6078_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_real,B2: real,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2))) )
       => ( ( groups8097168146408367636l_real(F,S) = B2 )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6079_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_real,B2: real,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2))) )
       => ( ( groups8778361861064173332t_real(F,S) = B2 )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6080_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_real,B2: real,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2))) )
       => ( ( groups5808333547571424918x_real(F,S) = B2 )
         => ( pp(aa_set_complex_bool(member_complex(I),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6081_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_rat,B2: rat,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2))) )
       => ( ( groups1300246762558778688al_rat(F,S) = B2 )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6082_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: fun_nat_rat,B2: rat,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2))) )
       => ( ( groups2906978787729119204at_rat(F,S) = B2 )
         => ( pp(aa_set_nat_bool(member_nat(I),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6083_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_rat,B2: rat,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2))) )
       => ( ( groups3906332499630173760nt_rat(F,S) = B2 )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6084_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_rat,B2: rat,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2))) )
       => ( ( groups5058264527183730370ex_rat(F,S) = B2 )
         => ( pp(aa_set_complex_bool(member_complex(I),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6085_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_nat,B2: nat,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2))) )
       => ( ( groups1935376822645274424al_nat(F,S) = B2 )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6086_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_nat,B2: nat,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2))) )
       => ( ( groups4541462559716669496nt_nat(F,S) = B2 )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6087_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_nat,B2: nat,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2))) )
       => ( ( groups5693394587270226106ex_nat(F,S) = B2 )
         => ( pp(aa_set_complex_bool(member_complex(I),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6088_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_real,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2))) )
       => ( ( groups8097168146408367636l_real(F,S) = zero_zero_real )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => ( aa_real_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6089_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_real,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2))) )
       => ( ( groups8778361861064173332t_real(F,S) = zero_zero_real )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => ( aa_int_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6090_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_real,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2))) )
       => ( ( groups5808333547571424918x_real(F,S) = zero_zero_real )
         => ( pp(aa_set_complex_bool(member_complex(I),S))
           => ( aa_complex_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6091_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_rat,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2))) )
       => ( ( groups1300246762558778688al_rat(F,S) = zero_zero_rat )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => ( aa_real_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6092_sum__nonneg__0,axiom,
    ! [S: set_nat,F: fun_nat_rat,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2))) )
       => ( ( groups2906978787729119204at_rat(F,S) = zero_zero_rat )
         => ( pp(aa_set_nat_bool(member_nat(I),S))
           => ( aa_nat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6093_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_rat,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2))) )
       => ( ( groups3906332499630173760nt_rat(F,S) = zero_zero_rat )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => ( aa_int_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6094_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_rat,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2))) )
       => ( ( groups5058264527183730370ex_rat(F,S) = zero_zero_rat )
         => ( pp(aa_set_complex_bool(member_complex(I),S))
           => ( aa_complex_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6095_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_nat,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2))) )
       => ( ( groups1935376822645274424al_nat(F,S) = zero_zero_nat )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => ( aa_real_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6096_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_nat,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2))) )
       => ( ( groups4541462559716669496nt_nat(F,S) = zero_zero_nat )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => ( aa_int_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6097_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_nat,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2))) )
       => ( ( groups5693394587270226106ex_nat(F,S) = zero_zero_nat )
         => ( pp(aa_set_complex_bool(member_complex(I),S))
           => ( aa_complex_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6098_finite__empty__induct,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,P: fun_se8353504738346895699t_bool] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A2))
     => ( pp(aa_set4148026973699692364t_bool(P,A2))
       => ( ! [A3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
              ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A4))
             => ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A3),A4))
               => ( pp(aa_set4148026973699692364t_bool(P,A4))
                 => pp(aa_set4148026973699692364t_bool(P,minus_3314409938677909166at_nat(A4,insert9069300056098147895at_nat(A3,bot_bo228742789529271731at_nat)))) ) ) )
         => pp(aa_set4148026973699692364t_bool(P,bot_bo228742789529271731at_nat)) ) ) ) ).

% finite_empty_induct
tff(fact_6099_finite__empty__induct,axiom,
    ! [A2: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( pp(aa_set_set_nat_bool(P,A2))
       => ( ! [A3: set_nat,A4: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A4))
             => ( pp(aa_set_set_nat_bool(member_set_nat(A3),A4))
               => ( pp(aa_set_set_nat_bool(P,A4))
                 => pp(aa_set_set_nat_bool(P,minus_2163939370556025621et_nat(A4,insert_set_nat(A3,bot_bot_set_set_nat)))) ) ) )
         => pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat)) ) ) ) ).

% finite_empty_induct
tff(fact_6100_finite__empty__induct,axiom,
    ! [A2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(P,A2))
       => ( ! [A3: complex,A4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
             => ( pp(aa_set_complex_bool(member_complex(A3),A4))
               => ( pp(aa_set_complex_bool(P,A4))
                 => pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A4,insert_complex(A3,bot_bot_set_complex)))) ) ) )
         => pp(aa_set_complex_bool(P,bot_bot_set_complex)) ) ) ) ).

% finite_empty_induct
tff(fact_6101_finite__empty__induct,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
     => ( pp(aa_set3135850010578026034t_bool(P,A2))
       => ( ! [A3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
              ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
             => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A3),A4))
               => ( pp(aa_set3135850010578026034t_bool(P,A4))
                 => pp(aa_set3135850010578026034t_bool(P,minus_1356011639430497352at_nat(A4,insert8211810215607154385at_nat(A3,bot_bo2099793752762293965at_nat)))) ) ) )
         => pp(aa_set3135850010578026034t_bool(P,bot_bo2099793752762293965at_nat)) ) ) ) ).

% finite_empty_induct
tff(fact_6102_finite__empty__induct,axiom,
    ! [A2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(P,A2))
       => ( ! [A3: real,A4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A4))
             => ( pp(aa_set_real_bool(member_real(A3),A4))
               => ( pp(aa_set_real_bool(P,A4))
                 => pp(aa_set_real_bool(P,minus_minus_set_real(A4,insert_real(A3,bot_bot_set_real)))) ) ) )
         => pp(aa_set_real_bool(P,bot_bot_set_real)) ) ) ) ).

% finite_empty_induct
tff(fact_6103_finite__empty__induct,axiom,
    ! [A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(P,A2))
       => ( ! [A3: int,A4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A4))
             => ( pp(aa_set_int_bool(member_int(A3),A4))
               => ( pp(aa_set_int_bool(P,A4))
                 => pp(aa_set_int_bool(P,minus_minus_set_int(A4,insert_int(A3,bot_bot_set_int)))) ) ) )
         => pp(aa_set_int_bool(P,bot_bot_set_int)) ) ) ) ).

% finite_empty_induct
tff(fact_6104_finite__empty__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(P,A2))
       => ( ! [A3: nat,A4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A4))
             => ( pp(aa_set_nat_bool(member_nat(A3),A4))
               => ( pp(aa_set_nat_bool(P,A4))
                 => pp(aa_set_nat_bool(P,minus_minus_set_nat(A4,insert_nat(A3,bot_bot_set_nat)))) ) ) )
         => pp(aa_set_nat_bool(P,bot_bot_set_nat)) ) ) ) ).

% finite_empty_induct
tff(fact_6105_infinite__coinduct,axiom,
    ! [X6: fun_se8353504738346895699t_bool,A2: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(X6,A2))
     => ( ! [A4: set_Pr4329608150637261639at_nat] :
            ( pp(aa_set4148026973699692364t_bool(X6,A4))
           => ? [X2: produc3843707927480180839at_nat] :
                ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X2),A4))
                & ( pp(aa_set4148026973699692364t_bool(X6,minus_3314409938677909166at_nat(A4,insert9069300056098147895at_nat(X2,bot_bo228742789529271731at_nat))))
                  | ~ pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,minus_3314409938677909166at_nat(A4,insert9069300056098147895at_nat(X2,bot_bo228742789529271731at_nat)))) ) ) )
       => ~ pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A2)) ) ) ).

% infinite_coinduct
tff(fact_6106_infinite__coinduct,axiom,
    ! [X6: fun_set_complex_bool,A2: set_complex] :
      ( pp(aa_set_complex_bool(X6,A2))
     => ( ! [A4: set_complex] :
            ( pp(aa_set_complex_bool(X6,A4))
           => ? [X2: complex] :
                ( pp(aa_set_complex_bool(member_complex(X2),A4))
                & ( pp(aa_set_complex_bool(X6,minus_811609699411566653omplex(A4,insert_complex(X2,bot_bot_set_complex))))
                  | ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A4,insert_complex(X2,bot_bot_set_complex)))) ) ) )
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ) ).

% infinite_coinduct
tff(fact_6107_infinite__coinduct,axiom,
    ! [X6: fun_se8074476271188067385t_bool,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(X6,A2))
     => ( ! [A4: set_Pr1261947904930325089at_nat] :
            ( pp(aa_set3135850010578026034t_bool(X6,A4))
           => ? [X2: product_prod_nat_nat] :
                ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),A4))
                & ( pp(aa_set3135850010578026034t_bool(X6,minus_1356011639430497352at_nat(A4,insert8211810215607154385at_nat(X2,bot_bo2099793752762293965at_nat))))
                  | ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A4,insert8211810215607154385at_nat(X2,bot_bo2099793752762293965at_nat)))) ) ) )
       => ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2)) ) ) ).

% infinite_coinduct
tff(fact_6108_infinite__coinduct,axiom,
    ! [X6: fun_set_real_bool,A2: set_real] :
      ( pp(aa_set_real_bool(X6,A2))
     => ( ! [A4: set_real] :
            ( pp(aa_set_real_bool(X6,A4))
           => ? [X2: real] :
                ( pp(aa_set_real_bool(member_real(X2),A4))
                & ( pp(aa_set_real_bool(X6,minus_minus_set_real(A4,insert_real(X2,bot_bot_set_real))))
                  | ~ pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(A4,insert_real(X2,bot_bot_set_real)))) ) ) )
       => ~ pp(aa_set_real_bool(finite_finite_real,A2)) ) ) ).

% infinite_coinduct
tff(fact_6109_infinite__coinduct,axiom,
    ! [X6: fun_set_int_bool,A2: set_int] :
      ( pp(aa_set_int_bool(X6,A2))
     => ( ! [A4: set_int] :
            ( pp(aa_set_int_bool(X6,A4))
           => ? [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),A4))
                & ( pp(aa_set_int_bool(X6,minus_minus_set_int(A4,insert_int(X2,bot_bot_set_int))))
                  | ~ pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A4,insert_int(X2,bot_bot_set_int)))) ) ) )
       => ~ pp(aa_set_int_bool(finite_finite_int,A2)) ) ) ).

% infinite_coinduct
tff(fact_6110_infinite__coinduct,axiom,
    ! [X6: fun_set_nat_bool,A2: set_nat] :
      ( pp(aa_set_nat_bool(X6,A2))
     => ( ! [A4: set_nat] :
            ( pp(aa_set_nat_bool(X6,A4))
           => ? [X2: nat] :
                ( pp(aa_set_nat_bool(member_nat(X2),A4))
                & ( pp(aa_set_nat_bool(X6,minus_minus_set_nat(A4,insert_nat(X2,bot_bot_set_nat))))
                  | ~ pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A4,insert_nat(X2,bot_bot_set_nat)))) ) ) )
       => ~ pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ) ).

% infinite_coinduct
tff(fact_6111_infinite__remove,axiom,
    ! [S2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ~ pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,S2))
     => ~ pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,minus_3314409938677909166at_nat(S2,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)))) ) ).

% infinite_remove
tff(fact_6112_infinite__remove,axiom,
    ! [S2: set_complex,A: complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) ).

% infinite_remove
tff(fact_6113_infinite__remove,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,S2))
     => ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(S2,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)))) ) ).

% infinite_remove
tff(fact_6114_infinite__remove,axiom,
    ! [S2: set_real,A: real] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,S2))
     => ~ pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) ).

% infinite_remove
tff(fact_6115_infinite__remove,axiom,
    ! [S2: set_int,A: int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
     => ~ pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int)))) ) ).

% infinite_remove
tff(fact_6116_infinite__remove,axiom,
    ! [S2: set_nat,A: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ~ pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(S2,insert_nat(A,bot_bot_set_nat)))) ) ).

% infinite_remove
tff(fact_6117_Diff__single__insert,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))),B2))
     => pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(A2),insert9069300056098147895at_nat(X,B2))) ) ).

% Diff_single_insert
tff(fact_6118_Diff__single__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B2))
     => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),insert8211810215607154385at_nat(X,B2))) ) ).

% Diff_single_insert
tff(fact_6119_Diff__single__insert,axiom,
    ! [A2: set_real,X: real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(minus_minus_set_real(A2,insert_real(X,bot_bot_set_real))),B2))
     => pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(X,B2))) ) ).

% Diff_single_insert
tff(fact_6120_Diff__single__insert,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,minus_minus_set_nat(A2,insert_nat(X,bot_bot_set_nat))),B2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(X,B2))) ) ).

% Diff_single_insert
tff(fact_6121_Diff__single__insert,axiom,
    ! [A2: set_int,X: int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(minus_minus_set_int(A2,insert_int(X,bot_bot_set_int))),B2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(X,B2))) ) ).

% Diff_single_insert
tff(fact_6122_subset__insert__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(A2),insert9069300056098147895at_nat(X,B2)))
    <=> ( ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),A2))
         => pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))),B2)) )
        & ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),A2))
         => pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(A2),B2)) ) ) ) ).

% subset_insert_iff
tff(fact_6123_subset__insert__iff,axiom,
    ! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),insert_set_nat(X,B2)))
    <=> ( ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
         => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(X,bot_bot_set_set_nat))),B2)) )
        & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A2))
         => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B2)) ) ) ) ).

% subset_insert_iff
tff(fact_6124_subset__insert__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),insert8211810215607154385at_nat(X,B2)))
    <=> ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2))
         => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B2)) )
        & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2))
         => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B2)) ) ) ) ).

% subset_insert_iff
tff(fact_6125_subset__insert__iff,axiom,
    ! [A2: set_real,X: real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(X,B2)))
    <=> ( ( pp(aa_set_real_bool(member_real(X),A2))
         => pp(aa_set_real_bool(ord_less_eq_set_real(minus_minus_set_real(A2,insert_real(X,bot_bot_set_real))),B2)) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A2))
         => pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2)) ) ) ) ).

% subset_insert_iff
tff(fact_6126_subset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(X,B2)))
    <=> ( ( pp(aa_set_nat_bool(member_nat(X),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,minus_minus_set_nat(A2,insert_nat(X,bot_bot_set_nat))),B2)) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2)) ) ) ) ).

% subset_insert_iff
tff(fact_6127_subset__insert__iff,axiom,
    ! [A2: set_int,X: int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(X,B2)))
    <=> ( ( pp(aa_set_int_bool(member_int(X),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(minus_minus_set_int(A2,insert_int(X,bot_bot_set_int))),B2)) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ) ) ).

% subset_insert_iff
tff(fact_6128_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_6129_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_6130_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_6131_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_6132_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups5754745047067104278omplex(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_la(G)))) = groups5754745047067104278omplex(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6133_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3049146728041665814omplex(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_lb(G)))) = groups3049146728041665814omplex(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6134_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups8097168146408367636l_real(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_lc(G)))) = groups8097168146408367636l_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6135_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups8778361861064173332t_real(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_ld(G)))) = groups8778361861064173332t_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6136_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_le(G)))) = groups5808333547571424918x_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6137_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1300246762558778688al_rat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_lf(G)))) = groups1300246762558778688al_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6138_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_lg(G)))) = groups3906332499630173760nt_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6139_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_lh(G)))) = groups5058264527183730370ex_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6140_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1935376822645274424al_nat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_li(G)))) = groups1935376822645274424al_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6141_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_lj(G)))) = groups4541462559716669496nt_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6142_set__update__subset__insert,axiom,
    ! [Xs: list_P6011104703257516679at_nat,I: nat,X: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,I,X))),insert8211810215607154385at_nat(X,set_Pr5648618587558075414at_nat(Xs)))) ).

% set_update_subset_insert
tff(fact_6143_set__update__subset__insert,axiom,
    ! [Xs: list_real,I: nat,X: real] : pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,X))),insert_real(X,set_real2(Xs)))) ).

% set_update_subset_insert
tff(fact_6144_set__update__subset__insert,axiom,
    ! [Xs: list_P5464809261938338413at_nat,I: nat,X: produc3843707927480180839at_nat] : pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(set_Pr3765526544606949372at_nat(list_u4696772448584712917at_nat(Xs,I,X))),insert9069300056098147895at_nat(X,set_Pr3765526544606949372at_nat(Xs)))) ).

% set_update_subset_insert
tff(fact_6145_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,X: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(list_update_nat(Xs,I,X))),insert_nat(X,set_nat2(Xs)))) ).

% set_update_subset_insert
tff(fact_6146_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,X)),insert_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))) ).

% set_update_subset_insert
tff(fact_6147_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,X: int] : pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X))),insert_int(X,set_int2(Xs)))) ).

% set_update_subset_insert
tff(fact_6148_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_6149_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_6150_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_6151_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_6152_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_6153_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_6154_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( pp(aa_set_complex_bool(member_complex(I),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I)))
         => ( ! [I2: complex] :
                ( pp(aa_set_complex_bool(member_complex(I2),I6))
               => 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,zero_zero_rat),groups5058264527183730370ex_rat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_6155_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( pp(aa_set_real_bool(member_real(I),I6))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F,I)))
         => ( ! [I2: real] :
                ( pp(aa_set_real_bool(member_real(I2),I6))
               => 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_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_6156_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( pp(aa_set_int_bool(member_int(I),I6))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(F,I)))
         => ( ! [I2: int] :
                ( pp(aa_set_int_bool(member_int(I2),I6))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_6157_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( pp(aa_set_complex_bool(member_complex(I),I6))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I)))
         => ( ! [I2: complex] :
                ( pp(aa_set_complex_bool(member_complex(I2),I6))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_6158_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_6159_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_6160_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_6161_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_6162_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_6163_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_6164_sum__pos,axiom,
    ! [I6: set_int,F: fun_int_rat] :
      ( 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_rat_bool(aa_rat_fun_rat_bool(ord_less_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_pos
tff(fact_6165_sum__pos,axiom,
    ! [I6: set_complex,F: fun_complex_nat] :
      ( 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_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_6166_sum__pos,axiom,
    ! [I6: set_real,F: fun_real_nat] :
      ( 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_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F,I2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_6167_sum__pos,axiom,
    ! [I6: set_int,F: fun_int_nat] :
      ( 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_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(F,I2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_6168_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_complex(G,X4) = zero_zero_complex ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
           => ( groups5754745047067104278omplex(G,T2) = groups5754745047067104278omplex(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6169_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups8097168146408367636l_real(G,T2) = groups8097168146408367636l_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6170_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups5808333547571424918x_real(G,T2) = groups5808333547571424918x_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6171_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_rat(G,X4) = zero_zero_rat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_rat(G,X4) = aa_real_rat(H,X4) ) )
           => ( groups1300246762558778688al_rat(G,T2) = groups1300246762558778688al_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6172_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_rat(G,X4) = zero_zero_rat ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_rat(G,X4) = aa_complex_rat(H,X4) ) )
           => ( groups5058264527183730370ex_rat(G,T2) = groups5058264527183730370ex_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6173_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_nat(G,X4) = zero_zero_nat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
           => ( groups1935376822645274424al_nat(G,T2) = groups1935376822645274424al_nat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6174_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_nat(G,X4) = aa_complex_nat(H,X4) ) )
           => ( groups5693394587270226106ex_nat(G,T2) = groups5693394587270226106ex_nat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6175_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_int,H: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_int(G,X4) = zero_zero_int ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_int(G,X4) = aa_real_int(H,X4) ) )
           => ( groups1932886352136224148al_int(G,T2) = groups1932886352136224148al_int(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6176_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_int(G,X4) = aa_complex_int(H,X4) ) )
           => ( groups5690904116761175830ex_int(G,T2) = groups5690904116761175830ex_int(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6177_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_complex(G,X4) = zero_zero_complex ) )
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => ( aa_nat_complex(G,X4) = aa_nat_complex(H,X4) ) )
           => ( groups2073611262835488442omplex(G,T2) = groups2073611262835488442omplex(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6178_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_complex(H,X4) = zero_zero_complex ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
           => ( groups5754745047067104278omplex(G,S2) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6179_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6180_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6181_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_rat(H,X4) = zero_zero_rat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_rat(G,X4) = aa_real_rat(H,X4) ) )
           => ( groups1300246762558778688al_rat(G,S2) = groups1300246762558778688al_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6182_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_rat(H,X4) = zero_zero_rat ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_rat(G,X4) = aa_complex_rat(H,X4) ) )
           => ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6183_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_nat(H,X4) = zero_zero_nat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
           => ( groups1935376822645274424al_nat(G,S2) = groups1935376822645274424al_nat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6184_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_nat(H,X4) = zero_zero_nat ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_nat(G,X4) = aa_complex_nat(H,X4) ) )
           => ( groups5693394587270226106ex_nat(G,S2) = groups5693394587270226106ex_nat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6185_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_int,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_int(H,X4) = zero_zero_int ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_int(G,X4) = aa_real_int(H,X4) ) )
           => ( groups1932886352136224148al_int(G,S2) = groups1932886352136224148al_int(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6186_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_int,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_int(H,X4) = zero_zero_int ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_int(G,X4) = aa_complex_int(H,X4) ) )
           => ( groups5690904116761175830ex_int(G,S2) = groups5690904116761175830ex_int(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6187_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_nat,S2: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_complex(H,X4) = zero_zero_complex ) )
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => ( aa_nat_complex(G,X4) = aa_nat_complex(H,X4) ) )
           => ( groups2073611262835488442omplex(G,S2) = groups2073611262835488442omplex(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6188_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T2) = groups5808333547571424918x_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6189_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_rat(G,X4) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,T2) = groups5058264527183730370ex_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6190_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T2) = groups5693394587270226106ex_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6191_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T2) = groups5690904116761175830ex_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6192_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_complex(G,X4) = zero_zero_complex ) )
         => ( groups2073611262835488442omplex(G,T2) = groups2073611262835488442omplex(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6193_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_rat(G,X4) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,T2) = groups2906978787729119204at_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6194_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_int(G,X4) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G,T2) = groups3539618377306564664at_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6195_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
             => ( aa_int_complex(G,X4) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,T2) = groups3049146728041665814omplex(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6196_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
             => ( aa_int_real(G,X4) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,T2) = groups8778361861064173332t_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6197_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
             => ( aa_int_rat(G,X4) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,T2) = groups3906332499630173760nt_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6198_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6199_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_rat(G,X4) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6200_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S2) = groups5693394587270226106ex_nat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6201_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S2) = groups5690904116761175830ex_int(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6202_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_complex(G,X4) = zero_zero_complex ) )
         => ( groups2073611262835488442omplex(G,S2) = groups2073611262835488442omplex(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6203_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_rat(G,X4) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,S2) = groups2906978787729119204at_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6204_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_int(G,X4) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G,S2) = groups3539618377306564664at_int(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6205_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
             => ( aa_int_complex(G,X4) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,S2) = groups3049146728041665814omplex(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6206_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
             => ( aa_int_real(G,X4) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6207_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
             => ( aa_int_rat(G,X4) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,S2) = groups3906332499630173760nt_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6208_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A2: set_real,B2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,C2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C2))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C2))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C2,A2)))
               => ( aa_real_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C2,B2)))
                 => ( aa_real_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,C2) = groups5754745047067104278omplex(H,C2) )
               => ( groups5754745047067104278omplex(G,A2) = groups5754745047067104278omplex(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6209_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A2: set_real,B2: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C2))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C2))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C2,A2)))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C2,B2)))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,C2) = groups8097168146408367636l_real(H,C2) )
               => ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6210_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C2))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C2))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C2,A2)))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C2,B2)))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,C2) = groups5808333547571424918x_real(H,C2) )
               => ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6211_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A2: set_real,B2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,C2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C2))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C2))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C2,A2)))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C2,B2)))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,C2) = groups1300246762558778688al_rat(H,C2) )
               => ( groups1300246762558778688al_rat(G,A2) = groups1300246762558778688al_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6212_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C2))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C2))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C2,A2)))
               => ( aa_complex_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C2,B2)))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,C2) = groups5058264527183730370ex_rat(H,C2) )
               => ( groups5058264527183730370ex_rat(G,A2) = groups5058264527183730370ex_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6213_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A2: set_real,B2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,C2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C2))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C2))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C2,A2)))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C2,B2)))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,C2) = groups1935376822645274424al_nat(H,C2) )
               => ( groups1935376822645274424al_nat(G,A2) = groups1935376822645274424al_nat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6214_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C2))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C2))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C2,A2)))
               => ( aa_complex_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C2,B2)))
                 => ( aa_complex_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups5693394587270226106ex_nat(G,C2) = groups5693394587270226106ex_nat(H,C2) )
               => ( groups5693394587270226106ex_nat(G,A2) = groups5693394587270226106ex_nat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6215_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A2: set_real,B2: set_real,G: fun_real_int,H: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,C2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C2))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C2))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C2,A2)))
               => ( aa_real_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C2,B2)))
                 => ( aa_real_int(H,B3) = zero_zero_int ) )
             => ( ( groups1932886352136224148al_int(G,C2) = groups1932886352136224148al_int(H,C2) )
               => ( groups1932886352136224148al_int(G,A2) = groups1932886352136224148al_int(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6216_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C2))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C2))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C2,A2)))
               => ( aa_complex_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C2,B2)))
                 => ( aa_complex_int(H,B3) = zero_zero_int ) )
             => ( ( groups5690904116761175830ex_int(G,C2) = groups5690904116761175830ex_int(H,C2) )
               => ( groups5690904116761175830ex_int(G,A2) = groups5690904116761175830ex_int(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6217_sum_Osame__carrierI,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,C2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),C2))
       => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),C2))
         => ( ! [A3: nat] :
                ( pp(aa_set_nat_bool(member_nat(A3),minus_minus_set_nat(C2,A2)))
               => ( aa_nat_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: nat] :
                  ( pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(C2,B2)))
                 => ( aa_nat_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups2073611262835488442omplex(G,C2) = groups2073611262835488442omplex(H,C2) )
               => ( groups2073611262835488442omplex(G,A2) = groups2073611262835488442omplex(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6218_sum_Osame__carrier,axiom,
    ! [C2: set_real,A2: set_real,B2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,C2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C2))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C2))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C2,A2)))
               => ( aa_real_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C2,B2)))
                 => ( aa_real_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,A2) = groups5754745047067104278omplex(H,B2) )
              <=> ( groups5754745047067104278omplex(G,C2) = groups5754745047067104278omplex(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6219_sum_Osame__carrier,axiom,
    ! [C2: set_real,A2: set_real,B2: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C2))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C2))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C2,A2)))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C2,B2)))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B2) )
              <=> ( groups8097168146408367636l_real(G,C2) = groups8097168146408367636l_real(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6220_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C2))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C2))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C2,A2)))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C2,B2)))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B2) )
              <=> ( groups5808333547571424918x_real(G,C2) = groups5808333547571424918x_real(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6221_sum_Osame__carrier,axiom,
    ! [C2: set_real,A2: set_real,B2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,C2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C2))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C2))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C2,A2)))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C2,B2)))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,A2) = groups1300246762558778688al_rat(H,B2) )
              <=> ( groups1300246762558778688al_rat(G,C2) = groups1300246762558778688al_rat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6222_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C2))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C2))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C2,A2)))
               => ( aa_complex_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C2,B2)))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,A2) = groups5058264527183730370ex_rat(H,B2) )
              <=> ( groups5058264527183730370ex_rat(G,C2) = groups5058264527183730370ex_rat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6223_sum_Osame__carrier,axiom,
    ! [C2: set_real,A2: set_real,B2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,C2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C2))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C2))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C2,A2)))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C2,B2)))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,A2) = groups1935376822645274424al_nat(H,B2) )
              <=> ( groups1935376822645274424al_nat(G,C2) = groups1935376822645274424al_nat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6224_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C2))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C2))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C2,A2)))
               => ( aa_complex_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C2,B2)))
                 => ( aa_complex_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups5693394587270226106ex_nat(G,A2) = groups5693394587270226106ex_nat(H,B2) )
              <=> ( groups5693394587270226106ex_nat(G,C2) = groups5693394587270226106ex_nat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6225_sum_Osame__carrier,axiom,
    ! [C2: set_real,A2: set_real,B2: set_real,G: fun_real_int,H: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,C2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C2))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C2))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C2,A2)))
               => ( aa_real_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C2,B2)))
                 => ( aa_real_int(H,B3) = zero_zero_int ) )
             => ( ( groups1932886352136224148al_int(G,A2) = groups1932886352136224148al_int(H,B2) )
              <=> ( groups1932886352136224148al_int(G,C2) = groups1932886352136224148al_int(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6226_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C2))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C2))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C2,A2)))
               => ( aa_complex_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C2,B2)))
                 => ( aa_complex_int(H,B3) = zero_zero_int ) )
             => ( ( groups5690904116761175830ex_int(G,A2) = groups5690904116761175830ex_int(H,B2) )
              <=> ( groups5690904116761175830ex_int(G,C2) = groups5690904116761175830ex_int(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6227_sum_Osame__carrier,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,C2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),C2))
       => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),C2))
         => ( ! [A3: nat] :
                ( pp(aa_set_nat_bool(member_nat(A3),minus_minus_set_nat(C2,A2)))
               => ( aa_nat_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: nat] :
                  ( pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(C2,B2)))
                 => ( aa_nat_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups2073611262835488442omplex(G,A2) = groups2073611262835488442omplex(H,B2) )
              <=> ( groups2073611262835488442omplex(G,C2) = groups2073611262835488442omplex(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6228_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups5808333547571424918x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,B2))),groups5808333547571424918x_real(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6229_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups5058264527183730370ex_rat(G,A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,B2))),groups5058264527183730370ex_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6230_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups5693394587270226106ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,B2))),groups5693394587270226106ex_nat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6231_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups5690904116761175830ex_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,B2))),groups5690904116761175830ex_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6232_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( groups2906978787729119204at_rat(G,A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,minus_minus_set_nat(A2,B2))),groups2906978787729119204at_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6233_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( groups3539618377306564664at_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,minus_minus_set_nat(A2,B2))),groups3539618377306564664at_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6234_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( pp(aa_set_int_bool(finite_finite_int,A2))
       => ( groups8778361861064173332t_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G,minus_minus_set_int(A2,B2))),groups8778361861064173332t_real(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6235_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( pp(aa_set_int_bool(finite_finite_int,A2))
       => ( groups3906332499630173760nt_rat(G,A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,B2))),groups3906332499630173760nt_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6236_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( pp(aa_set_int_bool(finite_finite_int,A2))
       => ( groups4541462559716669496nt_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,B2))),groups4541462559716669496nt_nat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6237_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( pp(aa_set_int_bool(finite_finite_int,A2))
       => ( groups4538972089207619220nt_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,minus_minus_set_int(A2,B2))),groups4538972089207619220nt_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6238_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
       => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6239_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
       => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A2,B2)) = aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6240_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
       => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A2,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6241_sum__diff,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A2))
       => ( groups2906978787729119204at_rat(F,minus_minus_set_nat(A2,B2)) = aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6242_sum__diff,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A2))
       => ( groups3539618377306564664at_int(F,minus_minus_set_nat(A2,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,A2)),groups3539618377306564664at_int(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6243_sum__diff,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
       => ( groups8778361861064173332t_real(F,minus_minus_set_int(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6244_sum__diff,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
       => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A2,B2)) = aa_rat_rat(minus_minus_rat(groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6245_sum__diff,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
       => ( groups4538972089207619220nt_int(F,minus_minus_set_int(A2,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6246_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
       => ( groups7754918857620584856omplex(F,minus_811609699411566653omplex(A2,B2)) = aa_complex_complex(minus_minus_complex(groups7754918857620584856omplex(F,A2)),groups7754918857620584856omplex(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6247_sum__diff,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A2))
       => ( groups6591440286371151544t_real(F,minus_minus_set_nat(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(F,A2)),groups6591440286371151544t_real(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6248_sum__diff__nat,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
       => ( groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B2)) ) ) ) ).

% sum_diff_nat
tff(fact_6249_sum__diff__nat,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(B2),A2))
       => ( groups977919841031483927at_nat(F,minus_1356011639430497352at_nat(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A2)),groups977919841031483927at_nat(F,B2)) ) ) ) ).

% sum_diff_nat
tff(fact_6250_sum__diff__nat,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(F,B2)) ) ) ) ).

% sum_diff_nat
tff(fact_6251_sum__diff__nat,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A2))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B2)) ) ) ) ).

% sum_diff_nat
tff(fact_6252_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_6253_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_6254_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_6255_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_6256_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_6257_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(aa_rat_fun_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_6258_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_6259_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_6260_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_6261_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(aa_rat_fun_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_6262_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_6263_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_6264_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_6265_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(aa_rat_fun_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_6266_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_6267_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_6268_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_6269_remove__induct,axiom,
    ! [P: fun_se8353504738346895699t_bool,B2: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(P,bot_bo228742789529271731at_nat))
     => ( ( ~ pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,B2))
         => pp(aa_set4148026973699692364t_bool(P,B2)) )
       => ( ! [A4: set_Pr4329608150637261639at_nat] :
              ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A4))
             => ( ( A4 != bot_bo228742789529271731at_nat )
               => ( pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(A4),B2))
                 => ( ! [X2: produc3843707927480180839at_nat] :
                        ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X2),A4))
                       => pp(aa_set4148026973699692364t_bool(P,minus_3314409938677909166at_nat(A4,insert9069300056098147895at_nat(X2,bot_bo228742789529271731at_nat)))) )
                   => pp(aa_set4148026973699692364t_bool(P,A4)) ) ) ) )
         => pp(aa_set4148026973699692364t_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_6270_remove__induct,axiom,
    ! [P: fun_set_set_nat_bool,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
     => ( ( ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B2))
         => pp(aa_set_set_nat_bool(P,B2)) )
       => ( ! [A4: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A4))
             => ( ( A4 != bot_bot_set_set_nat )
               => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A4),B2))
                 => ( ! [X2: set_nat] :
                        ( pp(aa_set_set_nat_bool(member_set_nat(X2),A4))
                       => pp(aa_set_set_nat_bool(P,minus_2163939370556025621et_nat(A4,insert_set_nat(X2,bot_bot_set_set_nat)))) )
                   => pp(aa_set_set_nat_bool(P,A4)) ) ) ) )
         => pp(aa_set_set_nat_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_6271_remove__induct,axiom,
    ! [P: fun_set_complex_bool,B2: set_complex] :
      ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
     => ( ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
         => pp(aa_set_complex_bool(P,B2)) )
       => ( ! [A4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
             => ( ( A4 != bot_bot_set_complex )
               => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B2))
                 => ( ! [X2: complex] :
                        ( pp(aa_set_complex_bool(member_complex(X2),A4))
                       => pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A4,insert_complex(X2,bot_bot_set_complex)))) )
                   => pp(aa_set_complex_bool(P,A4)) ) ) ) )
         => pp(aa_set_complex_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_6272_remove__induct,axiom,
    ! [P: fun_se8074476271188067385t_bool,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(P,bot_bo2099793752762293965at_nat))
     => ( ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
         => pp(aa_set3135850010578026034t_bool(P,B2)) )
       => ( ! [A4: set_Pr1261947904930325089at_nat] :
              ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
             => ( ( A4 != bot_bo2099793752762293965at_nat )
               => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A4),B2))
                 => ( ! [X2: product_prod_nat_nat] :
                        ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),A4))
                       => pp(aa_set3135850010578026034t_bool(P,minus_1356011639430497352at_nat(A4,insert8211810215607154385at_nat(X2,bot_bo2099793752762293965at_nat)))) )
                   => pp(aa_set3135850010578026034t_bool(P,A4)) ) ) ) )
         => pp(aa_set3135850010578026034t_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_6273_remove__induct,axiom,
    ! [P: fun_set_real_bool,B2: set_real] :
      ( pp(aa_set_real_bool(P,bot_bot_set_real))
     => ( ( ~ pp(aa_set_real_bool(finite_finite_real,B2))
         => pp(aa_set_real_bool(P,B2)) )
       => ( ! [A4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A4))
             => ( ( A4 != bot_bot_set_real )
               => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B2))
                 => ( ! [X2: real] :
                        ( pp(aa_set_real_bool(member_real(X2),A4))
                       => pp(aa_set_real_bool(P,minus_minus_set_real(A4,insert_real(X2,bot_bot_set_real)))) )
                   => pp(aa_set_real_bool(P,A4)) ) ) ) )
         => pp(aa_set_real_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_6274_remove__induct,axiom,
    ! [P: fun_set_nat_bool,B2: set_nat] :
      ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
     => ( ( ~ pp(aa_set_nat_bool(finite_finite_nat,B2))
         => pp(aa_set_nat_bool(P,B2)) )
       => ( ! [A4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A4))
             => ( ( A4 != bot_bot_set_nat )
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B2))
                 => ( ! [X2: nat] :
                        ( pp(aa_set_nat_bool(member_nat(X2),A4))
                       => pp(aa_set_nat_bool(P,minus_minus_set_nat(A4,insert_nat(X2,bot_bot_set_nat)))) )
                   => pp(aa_set_nat_bool(P,A4)) ) ) ) )
         => pp(aa_set_nat_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_6275_remove__induct,axiom,
    ! [P: fun_set_int_bool,B2: set_int] :
      ( pp(aa_set_int_bool(P,bot_bot_set_int))
     => ( ( ~ pp(aa_set_int_bool(finite_finite_int,B2))
         => pp(aa_set_int_bool(P,B2)) )
       => ( ! [A4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A4))
             => ( ( A4 != bot_bot_set_int )
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B2))
                 => ( ! [X2: int] :
                        ( pp(aa_set_int_bool(member_int(X2),A4))
                       => pp(aa_set_int_bool(P,minus_minus_set_int(A4,insert_int(X2,bot_bot_set_int)))) )
                   => pp(aa_set_int_bool(P,A4)) ) ) ) )
         => pp(aa_set_int_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_6276_finite__remove__induct,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,P: fun_se8353504738346895699t_bool] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,B2))
     => ( pp(aa_set4148026973699692364t_bool(P,bot_bo228742789529271731at_nat))
       => ( ! [A4: set_Pr4329608150637261639at_nat] :
              ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A4))
             => ( ( A4 != bot_bo228742789529271731at_nat )
               => ( pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(A4),B2))
                 => ( ! [X2: produc3843707927480180839at_nat] :
                        ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X2),A4))
                       => pp(aa_set4148026973699692364t_bool(P,minus_3314409938677909166at_nat(A4,insert9069300056098147895at_nat(X2,bot_bo228742789529271731at_nat)))) )
                   => pp(aa_set4148026973699692364t_bool(P,A4)) ) ) ) )
         => pp(aa_set4148026973699692364t_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_6277_finite__remove__induct,axiom,
    ! [B2: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B2))
     => ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
       => ( ! [A4: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A4))
             => ( ( A4 != bot_bot_set_set_nat )
               => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A4),B2))
                 => ( ! [X2: set_nat] :
                        ( pp(aa_set_set_nat_bool(member_set_nat(X2),A4))
                       => pp(aa_set_set_nat_bool(P,minus_2163939370556025621et_nat(A4,insert_set_nat(X2,bot_bot_set_set_nat)))) )
                   => pp(aa_set_set_nat_bool(P,A4)) ) ) ) )
         => pp(aa_set_set_nat_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_6278_finite__remove__induct,axiom,
    ! [B2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [A4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
             => ( ( A4 != bot_bot_set_complex )
               => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B2))
                 => ( ! [X2: complex] :
                        ( pp(aa_set_complex_bool(member_complex(X2),A4))
                       => pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A4,insert_complex(X2,bot_bot_set_complex)))) )
                   => pp(aa_set_complex_bool(P,A4)) ) ) ) )
         => pp(aa_set_complex_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_6279_finite__remove__induct,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,P: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
     => ( pp(aa_set3135850010578026034t_bool(P,bot_bo2099793752762293965at_nat))
       => ( ! [A4: set_Pr1261947904930325089at_nat] :
              ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
             => ( ( A4 != bot_bo2099793752762293965at_nat )
               => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A4),B2))
                 => ( ! [X2: product_prod_nat_nat] :
                        ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),A4))
                       => pp(aa_set3135850010578026034t_bool(P,minus_1356011639430497352at_nat(A4,insert8211810215607154385at_nat(X2,bot_bo2099793752762293965at_nat)))) )
                   => pp(aa_set3135850010578026034t_bool(P,A4)) ) ) ) )
         => pp(aa_set3135850010578026034t_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_6280_finite__remove__induct,axiom,
    ! [B2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [A4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A4))
             => ( ( A4 != bot_bot_set_real )
               => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B2))
                 => ( ! [X2: real] :
                        ( pp(aa_set_real_bool(member_real(X2),A4))
                       => pp(aa_set_real_bool(P,minus_minus_set_real(A4,insert_real(X2,bot_bot_set_real)))) )
                   => pp(aa_set_real_bool(P,A4)) ) ) ) )
         => pp(aa_set_real_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_6281_finite__remove__induct,axiom,
    ! [B2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [A4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A4))
             => ( ( A4 != bot_bot_set_nat )
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B2))
                 => ( ! [X2: nat] :
                        ( pp(aa_set_nat_bool(member_nat(X2),A4))
                       => pp(aa_set_nat_bool(P,minus_minus_set_nat(A4,insert_nat(X2,bot_bot_set_nat)))) )
                   => pp(aa_set_nat_bool(P,A4)) ) ) ) )
         => pp(aa_set_nat_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_6282_finite__remove__induct,axiom,
    ! [B2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [A4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A4))
             => ( ( A4 != bot_bot_set_int )
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B2))
                 => ( ! [X2: int] :
                        ( pp(aa_set_int_bool(member_int(X2),A4))
                       => pp(aa_set_int_bool(P,minus_minus_set_int(A4,insert_int(X2,bot_bot_set_int)))) )
                   => pp(aa_set_int_bool(P,A4)) ) ) ) )
         => pp(aa_set_int_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_6283_finite__induct__select,axiom,
    ! [S2: set_Pr4329608150637261639at_nat,P: fun_se8353504738346895699t_bool] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,S2))
     => ( pp(aa_set4148026973699692364t_bool(P,bot_bo228742789529271731at_nat))
       => ( ! [T5: set_Pr4329608150637261639at_nat] :
              ( pp(aa_set4148026973699692364t_bool(ord_le2604355607129572851at_nat(T5),S2))
             => ( pp(aa_set4148026973699692364t_bool(P,T5))
               => ? [X2: produc3843707927480180839at_nat] :
                    ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X2),minus_3314409938677909166at_nat(S2,T5)))
                    & pp(aa_set4148026973699692364t_bool(P,insert9069300056098147895at_nat(X2,T5))) ) ) )
         => pp(aa_set4148026973699692364t_bool(P,S2)) ) ) ) ).

% finite_induct_select
tff(fact_6284_finite__induct__select,axiom,
    ! [S2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [T5: set_complex] :
              ( pp(aa_set_complex_bool(ord_less_set_complex(T5),S2))
             => ( pp(aa_set_complex_bool(P,T5))
               => ? [X2: complex] :
                    ( pp(aa_set_complex_bool(member_complex(X2),minus_811609699411566653omplex(S2,T5)))
                    & pp(aa_set_complex_bool(P,insert_complex(X2,T5))) ) ) )
         => pp(aa_set_complex_bool(P,S2)) ) ) ) ).

% finite_induct_select
tff(fact_6285_finite__induct__select,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,P: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,S2))
     => ( pp(aa_set3135850010578026034t_bool(P,bot_bo2099793752762293965at_nat))
       => ( ! [T5: set_Pr1261947904930325089at_nat] :
              ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(T5),S2))
             => ( pp(aa_set3135850010578026034t_bool(P,T5))
               => ? [X2: product_prod_nat_nat] :
                    ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),minus_1356011639430497352at_nat(S2,T5)))
                    & pp(aa_set3135850010578026034t_bool(P,insert8211810215607154385at_nat(X2,T5))) ) ) )
         => pp(aa_set3135850010578026034t_bool(P,S2)) ) ) ) ).

% finite_induct_select
tff(fact_6286_finite__induct__select,axiom,
    ! [S2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [T5: set_real] :
              ( pp(aa_set_real_bool(ord_less_set_real(T5),S2))
             => ( pp(aa_set_real_bool(P,T5))
               => ? [X2: real] :
                    ( pp(aa_set_real_bool(member_real(X2),minus_minus_set_real(S2,T5)))
                    & pp(aa_set_real_bool(P,insert_real(X2,T5))) ) ) )
         => pp(aa_set_real_bool(P,S2)) ) ) ) ).

% finite_induct_select
tff(fact_6287_finite__induct__select,axiom,
    ! [S2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [T5: set_int] :
              ( pp(aa_set_int_bool(ord_less_set_int(T5),S2))
             => ( pp(aa_set_int_bool(P,T5))
               => ? [X2: int] :
                    ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(S2,T5)))
                    & pp(aa_set_int_bool(P,insert_int(X2,T5))) ) ) )
         => pp(aa_set_int_bool(P,S2)) ) ) ) ).

% finite_induct_select
tff(fact_6288_finite__induct__select,axiom,
    ! [S2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [T5: set_nat] :
              ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,T5),S2))
             => ( pp(aa_set_nat_bool(P,T5))
               => ? [X2: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X2),minus_minus_set_nat(S2,T5)))
                    & pp(aa_set_nat_bool(P,insert_nat(X2,T5))) ) ) )
         => pp(aa_set_nat_bool(P,S2)) ) ) ) ).

% finite_induct_select
tff(fact_6289_psubset__insert__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(ord_le2604355607129572851at_nat(A2),insert9069300056098147895at_nat(X,B2)))
    <=> ( ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),B2))
         => pp(aa_set4148026973699692364t_bool(ord_le2604355607129572851at_nat(A2),B2)) )
        & ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),B2))
         => ( ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),A2))
             => pp(aa_set4148026973699692364t_bool(ord_le2604355607129572851at_nat(minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))),B2)) )
            & ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),A2))
             => pp(aa_set4148026973699692364t_bool(ord_le1268244103169919719at_nat(A2),B2)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_6290_psubset__insert__iff,axiom,
    ! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A2),insert_set_nat(X,B2)))
    <=> ( ( pp(aa_set_set_nat_bool(member_set_nat(X),B2))
         => pp(aa_set_set_nat_bool(ord_less_set_set_nat(A2),B2)) )
        & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),B2))
         => ( ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
             => pp(aa_set_set_nat_bool(ord_less_set_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(X,bot_bot_set_set_nat))),B2)) )
            & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A2))
             => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B2)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_6291_psubset__insert__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A2),insert8211810215607154385at_nat(X,B2)))
    <=> ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B2))
         => pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A2),B2)) )
        & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B2))
         => ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2))
             => pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B2)) )
            & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2))
             => pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B2)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_6292_psubset__insert__iff,axiom,
    ! [A2: set_real,X: real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A2),insert_real(X,B2)))
    <=> ( ( pp(aa_set_real_bool(member_real(X),B2))
         => pp(aa_set_real_bool(ord_less_set_real(A2),B2)) )
        & ( ~ pp(aa_set_real_bool(member_real(X),B2))
         => ( ( pp(aa_set_real_bool(member_real(X),A2))
             => pp(aa_set_real_bool(ord_less_set_real(minus_minus_set_real(A2,insert_real(X,bot_bot_set_real))),B2)) )
            & ( ~ pp(aa_set_real_bool(member_real(X),A2))
             => pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_6293_psubset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),insert_nat(X,B2)))
    <=> ( ( pp(aa_set_nat_bool(member_nat(X),B2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B2)) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),B2))
         => ( ( pp(aa_set_nat_bool(member_nat(X),A2))
             => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,minus_minus_set_nat(A2,insert_nat(X,bot_bot_set_nat))),B2)) )
            & ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
             => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_6294_psubset__insert__iff,axiom,
    ! [A2: set_int,X: int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),insert_int(X,B2)))
    <=> ( ( pp(aa_set_int_bool(member_int(X),B2))
         => pp(aa_set_int_bool(ord_less_set_int(A2),B2)) )
        & ( ~ pp(aa_set_int_bool(member_int(X),B2))
         => ( ( pp(aa_set_int_bool(member_int(X),A2))
             => pp(aa_set_int_bool(ord_less_set_int(minus_minus_set_int(A2,insert_int(X,bot_bot_set_int))),B2)) )
            & ( ~ pp(aa_set_int_bool(member_int(X),A2))
             => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_6295_set__replicate__Suc,axiom,
    ! [N: nat,X: produc3843707927480180839at_nat] : set_Pr3765526544606949372at_nat(replic2264142908078655527at_nat(aa_nat_nat(suc,N),X)) = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ).

% set_replicate_Suc
tff(fact_6296_set__replicate__Suc,axiom,
    ! [N: nat,X: vEBT_VEBT] : set_VEBT_VEBT2(replicate_VEBT_VEBT(aa_nat_nat(suc,N),X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ).

% set_replicate_Suc
tff(fact_6297_set__replicate__Suc,axiom,
    ! [N: nat,X: product_prod_nat_nat] : set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(aa_nat_nat(suc,N),X)) = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ).

% set_replicate_Suc
tff(fact_6298_set__replicate__Suc,axiom,
    ! [N: nat,X: real] : set_real2(replicate_real(aa_nat_nat(suc,N),X)) = insert_real(X,bot_bot_set_real) ).

% set_replicate_Suc
tff(fact_6299_set__replicate__Suc,axiom,
    ! [N: nat,X: nat] : set_nat2(replicate_nat(aa_nat_nat(suc,N),X)) = insert_nat(X,bot_bot_set_nat) ).

% set_replicate_Suc
tff(fact_6300_set__replicate__Suc,axiom,
    ! [N: nat,X: int] : set_int2(replicate_int(aa_nat_nat(suc,N),X)) = insert_int(X,bot_bot_set_int) ).

% set_replicate_Suc
tff(fact_6301_set__replicate__conv__if,axiom,
    ! [N: nat,X: produc3843707927480180839at_nat] :
      ( ( ( N = zero_zero_nat )
       => ( set_Pr3765526544606949372at_nat(replic2264142908078655527at_nat(N,X)) = bot_bo228742789529271731at_nat ) )
      & ( ( N != zero_zero_nat )
       => ( set_Pr3765526544606949372at_nat(replic2264142908078655527at_nat(N,X)) = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ) ).

% set_replicate_conv_if
tff(fact_6302_set__replicate__conv__if,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N != zero_zero_nat )
       => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% set_replicate_conv_if
tff(fact_6303_set__replicate__conv__if,axiom,
    ! [N: nat,X: product_prod_nat_nat] :
      ( ( ( N = zero_zero_nat )
       => ( set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(N,X)) = bot_bo2099793752762293965at_nat ) )
      & ( ( N != zero_zero_nat )
       => ( set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(N,X)) = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ) ).

% set_replicate_conv_if
tff(fact_6304_set__replicate__conv__if,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero_nat )
       => ( set_real2(replicate_real(N,X)) = bot_bot_set_real ) )
      & ( ( N != zero_zero_nat )
       => ( set_real2(replicate_real(N,X)) = insert_real(X,bot_bot_set_real) ) ) ) ).

% set_replicate_conv_if
tff(fact_6305_set__replicate__conv__if,axiom,
    ! [N: nat,X: nat] :
      ( ( ( N = zero_zero_nat )
       => ( set_nat2(replicate_nat(N,X)) = bot_bot_set_nat ) )
      & ( ( N != zero_zero_nat )
       => ( set_nat2(replicate_nat(N,X)) = insert_nat(X,bot_bot_set_nat) ) ) ) ).

% set_replicate_conv_if
tff(fact_6306_set__replicate__conv__if,axiom,
    ! [N: nat,X: int] :
      ( ( ( N = zero_zero_nat )
       => ( set_int2(replicate_int(N,X)) = bot_bot_set_int ) )
      & ( ( N != zero_zero_nat )
       => ( set_int2(replicate_int(N,X)) = insert_int(X,bot_bot_set_int) ) ) ) ).

% set_replicate_conv_if
tff(fact_6307_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(aa_rat_fun_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(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,M)),groups2906978787729119204at_rat(aTP_Lamm_lk(G),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6308_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_ll(G),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6309_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_lm(G)),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6310_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_ln(G),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6311_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_lo(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_6312_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_lp(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_6313_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_lq(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_6314_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B2,A2)))
             => 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,A2)),groups8097168146408367636l_real(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6315_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B2,A2)))
             => 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,A2)),groups5808333547571424918x_real(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6316_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B2,A2)))
             => 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,A2)),groups1300246762558778688al_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6317_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B2,A2)))
             => 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,A2)),groups5058264527183730370ex_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6318_sum__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
       => ( ! [B3: nat] :
              ( pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(B2,A2)))
             => 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,A2)),groups2906978787729119204at_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6319_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B2,A2)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6320_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B2,A2)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6321_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B2,A2)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,B3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups1932886352136224148al_int(F,A2)),groups1932886352136224148al_int(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6322_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B2,A2)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_complex_int(F,B3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6323_sum__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
       => ( ! [B3: nat] :
              ( pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(B2,A2)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,B3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,A2)),groups3539618377306564664at_int(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6324_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(aa_rat_fun_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_6325_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_6326_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_6327_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_6328_sum__count__set,axiom,
    ! [Xs: list_complex,X6: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Xs)),X6))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,X6))
       => ( groups5693394587270226106ex_nat(count_list_complex(Xs),X6) = size_s3451745648224563538omplex(Xs) ) ) ) ).

% sum_count_set
tff(fact_6329_sum__count__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat,X6: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Xs)),X6))
     => ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,X6))
       => ( groups977919841031483927at_nat(count_4203492906077236349at_nat(Xs),X6) = size_s5460976970255530739at_nat(Xs) ) ) ) ).

% sum_count_set
tff(fact_6330_sum__count__set,axiom,
    ! [Xs: list_VEBT_VEBT,X6: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),X6)
     => ( finite5795047828879050333T_VEBT(X6)
       => ( groups771621172384141258BT_nat(count_list_VEBT_VEBT(Xs),X6) = size_s6755466524823107622T_VEBT(Xs) ) ) ) ).

% sum_count_set
tff(fact_6331_sum__count__set,axiom,
    ! [Xs: list_bool,X6: set_bool] :
      ( ord_less_eq_set_bool(set_bool2(Xs),X6)
     => ( finite_finite_bool(X6)
       => ( groups5768501273728744877ol_nat(count_list_bool(Xs),X6) = size_size_list_bool(Xs) ) ) ) ).

% sum_count_set
tff(fact_6332_sum__count__set,axiom,
    ! [Xs: list_int,X6: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),X6))
     => ( pp(aa_set_int_bool(finite_finite_int,X6))
       => ( groups4541462559716669496nt_nat(count_list_int(Xs),X6) = size_size_list_int(Xs) ) ) ) ).

% sum_count_set
tff(fact_6333_sum__count__set,axiom,
    ! [Xs: list_nat,X6: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Xs)),X6))
     => ( pp(aa_set_nat_bool(finite_finite_nat,X6))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_lis1299702591413342011at_nat(count_list_nat,Xs)),X6) = size_size_list_nat(Xs) ) ) ) ).

% sum_count_set
tff(fact_6334_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B2,A2)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,B)))
           => ( ! [X4: real] :
                  ( pp(aa_set_real_bool(member_real(X4),B2))
                 => 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,A2)),groups8097168146408367636l_real(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6335_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B2,A2)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,B)))
           => ( ! [X4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X4),B2))
                 => 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,A2)),groups5808333547571424918x_real(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6336_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B2,A2)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,B)))
           => ( ! [X4: real] :
                  ( pp(aa_set_real_bool(member_real(X4),B2))
                 => 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,A2)),groups1300246762558778688al_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6337_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B2,A2)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,B)))
           => ( ! [X4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X4),B2))
                 => 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,A2)),groups5058264527183730370ex_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6338_sum__strict__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,B: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
       => ( pp(aa_set_nat_bool(member_nat(B),minus_minus_set_nat(B2,A2)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(F,B)))
           => ( ! [X4: nat] :
                  ( pp(aa_set_nat_bool(member_nat(X4),B2))
                 => 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,A2)),groups2906978787729119204at_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6339_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B2,A2)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F,B)))
           => ( ! [X4: real] :
                  ( pp(aa_set_real_bool(member_real(X4),B2))
                 => 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_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6340_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B2,A2)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F,B)))
           => ( ! [X4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X4),B2))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X4))) )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6341_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B2,A2)))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_real_int(F,B)))
           => ( ! [X4: real] :
                  ( pp(aa_set_real_bool(member_real(X4),B2))
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,X4))) )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups1932886352136224148al_int(F,A2)),groups1932886352136224148al_int(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6342_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B2,A2)))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_complex_int(F,B)))
           => ( ! [X4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X4),B2))
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_complex_int(F,X4))) )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6343_sum__strict__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,B: nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
       => ( pp(aa_set_nat_bool(member_nat(B),minus_minus_set_nat(B2,A2)))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,B)))
           => ( ! [X4: nat] :
                  ( pp(aa_set_nat_bool(member_nat(X4),B2))
                 => 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_int,groups3539618377306564664at_int(F,A2)),groups3539618377306564664at_int(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6344_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_lr(F),set_or1269000886237332187st_nat(M,N)) = aa_complex_complex(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_lr(F),set_or1269000886237332187st_nat(M,N)) = zero_zero_complex ) ) ) ).

% sum_natinterval_diff
tff(fact_6345_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_ls(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_ls(F),set_or1269000886237332187st_nat(M,N)) = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
tff(fact_6346_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_lt(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_lt(F),set_or1269000886237332187st_nat(M,N)) = zero_zero_int ) ) ) ).

% sum_natinterval_diff
tff(fact_6347_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_lu(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_lu(F),set_or1269000886237332187st_nat(M,N)) = zero_zero_real ) ) ) ).

% sum_natinterval_diff
tff(fact_6348_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_lv(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_6349_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_lw(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_6350_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_lx(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_6351_exists__least__lemma,axiom,
    ! [P: fun_nat_bool] :
      ( ~ pp(aa_nat_bool(P,zero_zero_nat))
     => ( ? [X_12: nat] : pp(aa_nat_bool(P,X_12))
       => ? [N2: nat] :
            ( ~ pp(aa_nat_bool(P,N2))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) ) ).

% exists_least_lemma
tff(fact_6352_mask__eq__sum__exp,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) = groups3539618377306564664at_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bd,N))) ).

% mask_eq_sum_exp
tff(fact_6353_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_bd,N))) ).

% mask_eq_sum_exp
tff(fact_6354_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z3: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z3))) ).

% ex_le_of_int
tff(fact_6355_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z3: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z3))) ).

% ex_le_of_int
tff(fact_6356_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z3: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z3))) ).

% ex_less_of_int
tff(fact_6357_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z3: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z3))) ).

% ex_less_of_int
tff(fact_6358_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z3: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z3)),X)) ).

% ex_of_int_less
tff(fact_6359_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z3: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z3)),X)) ).

% ex_of_int_less
tff(fact_6360_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(power_power_complex(X),set_or1269000886237332187st_nat(M,N))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),M)),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_6361_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(one_one_rat),X)),groups2906978787729119204at_rat(power_power_rat(X),set_or1269000886237332187st_nat(M,N))) = aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),M)),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_6362_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: 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),X)),groups3539618377306564664at_int(power_power_int(X),set_or1269000886237332187st_nat(M,N))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),M)),aa_nat_int(power_power_int(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_6363_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: 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),X)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),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,X),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_6364_of__int__round__le,axiom,
    ! [X: 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,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),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_6365_of__int__round__le,axiom,
    ! [X: 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,X))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),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_6366_of__int__round__ge,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),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,X)))) ).

% of_int_round_ge
tff(fact_6367_of__int__round__ge,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(X),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,X)))) ).

% of_int_round_ge
tff(fact_6368_of__int__round__gt,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),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,X)))) ).

% of_int_round_gt
tff(fact_6369_of__int__round__gt,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(X),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,X)))) ).

% of_int_round_gt
tff(fact_6370_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_6371_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_bd,N))) ).

% mask_eq_sum_exp_nat
tff(fact_6372_gauss__sum__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ly),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_6373_sum__gp,axiom,
    ! [N: nat,M: nat,X: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2073611262835488442omplex(power_power_complex(X),set_or1269000886237332187st_nat(M,N)) = zero_zero_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( ( ( X = one_one_complex )
           => ( groups2073611262835488442omplex(power_power_complex(X),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)) ) )
          & ( ( X != one_one_complex )
           => ( groups2073611262835488442omplex(power_power_complex(X),set_or1269000886237332187st_nat(M,N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),M)),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N)))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_6374_sum__gp,axiom,
    ! [N: nat,M: nat,X: rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2906978787729119204at_rat(power_power_rat(X),set_or1269000886237332187st_nat(M,N)) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( ( ( X = one_one_rat )
           => ( groups2906978787729119204at_rat(power_power_rat(X),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)) ) )
          & ( ( X != one_one_rat )
           => ( groups2906978787729119204at_rat(power_power_rat(X),set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),M)),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N)))),aa_rat_rat(minus_minus_rat(one_one_rat),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_6375_sum__gp,axiom,
    ! [N: nat,M: nat,X: 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,X),set_or1269000886237332187st_nat(M,N)) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( ( ( X = one_one_real )
           => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),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)) ) )
          & ( ( X != one_one_real )
           => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),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,X),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_6376_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_6377_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(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_6378_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_6379_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_complex_complex(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(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6380_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N))) = times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N),plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N),one_on7984719198319812577d_enat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6381_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6382_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_6383_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_6384_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_6385_gauss__sum,axiom,
    ! [N: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(zero_zero_nat,N)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_6386_gauss__sum,axiom,
    ! [N: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_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_6387_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_6388_arith__series,axiom,
    ! [A: code_integer,D: code_integer,N: nat] : groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_lz(A),D),set_or1269000886237332187st_nat(zero_zero_nat,N)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),D))),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% arith_series
tff(fact_6389_arith__series,axiom,
    ! [A: int,D: int,N: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ma(A),D),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_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))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),D)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_6390_arith__series,axiom,
    ! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mb(A),D)),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))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_6391_double__gauss__sum,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)) ).

% double_gauss_sum
tff(fact_6392_double__gauss__sum,axiom,
    ! [N: nat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,set_or1269000886237332187st_nat(zero_zero_nat,N))) = times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N),plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N),one_on7984719198319812577d_enat)) ).

% double_gauss_sum
tff(fact_6393_double__gauss__sum,axiom,
    ! [N: nat] : aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)) ).

% double_gauss_sum
tff(fact_6394_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_6395_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_6396_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_6397_double__arith__series,axiom,
    ! [A: complex,D: complex,N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mc(A),D),set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),A)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),D))) ).

% double_arith_series
tff(fact_6398_double__arith__series,axiom,
    ! [A: extended_enat,D: extended_enat,N: nat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),groups7108830773950497114d_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_md(A),D),set_or1269000886237332187st_nat(zero_zero_nat,N))) = times_7803423173614009249d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N),one_on7984719198319812577d_enat),plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),A),times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N),D))) ).

% double_arith_series
tff(fact_6399_double__arith__series,axiom,
    ! [A: rat,D: rat,N: nat] : aa_rat_rat(aa_rat_fun_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_me(A),D),set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),D))) ).

% double_arith_series
tff(fact_6400_double__arith__series,axiom,
    ! [A: int,D: 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_ma(A),D),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))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),D))) ).

% double_arith_series
tff(fact_6401_double__arith__series,axiom,
    ! [A: nat,D: 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_mb(A),D)),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))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),D))) ).

% double_arith_series
tff(fact_6402_double__arith__series,axiom,
    ! [A: real,D: 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_mf(A),D),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))),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),D))) ).

% double_arith_series
tff(fact_6403_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_6404_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_6405_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,M) )
    <=> ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_6406_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(K),L)),zero_zero_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(ord_less_int,L),zero_zero_int)) ) ) ).

% xor_negative_int_iff
tff(fact_6407_of__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% of_nat_0
tff(fact_6408_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_6409_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_6410_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_6411_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_6412_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_6413_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_6414_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_6415_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_6416_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_6417_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_6418_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_6419_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_6420_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_6421_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_6422_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_6423_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_6424_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_6425_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_6426_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_6427_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_6428_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_6429_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_6430_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,W2)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,W2)),N)) ) ).

% numeral_less_real_of_nat_iff
tff(fact_6431_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_num_real(numeral_numeral_real,W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,W2))) ) ).

% real_of_nat_less_numeral_iff
tff(fact_6432_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(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_add
tff(fact_6433_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_6434_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_6435_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_6436_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_6437_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,N)) ).

% of_nat_mult
tff(fact_6438_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(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_mult
tff(fact_6439_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_6440_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_6441_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_6442_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,N) = one_one_complex )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6443_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_6444_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_6445_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_6446_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_6447_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6448_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_6449_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_6450_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_6451_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_6452_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_6453_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_6454_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_6455_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_6456_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_6457_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_6458_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_real(semiri5074537144036343181t_real,zero_n2687167440665602831ol_nat(P)) = zero_n3304061248610475627l_real(P) ).

% of_nat_of_bool
tff(fact_6459_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_nat(semiri1316708129612266289at_nat,zero_n2687167440665602831ol_nat(P)) = zero_n2687167440665602831ol_nat(P) ).

% of_nat_of_bool
tff(fact_6460_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_int(semiri1314217659103216013at_int,zero_n2687167440665602831ol_nat(P)) = aa_bool_int(zero_n2684676970156552555ol_int,P) ).

% of_nat_of_bool
tff(fact_6461_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_6462_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_6463_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_6464_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_6465_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,M)) ).

% of_nat_Suc
tff(fact_6466_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M)) ).

% of_nat_Suc
tff(fact_6467_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_6468_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_6469_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_6470_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_6471_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_6472_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_6473_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_6474_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6475_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: 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,B)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6476_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6477_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: 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,B)),W2)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6478_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6479_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6480_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6481_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6482_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: 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,B)),W2)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6483_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: 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,B)),W2)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6484_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: 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,B)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6485_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6486_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6487_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6488_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6489_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6490_of__nat__zero__less__power__iff,axiom,
    ! [X: 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,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6491_of__nat__zero__less__power__iff,axiom,
    ! [X: 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,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6492_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6493_of__nat__zero__less__power__iff,axiom,
    ! [X: 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,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6494_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),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,X),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_6495_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),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,X),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_6496_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_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,X),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_6497_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),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,X),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_6498_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: 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,X)))
    <=> 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)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6499_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: 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,X)))
    <=> 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)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6500_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> 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)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6501_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: 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,X)))
    <=> 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)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6502_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: 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,X)))
    <=> 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)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6503_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: 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,X)))
    <=> 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)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6504_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: 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,X)))
    <=> 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)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6505_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> 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)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6506_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),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,X),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_6507_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),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,X),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_6508_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),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,X),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_6509_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_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,X),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_6510_real__arch__simple,axiom,
    ! [X: real] :
    ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N2))) ).

% real_arch_simple
tff(fact_6511_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N2))) ).

% real_arch_simple
tff(fact_6512_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N2))) ).

% reals_Archimedean2
tff(fact_6513_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,N2))) ).

% reals_Archimedean2
tff(fact_6514_mult__of__nat__commute,axiom,
    ! [X: nat,Y: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,X)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(semiri8010041392384452111omplex,X)) ).

% mult_of_nat_commute
tff(fact_6515_mult__of__nat__commute,axiom,
    ! [X: nat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(semiri681578069525770553at_rat,X)) ).

% mult_of_nat_commute
tff(fact_6516_mult__of__nat__commute,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,X)) ).

% mult_of_nat_commute
tff(fact_6517_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,X)) ).

% mult_of_nat_commute
tff(fact_6518_mult__of__nat__commute,axiom,
    ! [X: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(semiri5074537144036343181t_real,X)) ).

% mult_of_nat_commute
tff(fact_6519_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: 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,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_6520_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: 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,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_6521_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: 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,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_6522_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_6523_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_6524_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_6525_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_6526_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_6527_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_6528_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_6529_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_6530_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,N)) != zero_zero_complex ).

% of_nat_neq_0
tff(fact_6531_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_6532_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_6533_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_6534_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_6535_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_6536_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_6537_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_6538_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_6539_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_6540_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_6541_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_6542_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_6543_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_6544_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_6545_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_6546_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_6547_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_6548_nat__int__comparison_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ) ).

% nat_int_comparison(2)
tff(fact_6549_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ) ).

% nat_int_comparison(3)
tff(fact_6550_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_6551_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M)))) ).

% not_int_zless_negative
tff(fact_6552_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(semiri1316708129612266289at_nat,Y)) ).

% of_nat_max
tff(fact_6553_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_int_int(ord_max_int(aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y)) ).

% of_nat_max
tff(fact_6554_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(semiri5074537144036343181t_real,Y)) ).

% of_nat_max
tff(fact_6555_nat__less__as__int,axiom,
    ! [X2: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).

% nat_less_as_int
tff(fact_6556_nat__leq__as__int,axiom,
    ! [X2: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).

% nat_leq_as_int
tff(fact_6557_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),X))) ) ).

% ex_less_of_nat_mult
tff(fact_6558_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),X))) ) ).

% ex_less_of_nat_mult
tff(fact_6559_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_6560_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_6561_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_6562_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_6563_reals__Archimedean3,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ! [Y3: real] :
        ? [N2: 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,N2)),X))) ) ).

% reals_Archimedean3
tff(fact_6564_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ) ) ).

% int_cases4
tff(fact_6565_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z))
    <=> ? [N3: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3))) ) ).

% zless_iff_Suc_zadd
tff(fact_6566_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_6567_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_mg,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_di,N))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_6568_sum__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( groups7754918857620584856omplex(aTP_Lamm_mg,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_dg(N),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_6569_pos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ~ ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% pos_int_cases
tff(fact_6570_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ? [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_6571_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) )
       => ~ ! [N2: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ) ).

% int_cases3
tff(fact_6572_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_6573_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_6574_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_6575_negD,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
     => ? [N2: nat] : X = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% negD
tff(fact_6576_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_6577_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = zero_zero_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ) ) ).

% int_ops(6)
tff(fact_6578_simp__from__to,axiom,
    ! [J: int,I: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,I),J) = bot_bot_set_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,I),J) = insert_int(I,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int)),J)) ) ) ) ).

% simp_from_to
tff(fact_6579_nat__approx__posE,axiom,
    ! [E2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
     => ~ ! [N2: 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,N2)))),E2)) ) ).

% nat_approx_posE
tff(fact_6580_nat__approx__posE,axiom,
    ! [E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
     => ~ ! [N2: 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,N2)))),E2)) ) ).

% nat_approx_posE
tff(fact_6581_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_6582_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(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_6583_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_6584_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_6585_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_6586_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => ( ! [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)),X)),C)) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_6587_neg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => ~ ! [N2: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% neg_int_cases
tff(fact_6588_zdiff__int__split,axiom,
    ! [P: fun_int_bool,X: nat,Y: nat] :
      ( pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Y))))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
         => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y)))) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
         => pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% zdiff_int_split
tff(fact_6589_XOR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_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,Y),aa_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(bit_se6526347334894502574or_int(X),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% XOR_upper
tff(fact_6590_of__nat__code__if,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(semiri8010041392384452111omplex,N) = zero_zero_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(semiri8010041392384452111omplex,N) = produc1917071388513777916omplex(aTP_Lamm_mh,divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_6591_of__nat__code__if,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_Extended_enat(semiri4216267220026989637d_enat,N) = zero_z5237406670263579293d_enat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_Extended_enat(semiri4216267220026989637d_enat,N) = produc2676513652042109336d_enat(aTP_Lamm_mi,divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_6592_of__nat__code__if,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(semiri681578069525770553at_rat,N) = zero_zero_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(semiri681578069525770553at_rat,N) = produc6207742614233964070at_rat(aTP_Lamm_mj,divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_6593_of__nat__code__if,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(semiri1316708129612266289at_nat,N) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(semiri1316708129612266289at_nat,N) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_mk),divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_6594_of__nat__code__if,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(semiri1314217659103216013at_int,N) = zero_zero_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(semiri1314217659103216013at_int,N) = produc6840382203811409530at_int(aTP_Lamm_ml,divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_6595_of__nat__code__if,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(semiri5074537144036343181t_real,N) = zero_zero_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(semiri5074537144036343181t_real,N) = aa_Pro6168199777849092593t_real(produc1703576794950452218t_real(aTP_Lamm_mm),divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_6596_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K4: 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)),K4))
       => ( 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))),K4))
         => 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,K4),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_6597_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K4: real,N: nat] :
      ( ( H != zero_zero_complex )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K4))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Z),H))),K4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),N)),aa_nat_complex(power_power_complex(Z),N))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(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,K4),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_6598_lemma__termdiff2,axiom,
    ! [H: complex,Z: complex,N: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),N)),aa_nat_complex(power_power_complex(Z),N))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(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(times_times_complex(H),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mo(H),Z),N),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_6599_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(aa_rat_fun_rat_rat(plus_plus_rat,Z),H)),N)),aa_nat_rat(power_power_rat(Z),N))),H)),aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,H),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_mq(H),Z),N),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_6600_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_ms(H),Z),N),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_6601_of__nat__code,axiom,
    ! [N: nat] : aa_nat_complex(semiri8010041392384452111omplex,N) = aa_complex_complex(aa_nat3829984526574172007omplex(semiri2816024913162550771omplex(aTP_Lamm_mt),N),zero_zero_complex) ).

% of_nat_code
tff(fact_6602_of__nat__code,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,N) = aa_rat_rat(semiri7787848453975740701ux_rat(aTP_Lamm_mu,N),zero_zero_rat) ).

% of_nat_code
tff(fact_6603_of__nat__code,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_mv),N),zero_zero_nat) ).

% of_nat_code
tff(fact_6604_of__nat__code,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_int_int(semiri8420488043553186161ux_int(aTP_Lamm_mw,N),zero_zero_int) ).

% of_nat_code
tff(fact_6605_of__nat__code,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(aa_nat_fun_real_real(semiri7260567687927622513x_real(aTP_Lamm_mx),N),zero_zero_real) ).

% of_nat_code
tff(fact_6606_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X),Xa2) = Y )
     => ( ( ( pp(aa_set_int_bool(member_int(X),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(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( Y = 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))),X))),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))),Xa2))))) ) )
        & ( ~ ( pp(aa_set_int_bool(member_int(X),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(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( Y = 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))),X))),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))),Xa2))))),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_se725231765392027082nd_int(aa_int_int(divide_divide_int(X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(Xa2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.elims
tff(fact_6607_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(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(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(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.simps
tff(fact_6608_lessThan__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(I),set_or890127255671739683et_nat(K)))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,I),K)) ) ).

% lessThan_iff
tff(fact_6609_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_6610_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_6611_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),set_ord_lessThan_nat(K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K)) ) ).

% lessThan_iff
tff(fact_6612_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_6613_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_6614_bit_Oconj__zero__right,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(X),zero_zero_int) = zero_zero_int ).

% bit.conj_zero_right
tff(fact_6615_bit_Oconj__zero__left,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),X) = zero_zero_int ).

% bit.conj_zero_left
tff(fact_6616_zero__and__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),A) = zero_zero_int ).

% zero_and_eq
tff(fact_6617_zero__and__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,zero_zero_nat),A) = zero_zero_nat ).

% zero_and_eq
tff(fact_6618_and__zero__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),zero_zero_int) = zero_zero_int ).

% and_zero_eq
tff(fact_6619_and__zero__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),zero_zero_nat) = zero_zero_nat ).

% and_zero_eq
tff(fact_6620_finite__lessThan,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,set_ord_lessThan_nat(K))) ).

% finite_lessThan
tff(fact_6621_lessThan__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_ord_lessThan_rat(X)),set_ord_lessThan_rat(Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_6622_lessThan__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_ord_lessThan_num(X)),set_ord_lessThan_num(Y)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_6623_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_ord_lessThan_nat(X)),set_ord_lessThan_nat(Y)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_6624_lessThan__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_lessThan_int(X)),set_ord_lessThan_int(Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_6625_lessThan__subset__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or5984915006950818249n_real(X)),set_or5984915006950818249n_real(Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_6626_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),zero_zero_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(ord_less_int,L),zero_zero_int)) ) ) ).

% and_negative_int_iff
tff(fact_6627_lessThan__0,axiom,
    set_ord_lessThan_nat(zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_6628_single__Diff__lessThan,axiom,
    ! [K: nat] : minus_minus_set_nat(insert_nat(K,bot_bot_set_nat),set_ord_lessThan_nat(K)) = insert_nat(K,bot_bot_set_nat) ).

% single_Diff_lessThan
tff(fact_6629_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_6630_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_6631_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = zero_zero_int ).

% and_numerals(1)
tff(fact_6632_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = zero_zero_nat ).

% and_numerals(1)
tff(fact_6633_and__numerals_I5_J,axiom,
    ! [X: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),one_one_int) = zero_zero_int ).

% and_numerals(5)
tff(fact_6634_and__numerals_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),one_one_nat) = zero_zero_nat ).

% and_numerals(5)
tff(fact_6635_lessThan__non__empty,axiom,
    ! [X: int] : set_ord_lessThan_int(X) != bot_bot_set_int ).

% lessThan_non_empty
tff(fact_6636_lessThan__non__empty,axiom,
    ! [X: real] : set_or5984915006950818249n_real(X) != bot_bot_set_real ).

% lessThan_non_empty
tff(fact_6637_infinite__Iio,axiom,
    ! [A: int] : ~ pp(aa_set_int_bool(finite_finite_int,set_ord_lessThan_int(A))) ).

% infinite_Iio
tff(fact_6638_infinite__Iio,axiom,
    ! [A: real] : ~ pp(aa_set_real_bool(finite_finite_real,set_or5984915006950818249n_real(A))) ).

% infinite_Iio
tff(fact_6639_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_my,U)) ).

% lessThan_def
tff(fact_6640_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_mz,U)) ).

% lessThan_def
tff(fact_6641_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_bool(aTP_Lamm_na,U)) ).

% lessThan_def
tff(fact_6642_lessThan__def,axiom,
    ! [U: nat] : set_ord_lessThan_nat(U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bd,U)) ).

% lessThan_def
tff(fact_6643_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_nb,U)) ).

% lessThan_def
tff(fact_6644_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_nc,U)) ).

% lessThan_def
tff(fact_6645_Iio__eq__empty__iff,axiom,
    ! [N: nat] :
      ( ( set_ord_lessThan_nat(N) = bot_bot_set_nat )
    <=> ( N = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
tff(fact_6646_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_6647_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_6648_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,set_ord_lessThan_nat(M)),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_6649_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_6650_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_6651_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( set_ord_lessThan_nat(N) = bot_bot_set_nat )
    <=> ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_6652_finite__nat__bounded,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ? [K2: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),set_ord_lessThan_nat(K2))) ) ).

% finite_nat_bounded
tff(fact_6653_finite__nat__iff__bounded,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
    <=> ? [K3: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),set_ord_lessThan_nat(K3))) ) ).

% finite_nat_iff_bounded
tff(fact_6654_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),Z)) ) ) ).

% AND_upper2''
tff(fact_6655_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Z)) ) ) ).

% AND_upper1''
tff(fact_6656_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(bit_se725231765392027082nd_int(K),L)),K)) ) ).

% and_less_eq
tff(fact_6657_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_ky(Q),P),set_ord_lessThan_int(N)) ) ) ).

% sum_diff_distrib
tff(fact_6658_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_kw(Q),P),set_or5984915006950818249n_real(N)) ) ) ).

% sum_diff_distrib
tff(fact_6659_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),set_ord_lessThan_nat(N))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),set_ord_lessThan_nat(N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kz(Q),P)),set_ord_lessThan_nat(N)) ) ) ).

% sum_diff_distrib
tff(fact_6660_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,set_ord_lessThan_nat(aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_lk(G),set_ord_lessThan_nat(N))) ).

% sum.lessThan_Suc_shift
tff(fact_6661_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,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_ll(G),set_ord_lessThan_nat(N))) ).

% sum.lessThan_Suc_shift
tff(fact_6662_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),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_lm(G)),set_ord_lessThan_nat(N))) ).

% sum.lessThan_Suc_shift
tff(fact_6663_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,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_ln(G),set_ord_lessThan_nat(N))) ).

% sum.lessThan_Suc_shift
tff(fact_6664_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_nd(F),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_6665_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_ne(F),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_6666_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,M: nat] : groups6591440286371151544t_real(aTP_Lamm_nf(F),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_6667_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_lo(F),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_6668_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_lp(F),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_6669_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,M: nat] : groups6591440286371151544t_real(aTP_Lamm_lq(F),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_6670_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_lm(G)),set_ord_lessThan_nat(N)) ).

% sum.atLeast1_atMost_eq
tff(fact_6671_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_ln(G),set_ord_lessThan_nat(N)) ).

% sum.atLeast1_atMost_eq
tff(fact_6672_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_rat,K4: 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)),K4)) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),K4))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),K4))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6673_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_int,K4: 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)),K4)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K4))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,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)),K4))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6674_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_nat,K4: 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)),K4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),K4))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),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)),K4))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6675_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_real,K4: 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)),K4)) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),K4))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(F,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)),K4))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6676_norm__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,X)),zero_zero_real))
    <=> ( X = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_6677_norm__le__zero__iff,axiom,
    ! [X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),zero_zero_real))
    <=> ( X = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_6678_zero__less__norm__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,X)))
    <=> ( X != zero_zero_real ) ) ).

% zero_less_norm_iff
tff(fact_6679_zero__less__norm__iff,axiom,
    ! [X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,X)))
    <=> ( X != zero_zero_complex ) ) ).

% zero_less_norm_iff
tff(fact_6680_norm__eq__zero,axiom,
    ! [X: real] :
      ( ( aa_real_real(real_V7735802525324610683m_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% norm_eq_zero
tff(fact_6681_norm__eq__zero,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,X) = zero_zero_real )
    <=> ( X = zero_zero_complex ) ) ).

% norm_eq_zero
tff(fact_6682_norm__zero,axiom,
    aa_real_real(real_V7735802525324610683m_real,zero_zero_real) = zero_zero_real ).

% norm_zero
tff(fact_6683_norm__zero,axiom,
    aa_complex_real(real_V1022390504157884413omplex,zero_zero_complex) = zero_zero_real ).

% norm_zero
tff(fact_6684_power__eq__1__iff,axiom,
    ! [W2: real,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W2),N) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W2) = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_6685_power__eq__1__iff,axiom,
    ! [W2: complex,N: nat] :
      ( ( aa_nat_complex(power_power_complex(W2),N) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W2) = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_6686_norm__diff__triangle__less,axiom,
    ! [X: real,Y: 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,X),Y))),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,Y),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,X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_less
tff(fact_6687_norm__diff__triangle__less,axiom,
    ! [X: complex,Y: 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(minus_minus_complex(X),Y))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_less
tff(fact_6688_and__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_6689_and__nat__numerals_I1_J,axiom,
    ! [Y: 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,Y))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_6690_and__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_6691_and__nat__numerals_I2_J,axiom,
    ! [Y: 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,Y))) = one_one_nat ).

% and_nat_numerals(2)
tff(fact_6692_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_6693_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_6694_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_6695_nonzero__norm__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ) ) ).

% nonzero_norm_divide
tff(fact_6696_nonzero__norm__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ) ) ).

% nonzero_norm_divide
tff(fact_6697_norm__not__less__zero,axiom,
    ! [X: complex] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),zero_zero_real)) ).

% norm_not_less_zero
tff(fact_6698_power__eq__imp__eq__norm,axiom,
    ! [W2: real,N: nat,Z: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W2),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,W2) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_6699_power__eq__imp__eq__norm,axiom,
    ! [W2: complex,N: nat,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W2),N) = aa_nat_complex(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,W2) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_6700_norm__mult__less,axiom,
    ! [X: real,R2: real,Y: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S))
       => 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,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S))) ) ) ).

% norm_mult_less
tff(fact_6701_norm__mult__less,axiom,
    ! [X: complex,R2: real,Y: complex,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S))) ) ) ).

% norm_mult_less
tff(fact_6702_norm__triangle__lt,axiom,
    ! [X: real,Y: real,E2: 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,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E2))
     => 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,X),Y))),E2)) ) ).

% norm_triangle_lt
tff(fact_6703_norm__triangle__lt,axiom,
    ! [X: complex,Y: complex,E2: 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,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E2)) ) ).

% norm_triangle_lt
tff(fact_6704_norm__add__less,axiom,
    ! [X: real,R2: real,Y: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S))
       => 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,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S))) ) ) ).

% norm_add_less
tff(fact_6705_norm__add__less,axiom,
    ! [X: complex,R2: real,Y: complex,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S))) ) ) ).

% norm_add_less
tff(fact_6706_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X),Xa2) = Y )
     => ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa2)))
       => ~ ( ( ( ( pp(aa_set_int_bool(member_int(X),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(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => ( Y = 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))),X))),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))),Xa2))))) ) )
              & ( ~ ( pp(aa_set_int_bool(member_int(X),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(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => ( Y = 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))),X))),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))),Xa2))))),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_se725231765392027082nd_int(aa_int_int(divide_divide_int(X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(Xa2),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(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa2))) ) ) ) ).

% and_int.pelims
tff(fact_6707_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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(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(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(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_6708_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_ng,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_6709_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(aTP_Lamm_nh(Z)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% geometric_deriv_sums
tff(fact_6710_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_6711_polyfun__diff,axiom,
    ! [N: nat,A: fun_nat_complex,X: complex,Y: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_complex_complex(minus_minus_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ni(A),X),set_ord_atMost_nat(N))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ni(A),Y),set_ord_atMost_nat(N))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_nk(N),A),X),Y),set_ord_lessThan_nat(N))) ) ) ).

% polyfun_diff
tff(fact_6712_polyfun__diff,axiom,
    ! [N: nat,A: fun_nat_rat,X: rat,Y: 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_nl(A),X),set_ord_atMost_nat(N))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_nl(A),Y),set_ord_atMost_nat(N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_nn(N),A),X),Y),set_ord_lessThan_nat(N))) ) ) ).

% polyfun_diff
tff(fact_6713_polyfun__diff,axiom,
    ! [N: nat,A: fun_nat_int,X: int,Y: 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_no(A),X),set_ord_atMost_nat(N))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_no(A),Y),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,X),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_nq(N),A),X),Y),set_ord_lessThan_nat(N))) ) ) ).

% polyfun_diff
tff(fact_6714_polyfun__diff,axiom,
    ! [N: nat,A: fun_nat_real,X: real,Y: 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_nr(A),X),set_ord_atMost_nat(N))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nr(A),Y),set_ord_atMost_nat(N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_nt(N),A),X),Y),set_ord_lessThan_nat(N))) ) ) ).

% polyfun_diff
tff(fact_6715_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(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(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(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),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_6716_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_6717_atMost__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(I),set_or4236626031148496127et_nat(K)))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,I),K)) ) ).

% atMost_iff
tff(fact_6718_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_6719_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_6720_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_6721_atMost__iff,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),set_ord_atMost_nat(K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K)) ) ).

% atMost_iff
tff(fact_6722_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_6723_sgn__0,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% sgn_0
tff(fact_6724_sgn__0,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_0
tff(fact_6725_sgn__0,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_0
tff(fact_6726_sgn__0,axiom,
    aa_rat_rat(sgn_sgn_rat,zero_zero_rat) = zero_zero_rat ).

% sgn_0
tff(fact_6727_sgn__0,axiom,
    aa_int_int(sgn_sgn_int,zero_zero_int) = zero_zero_int ).

% sgn_0
tff(fact_6728_sgn__zero,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_zero
tff(fact_6729_sgn__zero,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_zero
tff(fact_6730_finite__atMost,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,set_ord_atMost_nat(K))) ).

% finite_atMost
tff(fact_6731_sgn__greater,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% sgn_greater
tff(fact_6732_sgn__greater,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sgn_sgn_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% sgn_greater
tff(fact_6733_sgn__greater,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(sgn_sgn_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% sgn_greater
tff(fact_6734_sgn__greater,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(sgn_sgn_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% sgn_greater
tff(fact_6735_sgn__less,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% sgn_less
tff(fact_6736_sgn__less,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sgn_sgn_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% sgn_less
tff(fact_6737_sgn__less,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(sgn_sgn_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% sgn_less
tff(fact_6738_sgn__less,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(sgn_sgn_int,A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% sgn_less
tff(fact_6739_atMost__subset__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ord_le4403425263959731960et_int(set_or58775011639299419et_int(X),set_or58775011639299419et_int(Y))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(X),Y)) ) ).

% atMost_subset_iff
tff(fact_6740_atMost__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_ord_atMost_rat(X)),set_ord_atMost_rat(Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% atMost_subset_iff
tff(fact_6741_atMost__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_ord_atMost_num(X)),set_ord_atMost_num(Y)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% atMost_subset_iff
tff(fact_6742_atMost__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_ord_atMost_nat(X)),set_ord_atMost_nat(Y)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% atMost_subset_iff
tff(fact_6743_atMost__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_atMost_int(X)),set_ord_atMost_int(Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% atMost_subset_iff
tff(fact_6744_sgn__pos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = one_one_Code_integer ) ) ).

% sgn_pos
tff(fact_6745_sgn__pos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( aa_real_real(sgn_sgn_real,A) = one_one_real ) ) ).

% sgn_pos
tff(fact_6746_sgn__pos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat ) ) ).

% sgn_pos
tff(fact_6747_sgn__pos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( aa_int_int(sgn_sgn_int,A) = one_one_int ) ) ).

% sgn_pos
tff(fact_6748_Icc__subset__Iic__iff,axiom,
    ! [L: set_int,H: set_int,H2: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(L,H),set_or58775011639299419et_int(H2))
    <=> ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(L),H))
        | pp(aa_set_int_bool(ord_less_eq_set_int(H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6749_Icc__subset__Iic__iff,axiom,
    ! [L: rat,H: rat,H2: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or633870826150836451st_rat(L,H)),set_ord_atMost_rat(H2)))
    <=> ( ~ 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),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6750_Icc__subset__Iic__iff,axiom,
    ! [L: num,H: num,H2: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_or7049704709247886629st_num(L,H)),set_ord_atMost_num(H2)))
    <=> ( ~ 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),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6751_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H: nat,H2: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_or1269000886237332187st_nat(L,H)),set_ord_atMost_nat(H2)))
    <=> ( ~ 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),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6752_Icc__subset__Iic__iff,axiom,
    ! [L: int,H: int,H2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),H)),set_ord_atMost_int(H2)))
    <=> ( ~ 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),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6753_Icc__subset__Iic__iff,axiom,
    ! [L: real,H: real,H2: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or1222579329274155063t_real(L,H)),set_ord_atMost_real(H2)))
    <=> ( ~ 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),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6754_sgn__mult__self__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = zero_n356916108424825756nteger(aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,A),zero_z3403309356797280102nteger))) ).

% sgn_mult_self_eq
tff(fact_6755_sgn__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,A)) = zero_n3304061248610475627l_real(aa_bool_bool(fNot,aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,A),zero_zero_real))) ).

% sgn_mult_self_eq
tff(fact_6756_sgn__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(sgn_sgn_rat,A)) = zero_n2052037380579107095ol_rat(aa_bool_bool(fNot,aa_rat_bool(aa_rat_fun_rat_bool(fequal_rat,A),zero_zero_rat))) ).

% sgn_mult_self_eq
tff(fact_6757_sgn__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,A)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,A),zero_zero_int))) ).

% sgn_mult_self_eq
tff(fact_6758_atMost__0,axiom,
    set_ord_atMost_nat(zero_zero_nat) = insert_nat(zero_zero_nat,bot_bot_set_nat) ).

% atMost_0
tff(fact_6759_sgn__neg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% sgn_neg
tff(fact_6760_sgn__neg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% sgn_neg
tff(fact_6761_sgn__neg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% sgn_neg
tff(fact_6762_sgn__neg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% sgn_neg
tff(fact_6763_sgn__of__nat,axiom,
    ! [N: nat] : aa_rat_rat(sgn_sgn_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) = zero_n2052037380579107095ol_rat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% sgn_of_nat
tff(fact_6764_sgn__of__nat,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,N)) = zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% sgn_of_nat
tff(fact_6765_sgn__of__nat,axiom,
    ! [N: nat] : aa_real_real(sgn_sgn_real,aa_nat_real(semiri5074537144036343181t_real,N)) = zero_n3304061248610475627l_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% sgn_of_nat
tff(fact_6766_sgn__of__nat,axiom,
    ! [N: nat] : aa_int_int(sgn_sgn_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% sgn_of_nat
tff(fact_6767_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_complex,X: complex] :
      ( pp(aa_complex_bool(sums_complex(aTP_Lamm_gv(A)),X))
    <=> ( aa_nat_complex(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_6768_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_real,X: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_gx(A)),X))
    <=> ( aa_nat_real(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_6769_sgn__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_eq_0_iff
tff(fact_6770_sgn__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(sgn_sgn_complex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% sgn_eq_0_iff
tff(fact_6771_sgn__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% sgn_eq_0_iff
tff(fact_6772_sgn__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% sgn_eq_0_iff
tff(fact_6773_sgn__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% sgn_eq_0_iff
tff(fact_6774_sgn__0__0,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_0_0
tff(fact_6775_sgn__0__0,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% sgn_0_0
tff(fact_6776_sgn__0__0,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% sgn_0_0
tff(fact_6777_sgn__0__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% sgn_0_0
tff(fact_6778_sgn__zero__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(sgn_sgn_complex,X) = zero_zero_complex )
    <=> ( X = zero_zero_complex ) ) ).

% sgn_zero_iff
tff(fact_6779_sgn__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sgn_sgn_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% sgn_zero_iff
tff(fact_6780_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: real] : bot_bot_set_real != set_ord_atMost_real(H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_6781_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: nat] : bot_bot_set_nat != set_ord_atMost_nat(H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_6782_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: int] : bot_bot_set_int != set_ord_atMost_int(H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_6783_infinite__Iic,axiom,
    ! [A: int] : ~ pp(aa_set_int_bool(finite_finite_int,set_ord_atMost_int(A))) ).

% infinite_Iic
tff(fact_6784_atMost__def,axiom,
    ! [U: real] : set_ord_atMost_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_nu,U)) ).

% atMost_def
tff(fact_6785_atMost__def,axiom,
    ! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_ax,U)) ).

% atMost_def
tff(fact_6786_atMost__def,axiom,
    ! [U: set_int] : set_or58775011639299419et_int(U) = collect_set_int(aTP_Lamm_ba(U)) ).

% atMost_def
tff(fact_6787_atMost__def,axiom,
    ! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_nv,U)) ).

% atMost_def
tff(fact_6788_atMost__def,axiom,
    ! [U: num] : set_ord_atMost_num(U) = collect_num(aa_num_fun_num_bool(aTP_Lamm_nw,U)) ).

% atMost_def
tff(fact_6789_atMost__def,axiom,
    ! [U: nat] : set_ord_atMost_nat(U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_be,U)) ).

% atMost_def
tff(fact_6790_atMost__def,axiom,
    ! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_nx,U)) ).

% atMost_def
tff(fact_6791_sgn__not__eq__imp,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) != aa_int_int(sgn_sgn_int,A) )
     => ( ( aa_int_int(sgn_sgn_int,A) != zero_zero_int )
       => ( ( aa_int_int(sgn_sgn_int,B) != zero_zero_int )
         => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,aa_int_int(sgn_sgn_int,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_6792_sgn__not__eq__imp,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B) != aa_real_real(sgn_sgn_real,A) )
     => ( ( aa_real_real(sgn_sgn_real,A) != zero_zero_real )
       => ( ( aa_real_real(sgn_sgn_real,B) != zero_zero_real )
         => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,aa_real_real(sgn_sgn_real,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_6793_sgn__not__eq__imp,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,B) != aa_rat_rat(sgn_sgn_rat,A) )
     => ( ( aa_rat_rat(sgn_sgn_rat,A) != zero_zero_rat )
       => ( ( aa_rat_rat(sgn_sgn_rat,B) != zero_zero_rat )
         => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(sgn_sgn_rat,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_6794_sgn__not__eq__imp,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) != aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) )
     => ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) != zero_z3403309356797280102nteger )
       => ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) != zero_z3403309356797280102nteger )
         => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_6795_atMost__atLeast0,axiom,
    ! [N: nat] : set_ord_atMost_nat(N) = set_or1269000886237332187st_nat(zero_zero_nat,N) ).

% atMost_atLeast0
tff(fact_6796_not__Iic__le__Icc,axiom,
    ! [H: int,L2: int,H2: int] : ~ pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_atMost_int(H)),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L2),H2))) ).

% not_Iic_le_Icc
tff(fact_6797_not__Iic__le__Icc,axiom,
    ! [H: real,L2: real,H2: real] : ~ pp(aa_set_real_bool(ord_less_eq_set_real(set_ord_atMost_real(H)),set_or1222579329274155063t_real(L2,H2))) ).

% not_Iic_le_Icc
tff(fact_6798_finite__nat__iff__bounded__le,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
    <=> ? [K3: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),set_ord_atMost_nat(K3))) ) ).

% finite_nat_iff_bounded_le
tff(fact_6799_sgn__1__pos,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = one_one_Code_integer )
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% sgn_1_pos
tff(fact_6800_sgn__1__pos,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = one_one_real )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% sgn_1_pos
tff(fact_6801_sgn__1__pos,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat )
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% sgn_1_pos
tff(fact_6802_sgn__1__pos,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = one_one_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% sgn_1_pos
tff(fact_6803_Iic__subset__Iio__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_ord_atMost_rat(A)),set_ord_lessThan_rat(B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_6804_Iic__subset__Iio__iff,axiom,
    ! [A: num,B: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_ord_atMost_num(A)),set_ord_lessThan_num(B)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_6805_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_ord_atMost_nat(A)),set_ord_lessThan_nat(B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_6806_Iic__subset__Iio__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_atMost_int(A)),set_ord_lessThan_int(B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_6807_Iic__subset__Iio__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_ord_atMost_real(A)),set_or5984915006950818249n_real(B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_6808_powser__sums__zero,axiom,
    ! [A: fun_nat_complex] : pp(aa_complex_bool(sums_complex(aTP_Lamm_gv(A)),aa_nat_complex(A,zero_zero_nat))) ).

% powser_sums_zero
tff(fact_6809_powser__sums__zero,axiom,
    ! [A: fun_nat_real] : pp(aa_real_bool(sums_real(aTP_Lamm_gx(A)),aa_nat_real(A,zero_zero_nat))) ).

% powser_sums_zero
tff(fact_6810_sgn__1__neg,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% sgn_1_neg
tff(fact_6811_sgn__1__neg,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% sgn_1_neg
tff(fact_6812_sgn__1__neg,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% sgn_1_neg
tff(fact_6813_sgn__1__neg,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% sgn_1_neg
tff(fact_6814_sgn__if,axiom,
    ! [X: int] :
      ( ( ( X = zero_zero_int )
       => ( aa_int_int(sgn_sgn_int,X) = zero_zero_int ) )
      & ( ( X != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
           => ( aa_int_int(sgn_sgn_int,X) = one_one_int ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
           => ( aa_int_int(sgn_sgn_int,X) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ) ).

% sgn_if
tff(fact_6815_sgn__if,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( aa_real_real(sgn_sgn_real,X) = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
           => ( aa_real_real(sgn_sgn_real,X) = one_one_real ) )
          & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
           => ( aa_real_real(sgn_sgn_real,X) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ) ).

% sgn_if
tff(fact_6816_sgn__if,axiom,
    ! [X: rat] :
      ( ( ( X = zero_zero_rat )
       => ( aa_rat_rat(sgn_sgn_rat,X) = zero_zero_rat ) )
      & ( ( X != zero_zero_rat )
       => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
           => ( aa_rat_rat(sgn_sgn_rat,X) = one_one_rat ) )
          & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
           => ( aa_rat_rat(sgn_sgn_rat,X) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ) ) ).

% sgn_if
tff(fact_6817_sgn__if,axiom,
    ! [X: code_integer] :
      ( ( ( X = zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X) = zero_z3403309356797280102nteger ) )
      & ( ( X != zero_z3403309356797280102nteger )
       => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),X))
           => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X) = one_one_Code_integer ) )
          & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),X))
           => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ) ) ).

% sgn_if
tff(fact_6818_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_6819_norm__sgn,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(sgn_sgn_real,X)) = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(sgn_sgn_real,X)) = one_one_real ) ) ) ).

% norm_sgn
tff(fact_6820_norm__sgn,axiom,
    ! [X: complex] :
      ( ( ( X = zero_zero_complex )
       => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(sgn_sgn_complex,X)) = zero_zero_real ) )
      & ( ( X != zero_zero_complex )
       => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(sgn_sgn_complex,X)) = one_one_real ) ) ) ).

% norm_sgn
tff(fact_6821_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,set_ord_atMost_nat(aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_lk(G),set_ord_atMost_nat(N))) ).

% sum.atMost_Suc_shift
tff(fact_6822_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,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_ll(G),set_ord_atMost_nat(N))) ).

% sum.atMost_Suc_shift
tff(fact_6823_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),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_lm(G)),set_ord_atMost_nat(N))) ).

% sum.atMost_Suc_shift
tff(fact_6824_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,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_ln(G),set_ord_atMost_nat(N))) ).

% sum.atMost_Suc_shift
tff(fact_6825_sum__telescope,axiom,
    ! [F: fun_nat_rat,I: nat] : groups2906978787729119204at_rat(aTP_Lamm_nd(F),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_6826_sum__telescope,axiom,
    ! [F: fun_nat_int,I: nat] : groups3539618377306564664at_int(aTP_Lamm_ne(F),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_6827_sum__telescope,axiom,
    ! [F: fun_nat_real,I: nat] : groups6591440286371151544t_real(aTP_Lamm_nf(F),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_6828_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_complex,N: nat,D: fun_nat_complex] :
      ( ! [X3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ni(C),X3),set_ord_atMost_nat(N)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ni(D),X3),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(C,I4) = aa_nat_complex(D,I4) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_6829_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_real,N: nat,D: fun_nat_real] :
      ( ! [X3: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nr(C),X3),set_ord_atMost_nat(N)) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nr(D),X3),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(C,I4) = aa_nat_real(D,I4) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_6830_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_6831_binomial__mono,axiom,
    ! [K: nat,K5: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K5))
     => ( 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))),K5)),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),K5))) ) ) ).

% binomial_mono
tff(fact_6832_binomial__antimono,axiom,
    ! [K: nat,K5: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K5))
     => ( 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,K5),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K5)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ) ) ).

% binomial_antimono
tff(fact_6833_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_6834_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_complex,N: nat,K: nat] :
      ( ! [W: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ni(C),W),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(C,K) = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_6835_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_real,N: nat,K: nat] :
      ( ! [W: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nr(C),W),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(C,K) = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_6836_polyfun__eq__0,axiom,
    ! [C: fun_nat_complex,N: nat] :
      ( ! [X3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ni(C),X3),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(C,I4) = zero_zero_complex ) ) ) ).

% polyfun_eq_0
tff(fact_6837_polyfun__eq__0,axiom,
    ! [C: fun_nat_real,N: nat] :
      ( ! [X3: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nr(C),X3),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(C,I4) = zero_zero_real ) ) ) ).

% polyfun_eq_0
tff(fact_6838_sum_OatMost__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,set_ord_atMost_nat(N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_lk(G),set_ord_lessThan_nat(N))) ).

% sum.atMost_shift
tff(fact_6839_sum_OatMost__shift,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,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_ll(G),set_ord_lessThan_nat(N))) ).

% sum.atMost_shift
tff(fact_6840_sum_OatMost__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),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_lm(G)),set_ord_lessThan_nat(N))) ).

% sum.atMost_shift
tff(fact_6841_sum_OatMost__shift,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,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_ln(G),set_ord_lessThan_nat(N))) ).

% sum.atMost_shift
tff(fact_6842_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N) = minus_minus_set_nat(set_ord_atMost_nat(N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_6843_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ny(N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_oa(G)),set_ord_atMost_nat(N)) ).

% sum.triangle_reindex_eq
tff(fact_6844_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_real,N: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ny(N)))) = groups6591440286371151544t_real(aTP_Lamm_oc(G),set_ord_atMost_nat(N)) ).

% sum.triangle_reindex_eq
tff(fact_6845_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_6846_binomial__strict__antimono,axiom,
    ! [K: nat,K5: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K5))
     => ( 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,K5),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K5)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ) ) ).

% binomial_strict_antimono
tff(fact_6847_binomial__strict__mono,axiom,
    ! [K: nat,K5: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K5))
     => ( 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))),K5)),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),K5))) ) ) ).

% binomial_strict_mono
tff(fact_6848_polyfun__finite__roots,axiom,
    ! [C: fun_nat_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_od(C),N))))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
          & ( aa_nat_complex(C,I4) != zero_zero_complex ) ) ) ).

% polyfun_finite_roots
tff(fact_6849_polyfun__finite__roots,axiom,
    ! [C: fun_nat_real,N: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aTP_Lamm_oe(C),N))))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
          & ( aa_nat_real(C,I4) != zero_zero_real ) ) ) ).

% polyfun_finite_roots
tff(fact_6850_polyfun__roots__finite,axiom,
    ! [C: fun_nat_complex,K: nat,N: nat] :
      ( ( aa_nat_complex(C,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_od(C),N)))) ) ) ).

% polyfun_roots_finite
tff(fact_6851_polyfun__roots__finite,axiom,
    ! [C: fun_nat_real,K: nat,N: nat] :
      ( ( aa_nat_real(C,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_oe(C),N)))) ) ) ).

% polyfun_roots_finite
tff(fact_6852_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_complex,A: complex,N: nat] :
      ( ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ni(C),A),set_ord_atMost_nat(N)) = zero_zero_complex )
     => ~ ! [B3: fun_nat_complex] :
            ~ ! [Z4: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ni(C),Z4),set_ord_atMost_nat(N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z4),A)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ni(B3),Z4),set_ord_lessThan_nat(N))) ) ).

% polyfun_linear_factor_root
tff(fact_6853_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_rat,A: rat,N: nat] :
      ( ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_nl(C),A),set_ord_atMost_nat(N)) = zero_zero_rat )
     => ~ ! [B3: fun_nat_rat] :
            ~ ! [Z4: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_nl(C),Z4),set_ord_atMost_nat(N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(Z4),A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_nl(B3),Z4),set_ord_lessThan_nat(N))) ) ).

% polyfun_linear_factor_root
tff(fact_6854_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_int,A: int,N: nat] :
      ( ( groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_no(C),A),set_ord_atMost_nat(N)) = zero_zero_int )
     => ~ ! [B3: fun_nat_int] :
            ~ ! [Z4: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_no(C),Z4),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),A)),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_no(B3),Z4),set_ord_lessThan_nat(N))) ) ).

% polyfun_linear_factor_root
tff(fact_6855_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_real,A: real,N: nat] :
      ( ( groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nr(C),A),set_ord_atMost_nat(N)) = zero_zero_real )
     => ~ ! [B3: fun_nat_real] :
            ~ ! [Z4: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nr(C),Z4),set_ord_atMost_nat(N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z4),A)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nr(B3),Z4),set_ord_lessThan_nat(N))) ) ).

% polyfun_linear_factor_root
tff(fact_6856_sum__power__shift,axiom,
    ! [M: nat,N: nat,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2073611262835488442omplex(power_power_complex(X),set_or1269000886237332187st_nat(M,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),groups2073611262835488442omplex(power_power_complex(X),set_ord_atMost_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_6857_sum__power__shift,axiom,
    ! [M: nat,N: nat,X: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2906978787729119204at_rat(power_power_rat(X),set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),M)),groups2906978787729119204at_rat(power_power_rat(X),set_ord_atMost_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_6858_sum__power__shift,axiom,
    ! [M: nat,N: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups3539618377306564664at_int(power_power_int(X),set_or1269000886237332187st_nat(M,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),M)),groups3539618377306564664at_int(power_power_int(X),set_ord_atMost_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_6859_sum__power__shift,axiom,
    ! [M: nat,N: nat,X: 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,X),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,X),M)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),set_ord_atMost_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_6860_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_of(N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_oa(G)),set_ord_lessThan_nat(N)) ).

% sum.triangle_reindex
tff(fact_6861_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_real,N: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_of(N)))) = groups6591440286371151544t_real(aTP_Lamm_oc(G),set_ord_lessThan_nat(N)) ).

% sum.triangle_reindex
tff(fact_6862_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_complex,N: nat,B: fun_nat_complex,X: complex] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I2))
         => ( aa_nat_complex(A,I2) = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_complex(B,J2) = zero_zero_complex ) )
       => ( aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ni(A),X),set_ord_atMost_nat(M))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ni(B),X),set_ord_atMost_nat(N))) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_oh(A),B),X),set_ord_atMost_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product
tff(fact_6863_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_rat,N: nat,B: fun_nat_rat,X: rat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I2))
         => ( aa_nat_rat(A,I2) = zero_zero_rat ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_rat(B,J2) = zero_zero_rat ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_nl(A),X),set_ord_atMost_nat(M))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_nl(B),X),set_ord_atMost_nat(N))) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_oj(A),B),X),set_ord_atMost_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product
tff(fact_6864_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_int,N: nat,B: fun_nat_int,X: int] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I2))
         => ( aa_nat_int(A,I2) = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_int(B,J2) = zero_zero_int ) )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_no(A),X),set_ord_atMost_nat(M))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_no(B),X),set_ord_atMost_nat(N))) = groups3539618377306564664at_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_ol(A),B),X),set_ord_atMost_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product
tff(fact_6865_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_real,N: nat,B: fun_nat_real,X: real] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I2))
         => ( aa_nat_real(A,I2) = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_real(B,J2) = zero_zero_real ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nr(A),X),set_ord_atMost_nat(M))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nr(B),X),set_ord_atMost_nat(N))) = groups6591440286371151544t_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_on(A),B),X),set_ord_atMost_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product
tff(fact_6866_polyfun__eq__const,axiom,
    ! [C: fun_nat_complex,N: nat,K: complex] :
      ( ! [X3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ni(C),X3),set_ord_atMost_nat(N)) = K
    <=> ( ( aa_nat_complex(C,zero_zero_nat) = K )
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),set_or1269000886237332187st_nat(one_one_nat,N)))
           => ( aa_nat_complex(C,X3) = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
tff(fact_6867_polyfun__eq__const,axiom,
    ! [C: fun_nat_real,N: nat,K: real] :
      ( ! [X3: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nr(C),X3),set_ord_atMost_nat(N)) = K
    <=> ( ( aa_nat_real(C,zero_zero_nat) = K )
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),set_or1269000886237332187st_nat(one_one_nat,N)))
           => ( aa_nat_real(C,X3) = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
tff(fact_6868_polynomial__product__nat,axiom,
    ! [M: nat,A: fun_nat_nat,N: nat,B: fun_nat_nat,X: nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I2))
         => ( aa_nat_nat(A,I2) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_nat(B,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_oo(A),X)),set_ord_atMost_nat(M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oo(B),X)),set_ord_atMost_nat(N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_oq(A),B),X)),set_ord_atMost_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product_nat
tff(fact_6869_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(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1)))
     => ( ! [K2: int,L3: int] :
            ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K2),L3)))
           => ( ( ~ ( 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(L3),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(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P,K2),L3)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).

% and_int.pinduct
tff(fact_6870_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_or(K),G),H),set_ord_atMost_nat(P2)) = groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_os(K),G),H),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_6871_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_ot(K),G),H),set_ord_atMost_nat(P2)) = groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ou(K),G),H),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_6872_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_ov(K),G),H),set_ord_atMost_nat(P2)) = groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ow(K),G),H),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_6873_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_ox(K),G),H)),set_ord_atMost_nat(P2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_oy(K),G),H)),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_6874_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_oz(K),G),H),set_ord_atMost_nat(P2)) = groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_pa(K),G),H),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_6875_root__polyfun,axiom,
    ! [N: nat,Z: complex,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_complex(power_power_complex(Z),N) = A )
      <=> ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_pb(N),Z),A),set_ord_atMost_nat(N)) = zero_zero_complex ) ) ) ).

% root_polyfun
tff(fact_6876_root__polyfun,axiom,
    ! [N: nat,Z: int,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_int(power_power_int(Z),N) = A )
      <=> ( groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_pc(N),Z),A),set_ord_atMost_nat(N)) = zero_zero_int ) ) ) ).

% root_polyfun
tff(fact_6877_root__polyfun,axiom,
    ! [N: nat,Z: rat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_rat(power_power_rat(Z),N) = A )
      <=> ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_pd(N),Z),A),set_ord_atMost_nat(N)) = zero_zero_rat ) ) ) ).

% root_polyfun
tff(fact_6878_root__polyfun,axiom,
    ! [N: nat,Z: code_integer,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Z),N) = A )
      <=> ( groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_pe(N),Z),A),set_ord_atMost_nat(N)) = zero_z3403309356797280102nteger ) ) ) ).

% root_polyfun
tff(fact_6879_root__polyfun,axiom,
    ! [N: nat,Z: real,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N) = A )
      <=> ( groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_pf(N),Z),A),set_ord_atMost_nat(N)) = zero_zero_real ) ) ) ).

% root_polyfun
tff(fact_6880_polyfun__diff__alt,axiom,
    ! [N: nat,A: fun_nat_complex,X: complex,Y: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_complex_complex(minus_minus_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ni(A),X),set_ord_atMost_nat(N))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ni(A),Y),set_ord_atMost_nat(N))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_ph(N),A),X),Y),set_ord_lessThan_nat(N))) ) ) ).

% polyfun_diff_alt
tff(fact_6881_polyfun__diff__alt,axiom,
    ! [N: nat,A: fun_nat_rat,X: rat,Y: 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_nl(A),X),set_ord_atMost_nat(N))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_nl(A),Y),set_ord_atMost_nat(N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_pj(N),A),X),Y),set_ord_lessThan_nat(N))) ) ) ).

% polyfun_diff_alt
tff(fact_6882_polyfun__diff__alt,axiom,
    ! [N: nat,A: fun_nat_int,X: int,Y: 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_no(A),X),set_ord_atMost_nat(N))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_no(A),Y),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,X),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_pl(N),A),X),Y),set_ord_lessThan_nat(N))) ) ) ).

% polyfun_diff_alt
tff(fact_6883_polyfun__diff__alt,axiom,
    ! [N: nat,A: fun_nat_real,X: real,Y: 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_nr(A),X),set_ord_atMost_nat(N))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nr(A),Y),set_ord_atMost_nat(N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_pn(N),A),X),Y),set_ord_lessThan_nat(N))) ) ) ).

% polyfun_diff_alt
tff(fact_6884_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(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(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(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)),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_6885_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: fun_nat_complex,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
     => ? [M8: real] :
        ! [Z4: complex] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,M8),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_ni(C),Z4),set_ord_atMost_nat(N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,E2),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_6886_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: fun_nat_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
     => ? [M8: real] :
        ! [Z4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,M8),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_nr(C),Z4),set_ord_atMost_nat(N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,E2),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_6887_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(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aTP_Lamm_po(N),set_ord_atMost_nat(N))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_6888_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(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_pp(N),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_6889_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(aTP_Lamm_pq(N),set_ord_atMost_nat(N))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_6890_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_pr,N),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_6891_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(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aTP_Lamm_ps(N),set_ord_atMost_nat(N))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_6892_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(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_pt(N),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_6893_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(aTP_Lamm_pu(N),set_ord_atMost_nat(N))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_6894_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_pv,N),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_6895_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( groups2073611262835488442omplex(aTP_Lamm_pw(N),set_ord_atMost_nat(N)) = zero_zero_complex ) ) ).

% choose_alternating_sum
tff(fact_6896_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( groups2906978787729119204at_rat(aTP_Lamm_px(N),set_ord_atMost_nat(N)) = zero_zero_rat ) ) ).

% choose_alternating_sum
tff(fact_6897_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( groups7501900531339628137nteger(aTP_Lamm_py(N),set_ord_atMost_nat(N)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
tff(fact_6898_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( groups3539618377306564664at_int(aTP_Lamm_pz(N),set_ord_atMost_nat(N)) = zero_zero_int ) ) ).

% choose_alternating_sum
tff(fact_6899_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_qa,N),set_ord_atMost_nat(N)) = zero_zero_real ) ) ).

% choose_alternating_sum
tff(fact_6900_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( groups2073611262835488442omplex(aTP_Lamm_qb(N),set_ord_atMost_nat(N)) = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
tff(fact_6901_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( groups2906978787729119204at_rat(aTP_Lamm_qc(N),set_ord_atMost_nat(N)) = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
tff(fact_6902_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( groups7501900531339628137nteger(aTP_Lamm_qd(N),set_ord_atMost_nat(N)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
tff(fact_6903_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( groups3539618377306564664at_int(aTP_Lamm_qe(N),set_ord_atMost_nat(N)) = zero_zero_int ) ) ).

% choose_alternating_linear_sum
tff(fact_6904_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_qf,N),set_ord_atMost_nat(N)) = zero_zero_real ) ) ).

% choose_alternating_linear_sum
tff(fact_6905_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_6906_geometric__sums,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real))
     => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(power_power_real,C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),C)))) ) ).

% geometric_sums
tff(fact_6907_geometric__sums,axiom,
    ! [C: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real))
     => pp(aa_complex_bool(sums_complex(power_power_complex(C)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C)))) ) ).

% geometric_sums
tff(fact_6908_sums__zero,axiom,
    pp(aa_complex_bool(sums_complex(aTP_Lamm_qg),zero_zero_complex)) ).

% sums_zero
tff(fact_6909_sums__zero,axiom,
    pp(aa_real_bool(sums_real(aTP_Lamm_ft),zero_zero_real)) ).

% sums_zero
tff(fact_6910_sums__zero,axiom,
    pp(aa_nat_bool(sums_nat(aTP_Lamm_ck),zero_zero_nat)) ).

% sums_zero
tff(fact_6911_sums__zero,axiom,
    pp(aa_int_bool(sums_int(aTP_Lamm_qh),zero_zero_int)) ).

% sums_zero
tff(fact_6912_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_6913_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_6914_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_6915_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_6916_sgn__real__def,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( aa_real_real(sgn_sgn_real,A) = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
           => ( aa_real_real(sgn_sgn_real,A) = one_one_real ) )
          & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
           => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ) ).

% sgn_real_def
tff(fact_6917_sums__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,S: nat,T: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(G,N2)))
     => ( pp(aa_nat_bool(sums_nat(F),S))
       => ( pp(aa_nat_bool(sums_nat(G),T))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,S),T)) ) ) ) ).

% sums_le
tff(fact_6918_sums__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,S: int,T: int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(G,N2)))
     => ( pp(aa_int_bool(sums_int(F),S))
       => ( pp(aa_int_bool(sums_int(G),T))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,S),T)) ) ) ) ).

% sums_le
tff(fact_6919_sums__0,axiom,
    ! [F: fun_nat_complex] :
      ( ! [N2: nat] : aa_nat_complex(F,N2) = zero_zero_complex
     => pp(aa_complex_bool(sums_complex(F),zero_zero_complex)) ) ).

% sums_0
tff(fact_6920_sums__0,axiom,
    ! [F: fun_nat_real] :
      ( ! [N2: nat] : aa_nat_real(F,N2) = zero_zero_real
     => pp(aa_real_bool(sums_real(F),zero_zero_real)) ) ).

% sums_0
tff(fact_6921_sums__0,axiom,
    ! [F: fun_nat_nat] :
      ( ! [N2: nat] : aa_nat_nat(F,N2) = zero_zero_nat
     => pp(aa_nat_bool(sums_nat(F),zero_zero_nat)) ) ).

% sums_0
tff(fact_6922_sums__0,axiom,
    ! [F: fun_nat_int] :
      ( ! [N2: nat] : aa_nat_int(F,N2) = zero_zero_int
     => pp(aa_int_bool(sums_int(F),zero_zero_int)) ) ).

% sums_0
tff(fact_6923_sums__single,axiom,
    ! [I: nat,F: fun_nat_complex] : pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_gc(I),F)),aa_nat_complex(F,I))) ).

% sums_single
tff(fact_6924_sums__single,axiom,
    ! [I: nat,F: fun_nat_real] : pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_qi(I),F)),aa_nat_real(F,I))) ).

% sums_single
tff(fact_6925_sums__single,axiom,
    ! [I: nat,F: fun_nat_nat] : pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qj(I),F)),aa_nat_nat(F,I))) ).

% sums_single
tff(fact_6926_sums__single,axiom,
    ! [I: nat,F: fun_nat_int] : pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_qk(I),F)),aa_nat_int(F,I))) ).

% sums_single
tff(fact_6927_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_6928_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_6929_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_6930_sums__mult__iff,axiom,
    ! [C: complex,F: fun_nat_complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ql(C),F)),aa_complex_complex(times_times_complex(C),D)))
      <=> pp(aa_complex_bool(sums_complex(F),D)) ) ) ).

% sums_mult_iff
tff(fact_6931_sums__mult__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_qm(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D)))
      <=> pp(aa_real_bool(sums_real(F),D)) ) ) ).

% sums_mult_iff
tff(fact_6932_sums__mult2__iff,axiom,
    ! [C: complex,F: fun_nat_complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qn(C),F)),aa_complex_complex(times_times_complex(D),C)))
      <=> pp(aa_complex_bool(sums_complex(F),D)) ) ) ).

% sums_mult2_iff
tff(fact_6933_sums__mult2__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_qo(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,D),C)))
      <=> pp(aa_real_bool(sums_real(F),D)) ) ) ).

% sums_mult2_iff
tff(fact_6934_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_6935_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_6936_sums__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex,A: complex] :
      ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ql(C),F)),A))
     => ( ( C != zero_zero_complex )
       => pp(aa_complex_bool(sums_complex(F),aa_complex_complex(divide1717551699836669952omplex(A),C))) ) ) ).

% sums_mult_D
tff(fact_6937_sums__mult__D,axiom,
    ! [C: real,F: fun_nat_real,A: real] :
      ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_qm(C),F)),A))
     => ( ( C != zero_zero_real )
       => pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C))) ) ) ).

% sums_mult_D
tff(fact_6938_sums__Suc__imp,axiom,
    ! [F: fun_nat_complex,S: complex] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( pp(aa_complex_bool(sums_complex(aTP_Lamm_qp(F)),S))
       => pp(aa_complex_bool(sums_complex(F),S)) ) ) ).

% sums_Suc_imp
tff(fact_6939_sums__Suc__imp,axiom,
    ! [F: fun_nat_real,S: real] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aTP_Lamm_ln(F)),S))
       => pp(aa_real_bool(sums_real(F),S)) ) ) ).

% sums_Suc_imp
tff(fact_6940_sums__Suc,axiom,
    ! [F: fun_nat_real,L: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_ln(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_6941_sums__Suc,axiom,
    ! [F: fun_nat_nat,L: nat] :
      ( pp(aa_nat_bool(sums_nat(aTP_Lamm_lm(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_6942_sums__Suc,axiom,
    ! [F: fun_nat_int,L: int] :
      ( pp(aa_int_bool(sums_int(aTP_Lamm_ll(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_6943_sums__Suc__iff,axiom,
    ! [F: fun_nat_real,S: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_ln(F)),S))
    <=> pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S),aa_nat_real(F,zero_zero_nat)))) ) ).

% sums_Suc_iff
tff(fact_6944_sums__zero__iff__shift,axiom,
    ! [N: nat,F: fun_nat_complex,S: 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_qq(N),F)),S))
      <=> pp(aa_complex_bool(sums_complex(F),S)) ) ) ).

% sums_zero_iff_shift
tff(fact_6945_sums__zero__iff__shift,axiom,
    ! [N: nat,F: fun_nat_real,S: 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_qr(N),F)),S))
      <=> pp(aa_real_bool(sums_real(F),S)) ) ) ).

% sums_zero_iff_shift
tff(fact_6946_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_complex(F,N2) = zero_zero_complex ) )
       => pp(aa_complex_bool(sums_complex(F),groups2073611262835488442omplex(F,N4))) ) ) ).

% sums_finite
tff(fact_6947_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => pp(aa_int_bool(sums_int(F),groups3539618377306564664at_int(F,N4))) ) ) ).

% sums_finite
tff(fact_6948_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => pp(aa_nat_bool(sums_nat(F),aa_set_nat_nat(groups3542108847815614940at_nat(F),N4))) ) ) ).

% sums_finite
tff(fact_6949_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => pp(aa_real_bool(sums_real(F),groups6591440286371151544t_real(F,N4))) ) ) ).

% sums_finite
tff(fact_6950_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_qs(P),F)),groups2073611262835488442omplex(F,collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_6951_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_qt(P),F)),groups3539618377306564664at_int(F,collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_6952_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_qu(P),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_6953_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_qv(P),F)),groups6591440286371151544t_real(F,collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_6954_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qw(A2),F)),groups2073611262835488442omplex(F,A2))) ) ).

% sums_If_finite_set
tff(fact_6955_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_qx(A2),F)),groups3539618377306564664at_int(F,A2))) ) ).

% sums_If_finite_set
tff(fact_6956_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qy(A2),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2))) ) ).

% sums_If_finite_set
tff(fact_6957_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_qz(A2),F)),groups6591440286371151544t_real(F,A2))) ) ).

% sums_If_finite_set
tff(fact_6958_powser__sums__if,axiom,
    ! [M: nat,Z: complex] : pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_ra(M),Z)),aa_nat_complex(power_power_complex(Z),M))) ).

% powser_sums_if
tff(fact_6959_powser__sums__if,axiom,
    ! [M: nat,Z: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_rb(M),Z)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),M))) ).

% powser_sums_if
tff(fact_6960_powser__sums__if,axiom,
    ! [M: nat,Z: int] : pp(aa_int_bool(sums_int(aa_int_fun_nat_int(aTP_Lamm_rc(M),Z)),aa_nat_int(power_power_int(Z),M))) ).

% powser_sums_if
tff(fact_6961_sums__If__finite__set_H,axiom,
    ! [G: fun_nat_real,S2: real,A2: set_nat,S5: real,F: fun_nat_real] :
      ( pp(aa_real_bool(sums_real(G),S2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( ( S5 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S2),groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_rd(G),F),A2)) )
         => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_re(G),A2),F)),S5)) ) ) ) ).

% sums_If_finite_set'
tff(fact_6962_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_6963_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_6964_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_6965_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_6966_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_6967_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_rf(M),N)),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_6968_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_6969_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(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1)))
     => ( ! [I2: int,J2: int] :
            ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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_6970_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,M: nat] : groups2073611262835488442omplex(aTP_Lamm_rg(R2),set_or1269000886237332187st_nat(zero_zero_nat,M)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M))),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(R2),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_6971_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_rh(R2),set_or1269000886237332187st_nat(zero_zero_nat,M)) = aa_rat_rat(aa_rat_fun_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_6972_gchoose__row__sum__weighted,axiom,
    ! [R2: real,M: nat] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ri,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_6973_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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,Y),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),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
           => 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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U)) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_6974_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_6975_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)),X)) ) ).

% neg_numeral_le_ceiling
tff(fact_6976_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),X)) ) ).

% neg_numeral_le_ceiling
tff(fact_6977_ceiling__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real))) ) ).

% ceiling_less_neg_numeral
tff(fact_6978_ceiling__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat))) ) ).

% ceiling_less_neg_numeral
tff(fact_6979_real__sqrt__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% real_sqrt_less_iff
tff(fact_6980_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_complex(gbinomial_complex(zero_zero_complex),aa_nat_nat(suc,K)) = zero_zero_complex ).

% gbinomial_0(2)
tff(fact_6981_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_6982_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_6983_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_6984_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_int(gbinomial_int(zero_zero_int),aa_nat_nat(suc,K)) = zero_zero_int ).

% gbinomial_0(2)
tff(fact_6985_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y)) ) ).

% real_sqrt_gt_0_iff
tff(fact_6986_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% real_sqrt_lt_0_iff
tff(fact_6987_ceiling__zero,axiom,
    aa_rat_int(archim2889992004027027881ng_rat,zero_zero_rat) = zero_zero_int ).

% ceiling_zero
tff(fact_6988_ceiling__zero,axiom,
    aa_real_int(archim7802044766580827645g_real,zero_zero_real) = zero_zero_int ).

% ceiling_zero
tff(fact_6989_gbinomial__0_I1_J,axiom,
    ! [A: complex] : aa_nat_complex(gbinomial_complex(A),zero_zero_nat) = one_one_complex ).

% gbinomial_0(1)
tff(fact_6990_gbinomial__0_I1_J,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),zero_zero_nat) = one_one_real ).

% gbinomial_0(1)
tff(fact_6991_gbinomial__0_I1_J,axiom,
    ! [A: rat] : aa_nat_rat(gbinomial_rat(A),zero_zero_nat) = one_one_rat ).

% gbinomial_0(1)
tff(fact_6992_gbinomial__0_I1_J,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),zero_zero_nat) = one_one_nat ).

% gbinomial_0(1)
tff(fact_6993_gbinomial__0_I1_J,axiom,
    ! [A: int] : aa_nat_int(gbinomial_int(A),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_6994_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),Y)) ) ).

% real_sqrt_gt_1_iff
tff(fact_6995_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ).

% real_sqrt_lt_1_iff
tff(fact_6996_log__eq__one,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),A) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_6997_log__less__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_6998_log__less__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),one_one_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),A)) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_6999_one__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X)) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_7000_log__less__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),zero_zero_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_7001_zero__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_7002_ceiling__le__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% ceiling_le_zero
tff(fact_7003_ceiling__le__zero,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat)) ) ).

% ceiling_le_zero
tff(fact_7004_zero__less__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X)) ) ).

% zero_less_ceiling
tff(fact_7005_zero__less__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% zero_less_ceiling
tff(fact_7006_ceiling__le__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_num_real(numeral_numeral_real,V))) ) ).

% ceiling_le_numeral
tff(fact_7007_ceiling__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_num_rat(numeral_numeral_rat,V))) ) ).

% ceiling_le_numeral
tff(fact_7008_ceiling__less__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% ceiling_less_one
tff(fact_7009_ceiling__less__one,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat)) ) ).

% ceiling_less_one
tff(fact_7010_numeral__less__ceiling,axiom,
    ! [V: num,X: 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,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,V)),X)) ) ).

% numeral_less_ceiling
tff(fact_7011_numeral__less__ceiling,axiom,
    ! [V: num,X: 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,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,V)),X)) ) ).

% numeral_less_ceiling
tff(fact_7012_one__le__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X)) ) ).

% one_le_ceiling
tff(fact_7013_one__le__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% one_le_ceiling
tff(fact_7014_ceiling__le__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ).

% ceiling_le_one
tff(fact_7015_ceiling__le__one,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),one_one_rat)) ) ).

% ceiling_le_one
tff(fact_7016_one__less__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X)) ) ).

% one_less_ceiling
tff(fact_7017_one__less__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ).

% one_less_ceiling
tff(fact_7018_zero__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_7019_log__le__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),zero_zero_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_7020_one__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X)) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_7021_log__le__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),one_one_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),A)) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_7022_log__le__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_7023_ceiling__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(uminus_uminus_real,one_one_real))) ) ).

% ceiling_less_zero
tff(fact_7024_ceiling__less__zero,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ) ).

% ceiling_less_zero
tff(fact_7025_zero__le__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)) ) ).

% zero_le_ceiling
tff(fact_7026_zero__le__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X)) ) ).

% zero_le_ceiling
tff(fact_7027_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_7028_ceiling__less__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,V)),one_one_real))) ) ).

% ceiling_less_numeral
tff(fact_7029_ceiling__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat))) ) ).

% ceiling_less_numeral
tff(fact_7030_numeral__le__ceiling,axiom,
    ! [V: num,X: 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,X)))
    <=> 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)),X)) ) ).

% numeral_le_ceiling
tff(fact_7031_numeral__le__ceiling,axiom,
    ! [V: num,X: 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,X)))
    <=> 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)),X)) ) ).

% numeral_le_ceiling
tff(fact_7032_ceiling__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)))) ) ).

% ceiling_le_neg_numeral
tff(fact_7033_ceiling__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)))) ) ).

% ceiling_le_neg_numeral
tff(fact_7034_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),X)) ) ).

% neg_numeral_less_ceiling
tff(fact_7035_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),X)) ) ).

% neg_numeral_less_ceiling
tff(fact_7036_real__sqrt__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y))) ) ).

% real_sqrt_less_mono
tff(fact_7037_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sqrt,X))) ) ).

% real_sqrt_gt_zero
tff(fact_7038_ceiling__mono,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Y)),aa_real_int(archim7802044766580827645g_real,X))) ) ).

% ceiling_mono
tff(fact_7039_ceiling__mono,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Y)),aa_rat_int(archim2889992004027027881ng_rat,X))) ) ).

% ceiling_mono
tff(fact_7040_le__of__int__ceiling,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)))) ).

% le_of_int_ceiling
tff(fact_7041_le__of__int__ceiling,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X)))) ).

% le_of_int_ceiling
tff(fact_7042_ceiling__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,Y)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% ceiling_less_cancel
tff(fact_7043_ceiling__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim7802044766580827645g_real,Y)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% ceiling_less_cancel
tff(fact_7044_ceiling__le__iff,axiom,
    ! [X: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z))) ) ).

% ceiling_le_iff
tff(fact_7045_ceiling__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% ceiling_le_iff
tff(fact_7046_ceiling__le,axiom,
    ! [X: real,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,A)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),A)) ) ).

% ceiling_le
tff(fact_7047_ceiling__le,axiom,
    ! [X: rat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,A)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),A)) ) ).

% ceiling_le
tff(fact_7048_less__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)) ) ).

% less_ceiling_iff
tff(fact_7049_less__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),X)) ) ).

% less_ceiling_iff
tff(fact_7050_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_7051_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_7052_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_7053_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(aa_rat_fun_rat_rat(plus_plus_rat,R2),one_one_rat))) ).

% of_int_ceiling_le_add_one
tff(fact_7054_log__base__change,axiom,
    ! [A: real,B: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),B)) ) ) ) ).

% log_base_change
tff(fact_7055_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_7056_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_7057_log__of__power__eq,axiom,
    ! [M: nat,B: real,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ) ).

% log_of_power_eq
tff(fact_7058_less__log__of__power,axiom,
    ! [B: 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,B),N)),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B),M))) ) ) ).

% less_log_of_power
tff(fact_7059_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K)),A))
     => 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,A),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K))) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_7060_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),A))
     => 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(A),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(gbinomial_rat(A),K))) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_7061_real__less__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(sqrt,Y))) ) ).

% real_less_rsqrt
tff(fact_7062_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_7063_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_7064_ceiling__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,X) = A )
    <=> ( 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,A)),one_one_real)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,A))) ) ) ).

% ceiling_eq_iff
tff(fact_7065_ceiling__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( aa_rat_int(archim2889992004027027881ng_rat,X) = A )
    <=> ( 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,A)),one_one_rat)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,A))) ) ) ).

% ceiling_eq_iff
tff(fact_7066_ceiling__unique,axiom,
    ! [Z: int,X: 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)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z)))
       => ( aa_real_int(archim7802044766580827645g_real,X) = Z ) ) ) ).

% ceiling_unique
tff(fact_7067_ceiling__unique,axiom,
    ! [Z: int,X: 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)),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z)))
       => ( aa_rat_int(archim2889992004027027881ng_rat,X) = Z ) ) ) ).

% ceiling_unique
tff(fact_7068_ceiling__correct,axiom,
    ! [X: 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,X))),one_one_real)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)))) ) ).

% ceiling_correct
tff(fact_7069_ceiling__correct,axiom,
    ! [X: 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,X))),one_one_rat)),X))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X)))) ) ).

% ceiling_correct
tff(fact_7070_mult__ceiling__le,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => 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,A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B)))) ) ) ).

% mult_ceiling_le
tff(fact_7071_mult__ceiling__le,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B)))) ) ) ).

% mult_ceiling_le
tff(fact_7072_log__mult,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_7073_ceiling__less__iff,axiom,
    ! [X: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),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_7074_ceiling__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))) ) ).

% ceiling_less_iff
tff(fact_7075_le__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> 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)),X)) ) ).

% le_ceiling_iff
tff(fact_7076_le__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_real_int(archim7802044766580827645g_real,X)))
    <=> 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)),X)) ) ).

% le_ceiling_iff
tff(fact_7077_log__divide,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_7078_le__log__of__power,axiom,
    ! [B: 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,B),N)),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B),M))) ) ) ).

% le_log_of_power
tff(fact_7079_log__base__pow,axiom,
    ! [A: real,N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).

% log_base_pow
tff(fact_7080_log__nat__power,axiom,
    ! [X: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ).

% log_nat_power
tff(fact_7081_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),M)),aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),K)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_7082_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),M)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_7083_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: 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,A),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,A),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_7084_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_7085_ceiling__log__nat__eq__if,axiom,
    ! [B: 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,B),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,B),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))),B))
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),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_7086_ceiling__divide__upper,axiom,
    ! [Q4: real,P2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q4))
     => 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),Q4)))),Q4))) ) ).

% ceiling_divide_upper
tff(fact_7087_ceiling__divide__upper,axiom,
    ! [Q4: rat,P2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q4))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,P2),aa_rat_rat(aa_rat_fun_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),Q4)))),Q4))) ) ).

% ceiling_divide_upper
tff(fact_7088_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_7089_log__of__power__less,axiom,
    ! [M: nat,B: 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,B),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( 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,B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_less
tff(fact_7090_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: 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))),B))
     => ( 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,B)),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,B),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,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_7091_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),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,X)),Y)) ) ) ) ).

% real_less_lsqrt
tff(fact_7092_ceiling__divide__lower,axiom,
    ! [Q4: real,P2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q4))
     => 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),Q4)))),one_one_real)),Q4)),P2)) ) ).

% ceiling_divide_lower
tff(fact_7093_ceiling__divide__lower,axiom,
    ! [Q4: rat,P2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q4))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_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),Q4)))),one_one_rat)),Q4)),P2)) ) ).

% ceiling_divide_lower
tff(fact_7094_log__of__power__le,axiom,
    ! [M: nat,B: 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,B),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( 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,B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_le
tff(fact_7095_ceiling__eq,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),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,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_7096_ceiling__eq,axiom,
    ! [N: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,N)),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,N)),one_one_rat)))
       => ( aa_rat_int(archim2889992004027027881ng_rat,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_7097_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_7098_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: 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,A),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,A),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,A),one_one_real)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_7099_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),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(A),one_one_rat)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_7100_arsinh__real__aux,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))))) ).

% arsinh_real_aux
tff(fact_7101_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_7102_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_7103_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : groups2073611262835488442omplex(aTP_Lamm_rj(K),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_7104_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : groups2906978787729119204at_rat(aTP_Lamm_rk(K),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_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_7105_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_rl,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_7106_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_7107_gbinomial__absorption_H,axiom,
    ! [K: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_7108_gbinomial__absorption_H,axiom,
    ! [K: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(divide_divide_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_7109_gbinomial__absorption_H,axiom,
    ! [K: nat,A: 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,A),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),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,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_7110_floor__log__nat__eq__powr__iff,axiom,
    ! [B: 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))),B))
     => ( 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,B)),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,B),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,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_7111_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = 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,B),aa_nat_real(semiri5074537144036343181t_real,K))),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),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_7112_floor__log__nat__eq__if,axiom,
    ! [B: 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,B),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,B),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))),B))
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_7113_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_7114_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),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,Y)),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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U)) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_7115_dbl__dec__simps_I2_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,zero_zero_complex) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% dbl_dec_simps(2)
tff(fact_7116_dbl__dec__simps_I2_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_dec_simps(2)
tff(fact_7117_dbl__dec__simps_I2_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,zero_zero_real) = aa_real_real(uminus_uminus_real,one_one_real) ).

% dbl_dec_simps(2)
tff(fact_7118_dbl__dec__simps_I2_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,zero_zero_rat) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_dec_simps(2)
tff(fact_7119_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_7120_abs__idempotent,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_idempotent
tff(fact_7121_abs__idempotent,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_idempotent
tff(fact_7122_abs__idempotent,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_idempotent
tff(fact_7123_abs__idempotent,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_idempotent
tff(fact_7124_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_7125_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(abs_abs_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% abs_0_eq
tff(fact_7126_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(abs_abs_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% abs_0_eq
tff(fact_7127_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(abs_abs_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% abs_0_eq
tff(fact_7128_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_7129_abs__eq__0,axiom,
    ! [A: real] :
      ( ( aa_real_real(abs_abs_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% abs_eq_0
tff(fact_7130_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0
tff(fact_7131_abs__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% abs_eq_0
tff(fact_7132_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_7133_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_7134_abs__zero,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_7135_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_7136_abs__0,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_7137_abs__0,axiom,
    aa_complex_complex(abs_abs_complex,zero_zero_complex) = zero_zero_complex ).

% abs_0
tff(fact_7138_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_7139_abs__0,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_7140_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_7141_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) = plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)) ).

% abs_add_abs
tff(fact_7142_abs__add__abs,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_add_abs
tff(fact_7143_abs__add__abs,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_add_abs
tff(fact_7144_abs__add__abs,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_add_abs
tff(fact_7145_abs__minus__cancel,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_minus_cancel
tff(fact_7146_abs__minus__cancel,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_minus_cancel
tff(fact_7147_abs__minus__cancel,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_minus_cancel
tff(fact_7148_abs__minus__cancel,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).

% abs_minus_cancel
tff(fact_7149_abs__of__nat,axiom,
    ! [N: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ).

% abs_of_nat
tff(fact_7150_abs__of__nat,axiom,
    ! [N: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,N)) = aa_nat_Code_integer(semiri4939895301339042750nteger,N) ).

% abs_of_nat
tff(fact_7151_abs__of__nat,axiom,
    ! [N: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% abs_of_nat
tff(fact_7152_abs__of__nat,axiom,
    ! [N: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).

% abs_of_nat
tff(fact_7153_powr__0,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,zero_zero_real),Z) = zero_zero_real ).

% powr_0
tff(fact_7154_powr__eq__0__iff,axiom,
    ! [W2: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,W2),Z) = zero_zero_real )
    <=> ( W2 = zero_zero_real ) ) ).

% powr_eq_0_iff
tff(fact_7155_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger))
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_7156_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),zero_zero_real))
    <=> ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
tff(fact_7157_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat))
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_7158_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int))
    <=> ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_7159_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),A))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% abs_le_self_iff
tff(fact_7160_abs__le__self__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% abs_le_self_iff
tff(fact_7161_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% abs_le_self_iff
tff(fact_7162_abs__le__self__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% abs_le_self_iff
tff(fact_7163_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_nonneg
tff(fact_7164_abs__of__nonneg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_nonneg
tff(fact_7165_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_nonneg
tff(fact_7166_abs__of__nonneg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_nonneg
tff(fact_7167_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A)))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_7168_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,A)))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
tff(fact_7169_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A)))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_7170_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A)))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_7171_powr__zero__eq__one,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),zero_zero_real) = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),zero_zero_real) = one_one_real ) ) ) ).

% powr_zero_eq_one
tff(fact_7172_floor__zero,axiom,
    aa_real_int(archim6058952711729229775r_real,zero_zero_real) = zero_zero_int ).

% floor_zero
tff(fact_7173_floor__zero,axiom,
    aa_rat_int(archim3151403230148437115or_rat,zero_zero_rat) = zero_zero_int ).

% floor_zero
tff(fact_7174_powr__gt__zero,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)))
    <=> ( X != zero_zero_real ) ) ).

% powr_gt_zero
tff(fact_7175_powr__less__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% powr_less_cancel_iff
tff(fact_7176_sum__abs,axiom,
    ! [F: fun_nat_real,A2: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,groups6591440286371151544t_real(F,A2))),groups6591440286371151544t_real(aTP_Lamm_rm(F),A2))) ).

% sum_abs
tff(fact_7177_zero__le__divide__abs__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B))))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_7178_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_rat_rat(abs_abs_rat,B))))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_7179_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B))),zero_zero_real))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_7180_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),aa_rat_rat(abs_abs_rat,B))),zero_zero_rat))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_7181_abs__of__nonpos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_nonpos
tff(fact_7182_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_nonpos
tff(fact_7183_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_nonpos
tff(fact_7184_abs__of__nonpos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_nonpos
tff(fact_7185_abs__sgn__eq__1,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = one_one_Code_integer ) ) ).

% abs_sgn_eq_1
tff(fact_7186_abs__sgn__eq__1,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = one_one_real ) ) ).

% abs_sgn_eq_1
tff(fact_7187_abs__sgn__eq__1,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = one_one_rat ) ) ).

% abs_sgn_eq_1
tff(fact_7188_abs__sgn__eq__1,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = one_one_int ) ) ).

% abs_sgn_eq_1
tff(fact_7189_powr__eq__one__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X) = one_one_real )
      <=> ( X = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_7190_powr__le__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% powr_le_cancel_iff
tff(fact_7191_artanh__minus__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,X)) ) ) ).

% artanh_minus_real
tff(fact_7192_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,abs_abs_Code_integer(A)) = zero_n356916108424825756nteger(aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,A),zero_z3403309356797280102nteger))) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_7193_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(abs_abs_complex,A)) = zero_n1201886186963655149omplex(aa_bool_bool(fNot,aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,A),zero_zero_complex))) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_7194_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(abs_abs_real,A)) = zero_n3304061248610475627l_real(aa_bool_bool(fNot,aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,A),zero_zero_real))) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_7195_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(abs_abs_rat,A)) = zero_n2052037380579107095ol_rat(aa_bool_bool(fNot,aa_rat_bool(aa_rat_fun_rat_bool(fequal_rat,A),zero_zero_rat))) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_7196_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(abs_abs_int,A)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,A),zero_zero_int))) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_7197_sgn__abs,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = zero_n356916108424825756nteger(aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,A),zero_z3403309356797280102nteger))) ).

% sgn_abs
tff(fact_7198_sgn__abs,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(sgn_sgn_complex,A)) = zero_n1201886186963655149omplex(aa_bool_bool(fNot,aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,A),zero_zero_complex))) ).

% sgn_abs
tff(fact_7199_sgn__abs,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = zero_n3304061248610475627l_real(aa_bool_bool(fNot,aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,A),zero_zero_real))) ).

% sgn_abs
tff(fact_7200_sgn__abs,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = zero_n2052037380579107095ol_rat(aa_bool_bool(fNot,aa_rat_bool(aa_rat_fun_rat_bool(fequal_rat,A),zero_zero_rat))) ).

% sgn_abs
tff(fact_7201_sgn__abs,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,A),zero_zero_int))) ).

% sgn_abs
tff(fact_7202_sum__abs__ge__zero,axiom,
    ! [F: fun_nat_real,A2: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups6591440286371151544t_real(aTP_Lamm_rm(F),A2))) ).

% sum_abs_ge_zero
tff(fact_7203_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),N)))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_7204_zero__less__power__abs__iff,axiom,
    ! [A: 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,aa_real_real(abs_abs_real,A)),N)))
    <=> ( ( A != zero_zero_real )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_7205_zero__less__power__abs__iff,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),N)))
    <=> ( ( A != zero_zero_rat )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_7206_zero__less__power__abs__iff,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N)))
    <=> ( ( A != zero_zero_int )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_7207_zero__le__floor,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% zero_le_floor
tff(fact_7208_zero__le__floor,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X)) ) ).

% zero_le_floor
tff(fact_7209_floor__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% floor_less_zero
tff(fact_7210_floor__less__zero,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat)) ) ).

% floor_less_zero
tff(fact_7211_numeral__le__floor,axiom,
    ! [V: num,X: 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,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,V)),X)) ) ).

% numeral_le_floor
tff(fact_7212_numeral__le__floor,axiom,
    ! [V: num,X: 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,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,V)),X)) ) ).

% numeral_le_floor
tff(fact_7213_zero__less__floor,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ).

% zero_less_floor
tff(fact_7214_zero__less__floor,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ).

% zero_less_floor
tff(fact_7215_floor__le__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ).

% floor_le_zero
tff(fact_7216_floor__le__zero,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),one_one_rat)) ) ).

% floor_le_zero
tff(fact_7217_floor__less__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_num_real(numeral_numeral_real,V))) ) ).

% floor_less_numeral
tff(fact_7218_floor__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_num_rat(numeral_numeral_rat,V))) ) ).

% floor_less_numeral
tff(fact_7219_one__le__floor,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ).

% one_le_floor
tff(fact_7220_one__le__floor,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ).

% one_le_floor
tff(fact_7221_floor__less__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ).

% floor_less_one
tff(fact_7222_floor__less__one,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),one_one_rat)) ) ).

% floor_less_one
tff(fact_7223_log__powr__cancel,axiom,
    ! [A: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_7224_powr__log__cancel,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)) = X ) ) ) ) ).

% powr_log_cancel
tff(fact_7225_numeral__less__floor,axiom,
    ! [V: num,X: 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,X)))
    <=> 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)),X)) ) ).

% numeral_less_floor
tff(fact_7226_numeral__less__floor,axiom,
    ! [V: num,X: 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,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),X)) ) ).

% numeral_less_floor
tff(fact_7227_floor__le__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),one_one_real))) ) ).

% floor_le_numeral
tff(fact_7228_floor__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,V)),one_one_rat))) ) ).

% floor_le_numeral
tff(fact_7229_one__less__floor,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) ) ).

% one_less_floor
tff(fact_7230_one__less__floor,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)) ) ).

% one_less_floor
tff(fact_7231_floor__le__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% floor_le_one
tff(fact_7232_floor__le__one,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% floor_le_one
tff(fact_7233_neg__numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),X)) ) ).

% neg_numeral_le_floor
tff(fact_7234_neg__numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),X)) ) ).

% neg_numeral_le_floor
tff(fact_7235_floor__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)))) ) ).

% floor_less_neg_numeral
tff(fact_7236_floor__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)))) ) ).

% floor_less_neg_numeral
tff(fact_7237_neg__numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)),X)) ) ).

% neg_numeral_less_floor
tff(fact_7238_neg__numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),X)) ) ).

% neg_numeral_less_floor
tff(fact_7239_floor__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real))) ) ).

% floor_le_neg_numeral
tff(fact_7240_floor__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat))) ) ).

% floor_le_neg_numeral
tff(fact_7241_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% abs_le_D1
tff(fact_7242_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).

% abs_le_D1
tff(fact_7243_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% abs_le_D1
tff(fact_7244_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% abs_le_D1
tff(fact_7245_abs__ge__self,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(abs_abs_real,A))) ).

% abs_ge_self
tff(fact_7246_abs__ge__self,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),abs_abs_Code_integer(A))) ).

% abs_ge_self
tff(fact_7247_abs__ge__self,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(abs_abs_rat,A))) ).

% abs_ge_self
tff(fact_7248_abs__ge__self,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(abs_abs_int,A))) ).

% abs_ge_self
tff(fact_7249_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_7250_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(abs_abs_complex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
tff(fact_7251_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(abs_abs_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
tff(fact_7252_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
tff(fact_7253_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
tff(fact_7254_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)) = abs_abs_Code_integer(minus_8373710615458151222nteger(B,A)) ).

% abs_minus_commute
tff(fact_7255_abs__minus__commute,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ).

% abs_minus_commute
tff(fact_7256_abs__minus__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A)) ).

% abs_minus_commute
tff(fact_7257_abs__minus__commute,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)) ).

% abs_minus_commute
tff(fact_7258_abs__ge__zero,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A))) ).

% abs_ge_zero
tff(fact_7259_abs__ge__zero,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,A))) ).

% abs_ge_zero
tff(fact_7260_abs__ge__zero,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A))) ).

% abs_ge_zero
tff(fact_7261_abs__ge__zero,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A))) ).

% abs_ge_zero
tff(fact_7262_floor__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y))) ) ).

% floor_mono
tff(fact_7263_floor__mono,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))) ) ).

% floor_mono
tff(fact_7264_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_pos
tff(fact_7265_abs__of__pos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_pos
tff(fact_7266_abs__of__pos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_pos
tff(fact_7267_abs__of__pos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_pos
tff(fact_7268_abs__not__less__zero,axiom,
    ! [A: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger)) ).

% abs_not_less_zero
tff(fact_7269_abs__not__less__zero,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),zero_zero_real)) ).

% abs_not_less_zero
tff(fact_7270_abs__not__less__zero,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat)) ).

% abs_not_less_zero
tff(fact_7271_abs__not__less__zero,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int)) ).

% abs_not_less_zero
tff(fact_7272_of__int__floor__le,axiom,
    ! [X: 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,X))),X)) ).

% of_int_floor_le
tff(fact_7273_of__int__floor__le,axiom,
    ! [X: 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,X))),X)) ).

% of_int_floor_le
tff(fact_7274_floor__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% floor_less_cancel
tff(fact_7275_floor__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% floor_less_cancel
tff(fact_7276_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(plus_p5714425477246183910nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))) ).

% abs_triangle_ineq
tff(fact_7277_abs__triangle__ineq,axiom,
    ! [A: real,B: 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,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))) ).

% abs_triangle_ineq
tff(fact_7278_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))) ).

% abs_triangle_ineq
tff(fact_7279_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : 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(plus_plus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))) ).

% abs_triangle_ineq
tff(fact_7280_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),C))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(B)),D))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D))) ) ) ).

% abs_mult_less
tff(fact_7281_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,B)),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D))) ) ) ).

% abs_mult_less
tff(fact_7282_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,B)),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),D))) ) ) ).

% abs_mult_less
tff(fact_7283_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,B)),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))) ) ) ).

% abs_mult_less
tff(fact_7284_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(B,A)))) ).

% abs_triangle_ineq2_sym
tff(fact_7285_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_7286_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_7287_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: 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,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_7288_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)))) ).

% abs_triangle_ineq3
tff(fact_7289_abs__triangle__ineq3,axiom,
    ! [A: real,B: 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(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))) ).

% abs_triangle_ineq3
tff(fact_7290_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B)))) ).

% abs_triangle_ineq3
tff(fact_7291_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : 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_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))) ).

% abs_triangle_ineq3
tff(fact_7292_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)))) ).

% abs_triangle_ineq2
tff(fact_7293_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))) ).

% abs_triangle_ineq2
tff(fact_7294_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B)))) ).

% abs_triangle_ineq2
tff(fact_7295_abs__triangle__ineq2,axiom,
    ! [A: int,B: 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,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))) ).

% abs_triangle_ineq2
tff(fact_7296_nonzero__abs__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% nonzero_abs_divide
tff(fact_7297_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% nonzero_abs_divide
tff(fact_7298_abs__leI,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)) ) ) ).

% abs_leI
tff(fact_7299_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)) ) ) ).

% abs_leI
tff(fact_7300_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)) ) ) ).

% abs_leI
tff(fact_7301_abs__leI,axiom,
    ! [A: int,B: 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_eq_int,aa_int_int(uminus_uminus_int,A)),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)) ) ) ).

% abs_leI
tff(fact_7302_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)) ) ).

% abs_le_D2
tff(fact_7303_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ).

% abs_le_D2
tff(fact_7304_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) ) ).

% abs_le_D2
tff(fact_7305_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)) ) ).

% abs_le_D2
tff(fact_7306_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)) ) ) ).

% abs_le_iff
tff(fact_7307_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ) ).

% abs_le_iff
tff(fact_7308_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) ) ) ).

% abs_le_iff
tff(fact_7309_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
    <=> ( 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_eq_int,aa_int_int(uminus_uminus_int,A)),B)) ) ) ).

% abs_le_iff
tff(fact_7310_abs__ge__minus__self,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(abs_abs_real,A))) ).

% abs_ge_minus_self
tff(fact_7311_abs__ge__minus__self,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),abs_abs_Code_integer(A))) ).

% abs_ge_minus_self
tff(fact_7312_abs__ge__minus__self,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(abs_abs_rat,A))) ).

% abs_ge_minus_self
tff(fact_7313_abs__ge__minus__self,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A))) ).

% abs_ge_minus_self
tff(fact_7314_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),B))
    <=> ( 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,aa_int_int(uminus_uminus_int,A)),B)) ) ) ).

% abs_less_iff
tff(fact_7315_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),B)) ) ) ).

% abs_less_iff
tff(fact_7316_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) ) ) ).

% abs_less_iff
tff(fact_7317_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),B))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ) ).

% abs_less_iff
tff(fact_7318_powr__less__mono2__neg,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A))) ) ) ) ).

% powr_less_mono2_neg
tff(fact_7319_powr__non__neg,axiom,
    ! [A: real,X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X)),zero_zero_real)) ).

% powr_non_neg
tff(fact_7320_powr__less__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B))) ) ) ).

% powr_less_mono
tff(fact_7321_powr__less__cancel,axiom,
    ! [X: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% powr_less_cancel
tff(fact_7322_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),E)) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_7323_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),E)) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_7324_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
          | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
          | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)) ) )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ) ) ).

% abs_eq_mult
tff(fact_7325_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
          | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% abs_eq_mult
tff(fact_7326_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
          | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% abs_eq_mult
tff(fact_7327_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
          | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).

% abs_eq_mult
tff(fact_7328_abs__mult__pos,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(Y)),X) = abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_7329_abs__mult__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Y)),X) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_7330_abs__mult__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,Y)),X) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_7331_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),X) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_7332_abs__minus__le__zero,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(abs_abs_real,A))),zero_zero_real)) ).

% abs_minus_le_zero
tff(fact_7333_abs__minus__le__zero,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,abs_abs_Code_integer(A))),zero_z3403309356797280102nteger)) ).

% abs_minus_le_zero
tff(fact_7334_abs__minus__le__zero,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(abs_abs_rat,A))),zero_zero_rat)) ).

% abs_minus_le_zero
tff(fact_7335_abs__minus__le__zero,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,A))),zero_zero_int)) ).

% abs_minus_le_zero
tff(fact_7336_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(abs_abs_real,B) )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
        & ( ( B = A )
          | ( B = aa_real_real(uminus_uminus_real,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_7337_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = abs_abs_Code_integer(B) )
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
        & ( ( B = A )
          | ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_7338_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(abs_abs_rat,B) )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
        & ( ( B = A )
          | ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_7339_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(abs_abs_int,B) )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
        & ( ( B = A )
          | ( B = aa_int_int(uminus_uminus_int,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_7340_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(abs_abs_real,A) = B )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
        & ( ( A = B )
          | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_7341_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer(A) = B )
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
        & ( ( A = B )
          | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_7342_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = B )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
        & ( ( A = B )
          | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_7343_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(abs_abs_int,A) = B )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
        & ( ( A = B )
          | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_7344_abs__div__pos,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,X)),Y) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% abs_div_pos
tff(fact_7345_abs__div__pos,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,X)),Y) = aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(X),Y)) ) ) ).

% abs_div_pos
tff(fact_7346_zero__le__power__abs,axiom,
    ! [A: code_integer,N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),N))) ).

% zero_le_power_abs
tff(fact_7347_zero__le__power__abs,axiom,
    ! [A: 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,aa_real_real(abs_abs_real,A)),N))) ).

% zero_le_power_abs
tff(fact_7348_zero__le__power__abs,axiom,
    ! [A: 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(aa_rat_rat(abs_abs_rat,A)),N))) ).

% zero_le_power_abs
tff(fact_7349_zero__le__power__abs,axiom,
    ! [A: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N))) ).

% zero_le_power_abs
tff(fact_7350_abs__if__raw,axiom,
    ! [X2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X2) = aa_int_int(uminus_uminus_int,X2) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X2) = X2 ) ) ) ).

% abs_if_raw
tff(fact_7351_abs__if__raw,axiom,
    ! [X2: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X2) = aa_real_real(uminus_uminus_real,X2) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X2) = X2 ) ) ) ).

% abs_if_raw
tff(fact_7352_abs__if__raw,axiom,
    ! [X2: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,X2) = aa_rat_rat(uminus_uminus_rat,X2) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,X2) = X2 ) ) ) ).

% abs_if_raw
tff(fact_7353_abs__if__raw,axiom,
    ! [X2: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(X2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X2) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(X2) = X2 ) ) ) ).

% abs_if_raw
tff(fact_7354_abs__of__neg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_neg
tff(fact_7355_abs__of__neg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_neg
tff(fact_7356_abs__of__neg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_neg
tff(fact_7357_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_neg
tff(fact_7358_abs__if,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
       => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
       => ( aa_int_int(abs_abs_int,A) = A ) ) ) ).

% abs_if
tff(fact_7359_abs__if,axiom,
    ! [A: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = A ) ) ) ).

% abs_if
tff(fact_7360_abs__if,axiom,
    ! [A: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ) ).

% abs_if
tff(fact_7361_abs__if,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(A) = A ) ) ) ).

% abs_if
tff(fact_7362_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,B),plus_p5714425477246183910nteger(C,D)))),plus_p5714425477246183910nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(A,C)),abs_abs_Code_integer(minus_8373710615458151222nteger(B,D))))) ).

% abs_diff_triangle_ineq
tff(fact_7363_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: 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(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D))))) ).

% abs_diff_triangle_ineq
tff(fact_7364_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),D)))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),C))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),D))))) ).

% abs_diff_triangle_ineq
tff(fact_7365_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : 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_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),D)))),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,A),C))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))))) ).

% abs_diff_triangle_ineq
tff(fact_7366_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))) ).

% abs_triangle_ineq4
tff(fact_7367_abs__triangle__ineq4,axiom,
    ! [A: real,B: 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,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))) ).

% abs_triangle_ineq4
tff(fact_7368_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))) ).

% abs_triangle_ineq4
tff(fact_7369_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : 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,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))) ).

% abs_triangle_ineq4
tff(fact_7370_abs__diff__le__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,A))),R2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R2)),X))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),plus_p5714425477246183910nteger(A,R2))) ) ) ).

% abs_diff_le_iff
tff(fact_7371_abs__diff__le__iff,axiom,
    ! [X: real,A: real,R2: 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,X),A))),R2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_7372_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),A))),R2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),R2)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_7373_abs__diff__le__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( 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,X),A))),R2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),X))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_7374_abs__diff__less__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,A))),R2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R2)),X))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),plus_p5714425477246183910nteger(A,R2))) ) ) ).

% abs_diff_less_iff
tff(fact_7375_abs__diff__less__iff,axiom,
    ! [X: real,A: real,R2: 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,X),A))),R2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_7376_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),A))),R2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),R2)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_7377_abs__diff__less__iff,axiom,
    ! [X: int,A: int,R2: 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(minus_minus_int,X),A))),R2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),X))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_7378_le__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),X)) ) ).

% le_floor_iff
tff(fact_7379_le__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)) ) ).

% le_floor_iff
tff(fact_7380_floor__less__iff,axiom,
    ! [X: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z))) ) ).

% floor_less_iff
tff(fact_7381_floor__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% floor_less_iff
tff(fact_7382_powr__mono2_H,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A))) ) ) ) ).

% powr_mono2'
tff(fact_7383_powr__less__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A))) ) ) ) ).

% powr_less_mono2
tff(fact_7384_gr__one__powr,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y))) ) ) ).

% gr_one_powr
tff(fact_7385_powr__inj,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y) )
        <=> ( X = Y ) ) ) ) ).

% powr_inj
tff(fact_7386_abs__sgn__eq,axiom,
    ! [A: code_integer] :
      ( ( ( A = zero_z3403309356797280102nteger )
       => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = zero_z3403309356797280102nteger ) )
      & ( ( A != zero_z3403309356797280102nteger )
       => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = one_one_Code_integer ) ) ) ).

% abs_sgn_eq
tff(fact_7387_abs__sgn__eq,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = one_one_real ) ) ) ).

% abs_sgn_eq
tff(fact_7388_abs__sgn__eq,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = one_one_rat ) ) ) ).

% abs_sgn_eq
tff(fact_7389_abs__sgn__eq,axiom,
    ! [A: int] :
      ( ( ( A = zero_zero_int )
       => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = zero_zero_int ) )
      & ( ( A != zero_zero_int )
       => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = one_one_int ) ) ) ).

% abs_sgn_eq
tff(fact_7390_abs__real__def,axiom,
    ! [A: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = A ) ) ) ).

% abs_real_def
tff(fact_7391_abs__add__one__gt__zero,axiom,
    ! [X: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(X)))) ).

% abs_add_one_gt_zero
tff(fact_7392_abs__add__one__gt__zero,axiom,
    ! [X: real] : 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),aa_real_real(abs_abs_real,X)))) ).

% abs_add_one_gt_zero
tff(fact_7393_abs__add__one__gt__zero,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(abs_abs_rat,X)))) ).

% abs_add_one_gt_zero
tff(fact_7394_abs__add__one__gt__zero,axiom,
    ! [X: 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,one_one_int),aa_int_int(abs_abs_int,X)))) ).

% abs_add_one_gt_zero
tff(fact_7395_floor__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = K )
        <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,K))),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),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_7396_of__int__leD,axiom,
    ! [N: int,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),X)) ) ) ).

% of_int_leD
tff(fact_7397_of__int__leD,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

% of_int_leD
tff(fact_7398_of__int__leD,axiom,
    ! [N: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ) ).

% of_int_leD
tff(fact_7399_of__int__leD,axiom,
    ! [N: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),X)) ) ) ).

% of_int_leD
tff(fact_7400_of__int__lessD,axiom,
    ! [N: int,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),X)) ) ) ).

% of_int_lessD
tff(fact_7401_of__int__lessD,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).

% of_int_lessD
tff(fact_7402_of__int__lessD,axiom,
    ! [N: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X)) ) ) ).

% of_int_lessD
tff(fact_7403_of__int__lessD,axiom,
    ! [N: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),X)) ) ) ).

% of_int_lessD
tff(fact_7404_powr__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N) ) ) ).

% powr_realpow
tff(fact_7405_less__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ).

% less_log_iff
tff(fact_7406_log__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ).

% log_less_iff
tff(fact_7407_less__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y)) ) ) ) ).

% less_powr_iff
tff(fact_7408_powr__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X))) ) ) ) ).

% powr_less_iff
tff(fact_7409_floor__eq,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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,X) = N ) ) ) ).

% floor_eq
tff(fact_7410_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_7411_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_7412_round__diff__minimal,axiom,
    ! [Z: real,M: int] : 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,Z),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Z))))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),aa_int_real(ring_1_of_int_real,M))))) ).

% round_diff_minimal
tff(fact_7413_round__diff__minimal,axiom,
    ! [Z: rat,M: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Z))))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z),aa_int_rat(ring_1_of_int_rat,M))))) ).

% round_diff_minimal
tff(fact_7414_floor__unique,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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,X) = Z ) ) ) ).

% floor_unique
tff(fact_7415_floor__unique,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)))
       => ( aa_rat_int(archim3151403230148437115or_rat,X) = Z ) ) ) ).

% floor_unique
tff(fact_7416_floor__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( aa_real_int(archim6058952711729229775r_real,X) = A )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,A)),one_one_real))) ) ) ).

% floor_eq_iff
tff(fact_7417_floor__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( aa_rat_int(archim3151403230148437115or_rat,X) = A )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,A)),one_one_rat))) ) ) ).

% floor_eq_iff
tff(fact_7418_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_7419_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(aa_rat_fun_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_7420_le__mult__floor,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => 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,A)),aa_real_int(archim6058952711729229775r_real,B))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))) ) ) ).

% le_mult_floor
tff(fact_7421_le__mult__floor,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => 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,A)),aa_rat_int(archim3151403230148437115or_rat,B))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))) ) ) ).

% le_mult_floor
tff(fact_7422_less__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_real_int(archim6058952711729229775r_real,X)))
    <=> 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)),X)) ) ).

% less_floor_iff
tff(fact_7423_less__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X)) ) ).

% less_floor_iff
tff(fact_7424_floor__le__iff,axiom,
    ! [X: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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_7425_floor__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))) ) ).

% floor_le_iff
tff(fact_7426_floor__correct,axiom,
    ! [X: 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,X))),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int)))) ) ).

% floor_correct
tff(fact_7427_floor__correct,axiom,
    ! [X: 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,X))),X))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int)))) ) ).

% floor_correct
tff(fact_7428_abs__le__square__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),abs_abs_Code_integer(Y)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_7429_abs__le__square__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_7430_abs__le__square__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),aa_rat_rat(abs_abs_rat,Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_7431_abs__le__square__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),aa_int_int(abs_abs_int,Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_7432_powr__neg__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X) ) ) ).

% powr_neg_one
tff(fact_7433_powr__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X))) ) ) ) ).

% powr_le_iff
tff(fact_7434_le__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y)) ) ) ) ).

% le_powr_iff
tff(fact_7435_log__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ).

% log_le_iff
tff(fact_7436_le__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ).

% le_log_iff
tff(fact_7437_floor__eq2,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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,X) = N ) ) ) ).

% floor_eq2
tff(fact_7438_sgn__power__injE,axiom,
    ! [A: real,N: nat,X: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),N)) = X )
     => ( ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B)),N)) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_7439_floor__divide__lower,axiom,
    ! [Q4: real,P2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q4))
     => 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),Q4)))),Q4)),P2)) ) ).

% floor_divide_lower
tff(fact_7440_floor__divide__lower,axiom,
    ! [Q4: rat,P2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q4))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_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),Q4)))),Q4)),P2)) ) ).

% floor_divide_lower
tff(fact_7441_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_7442_power2__le__iff__abs__le,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),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(abs_abs_real,X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_7443_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_7444_power2__le__iff__abs__le,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_7445_abs__sqrt__wlog,axiom,
    ! [P: fun_Co8310303586393106575r_bool,X: code_integer] :
      ( ! [X4: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X4))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,X4),aa_nat_Code_integer(power_8256067586552552935nteger(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,abs_abs_Code_integer(X)),aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_7446_abs__sqrt__wlog,axiom,
    ! [P: fun_re1255401033344146673l_bool,X: 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(P,X4),aa_nat_real(aa_real_fun_nat_real(power_power_real,X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,aa_real_real(abs_abs_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_7447_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,X: rat] :
      ( ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,X4),aa_nat_rat(power_power_rat(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,aa_rat_rat(abs_abs_rat,X)),aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_7448_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_bool,X: int] :
      ( ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X4))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,X4),aa_nat_int(power_power_int(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(abs_abs_int,X)),aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_7449_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),one_one_Code_integer)) ) ).

% abs_square_le_1
tff(fact_7450_abs__square__le__1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)) ) ).

% abs_square_le_1
tff(fact_7451_abs__square__le__1,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat)) ) ).

% abs_square_le_1
tff(fact_7452_abs__square__le__1,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),one_one_int)) ) ).

% abs_square_le_1
tff(fact_7453_abs__square__less__1,axiom,
    ! [X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(X)),one_one_Code_integer)) ) ).

% abs_square_less_1
tff(fact_7454_abs__square__less__1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)) ) ).

% abs_square_less_1
tff(fact_7455_abs__square__less__1,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat)) ) ).

% abs_square_less_1
tff(fact_7456_abs__square__less__1,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int)) ) ).

% abs_square_less_1
tff(fact_7457_power__mono__even,axiom,
    ! [N: nat,A: code_integer,B: 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,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N))) ) ) ).

% power_mono_even
tff(fact_7458_power__mono__even,axiom,
    ! [N: nat,A: real,B: 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(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N))) ) ) ).

% power_mono_even
tff(fact_7459_power__mono__even,axiom,
    ! [N: nat,A: rat,B: 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,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ).

% power_mono_even
tff(fact_7460_power__mono__even,axiom,
    ! [N: nat,A: int,B: 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,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ).

% power_mono_even
tff(fact_7461_add__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_7462_log__add__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_7463_minus__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_7464_convex__sum__bound__le,axiom,
    ! [I6: set_real,X: fun_re6978948838062957830nteger,A: fun_re6978948838062957830nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),I6))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(X,I2))) )
     => ( ( groups7713935264441627589nteger(X,I6) = one_one_Code_integer )
       => ( ! [I2: real] :
              ( pp(aa_set_real_bool(member_real(I2),I6))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_real_Code_integer(A,I2),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7713935264441627589nteger(aa_fun7923231501384339402nteger(aTP_Lamm_rn(X),A),I6),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_7465_convex__sum__bound__le,axiom,
    ! [I6: set_nat,X: fun_nat_Code_integer,A: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),I6))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(X,I2))) )
     => ( ( groups7501900531339628137nteger(X,I6) = one_one_Code_integer )
       => ( ! [I2: nat] :
              ( pp(aa_set_nat_bool(member_nat(I2),I6))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_nat_Code_integer(A,I2),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_ro(X),A),I6),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_7466_convex__sum__bound__le,axiom,
    ! [I6: set_int,X: fun_int_Code_integer,A: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),I6))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(X,I2))) )
     => ( ( groups7873554091576472773nteger(X,I6) = one_one_Code_integer )
       => ( ! [I2: int] :
              ( pp(aa_set_int_bool(member_int(I2),I6))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_int_Code_integer(A,I2),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_rp(X),A),I6),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_7467_convex__sum__bound__le,axiom,
    ! [I6: set_real,X: fun_real_real,A: fun_real_real,B: real,Delta: real] :
      ( ! [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(X,I2))) )
     => ( ( groups8097168146408367636l_real(X,I6) = 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(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(A,I2)),B))),Delta)) )
         => 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,groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_rq(X),A),I6)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_7468_convex__sum__bound__le,axiom,
    ! [I6: set_int,X: fun_int_real,A: fun_int_real,B: real,Delta: real] :
      ( ! [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(X,I2))) )
     => ( ( groups8778361861064173332t_real(X,I6) = 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(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(A,I2)),B))),Delta)) )
         => 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,groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_rr(X),A),I6)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_7469_convex__sum__bound__le,axiom,
    ! [I6: set_real,X: fun_real_rat,A: fun_real_rat,B: rat,Delta: rat] :
      ( ! [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(X,I2))) )
     => ( ( groups1300246762558778688al_rat(X,I6) = one_one_rat )
       => ( ! [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_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_real_rat(A,I2)),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_rs(X),A),I6)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_7470_convex__sum__bound__le,axiom,
    ! [I6: set_nat,X: fun_nat_rat,A: fun_nat_rat,B: rat,Delta: rat] :
      ( ! [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(X,I2))) )
     => ( ( groups2906978787729119204at_rat(X,I6) = one_one_rat )
       => ( ! [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_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(A,I2)),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_rt(X),A),I6)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_7471_convex__sum__bound__le,axiom,
    ! [I6: set_int,X: fun_int_rat,A: fun_int_rat,B: rat,Delta: rat] :
      ( ! [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(X,I2))) )
     => ( ( groups3906332499630173760nt_rat(X,I6) = one_one_rat )
       => ( ! [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_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(A,I2)),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ru(X),A),I6)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_7472_convex__sum__bound__le,axiom,
    ! [I6: set_real,X: fun_real_int,A: fun_real_int,B: int,Delta: int] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),I6))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(X,I2))) )
     => ( ( groups1932886352136224148al_int(X,I6) = one_one_int )
       => ( ! [I2: real] :
              ( pp(aa_set_real_bool(member_real(I2),I6))
             => 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_real_int(A,I2)),B))),Delta)) )
         => 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,groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aTP_Lamm_rv(X),A),I6)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_7473_convex__sum__bound__le,axiom,
    ! [I6: set_nat,X: fun_nat_int,A: fun_nat_int,B: int,Delta: int] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),I6))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(X,I2))) )
     => ( ( groups3539618377306564664at_int(X,I6) = one_one_int )
       => ( ! [I2: nat] :
              ( pp(aa_set_nat_bool(member_nat(I2),I6))
             => 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(A,I2)),B))),Delta)) )
         => 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,groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_rw(X),A),I6)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_7474_floor__divide__upper,axiom,
    ! [Q4: real,P2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q4))
     => 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),Q4)))),one_one_real)),Q4))) ) ).

% floor_divide_upper
tff(fact_7475_floor__divide__upper,axiom,
    ! [Q4: rat,P2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q4))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,P2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_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),Q4)))),one_one_rat)),Q4))) ) ).

% floor_divide_upper
tff(fact_7476_log__minus__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_7477_powr__neg__numeral,axiom,
    ! [X: real,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),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,X),aa_num_nat(numeral_numeral_nat,N))) ) ) ).

% powr_neg_numeral
tff(fact_7478_of__int__round__abs__le,axiom,
    ! [X: 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_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X))),X))),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_abs_le
tff(fact_7479_of__int__round__abs__le,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))),X))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% of_int_round_abs_le
tff(fact_7480_round__unique_H,axiom,
    ! [X: real,N: int] :
      ( 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,X),aa_int_real(ring_1_of_int_real,N)))),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,X) = N ) ) ).

% round_unique'
tff(fact_7481_round__unique_H,axiom,
    ! [X: rat,N: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),aa_int_rat(ring_1_of_int_rat,N)))),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,X) = N ) ) ).

% round_unique'
tff(fact_7482_lemma__interval,axiom,
    ! [A: real,X: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [Y3: 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,X),Y3))),D4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Y3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),B)) ) ) ) ) ) ).

% lemma_interval
tff(fact_7483_lemma__interval__lt,axiom,
    ! [A: real,X: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [Y3: 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,X),Y3))),D4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Y3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),B)) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_7484_round__altdef,axiom,
    ! [X: 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,X)))
       => ( aa_real_int(archim8280529875227126926d_real,X) = aa_real_int(archim7802044766580827645g_real,X) ) )
      & ( ~ 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,X)))
       => ( aa_real_int(archim8280529875227126926d_real,X) = aa_real_int(archim6058952711729229775r_real,X) ) ) ) ).

% round_altdef
tff(fact_7485_round__altdef,axiom,
    ! [X: 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,X)))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X) = aa_rat_int(archim2889992004027027881ng_rat,X) ) )
      & ( ~ 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,X)))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X) = aa_rat_int(archim3151403230148437115or_rat,X) ) ) ) ).

% round_altdef
tff(fact_7486_arctan__double,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),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,X)) = 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))),X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_7487_powr__int,axiom,
    ! [X: real,I: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( 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,X),aa_int_real(ring_1_of_int_real,I)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),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,X),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,X),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I)))) ) ) ) ) ).

% powr_int
tff(fact_7488_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(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_7489_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_7490_arctan__less__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% arctan_less_zero_iff
tff(fact_7491_zero__less__arctan__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(arctan,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% zero_less_arctan_iff
tff(fact_7492_frac__of__int,axiom,
    ! [Z: int] : aa_real_real(archim2898591450579166408c_real,aa_int_real(ring_1_of_int_real,Z)) = zero_zero_real ).

% frac_of_int
tff(fact_7493_frac__of__int,axiom,
    ! [Z: int] : aa_rat_rat(archimedean_frac_rat,aa_int_rat(ring_1_of_int_rat,Z)) = zero_zero_rat ).

% frac_of_int
tff(fact_7494_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_7495_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_7496_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_7497_zless__nat__conj,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),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,W2),Z)) ) ) ).

% zless_nat_conj
tff(fact_7498_nat__neg__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = zero_zero_nat ).

% nat_neg_numeral
tff(fact_7499_nat__zminus__int,axiom,
    ! [N: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_7500_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_7501_nat__ceiling__le__eq,axiom,
    ! [X: real,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,A))) ) ).

% nat_ceiling_le_eq
tff(fact_7502_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_7503_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: 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,X)),N)),aa_int_nat(nat2,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_7504_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_7505_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: 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,X)),N)),aa_int_nat(nat2,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_7506_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_7507_arctan__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% arctan_less_iff
tff(fact_7508_arctan__monotone,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y))) ) ).

% arctan_monotone
tff(fact_7509_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_7510_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( 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,A)),aa_nat_int(semiri1314217659103216013at_int,B)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( 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,A)),aa_nat_int(semiri1314217659103216013at_int,B)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ) ).

% nat_abs_int_diff
tff(fact_7511_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_7512_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y))) ) ).

% nat_mono
tff(fact_7513_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ! [M3: int] :
        ? [N3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M3),aa_int_int(abs_abs_int,N3)))
          & pp(aa_set_int_bool(member_int(N3),S2)) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_7514_infinite__int__iff__unbounded,axiom,
    ! [S2: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ! [M3: int] :
        ? [N3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M3),aa_int_int(abs_abs_int,N3)))
          & pp(aa_set_int_bool(member_int(N3),S2)) ) ) ).

% infinite_int_iff_unbounded
tff(fact_7515_frac__ge__0,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,X))) ).

% frac_ge_0
tff(fact_7516_frac__ge__0,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,X))) ).

% frac_ge_0
tff(fact_7517_frac__lt__1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(archim2898591450579166408c_real,X)),one_one_real)) ).

% frac_lt_1
tff(fact_7518_frac__lt__1,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(archimedean_frac_rat,X)),one_one_rat)) ).

% frac_lt_1
tff(fact_7519_nat__mono__iff,axiom,
    ! [Z: int,W2: 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,W2)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ) ).

% nat_mono_iff
tff(fact_7520_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_7521_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_7522_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_7523_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X)),N))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% nat_le_iff
tff(fact_7524_zabs__def,axiom,
    ! [I: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
       => ( aa_int_int(abs_abs_int,I) = aa_int_int(uminus_uminus_int,I) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
       => ( aa_int_int(abs_abs_int,I) = I ) ) ) ).

% zabs_def
tff(fact_7525_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(modulo_modulo_int(K),L))),aa_int_int(abs_abs_int,L))) ) ).

% abs_mod_less
tff(fact_7526_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_7527_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_7528_nat__less__eq__zless,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ) ).

% nat_less_eq_zless
tff(fact_7529_nat__le__eq__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),W2))
        | 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,W2)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z)) ) ) ).

% nat_le_eq_zle
tff(fact_7530_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M = aa_int_nat(nat2,W2) )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( W2 = aa_nat_int(semiri1314217659103216013at_int,M) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
tff(fact_7531_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( aa_int_nat(nat2,W2) = M )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( W2 = aa_nat_int(semiri1314217659103216013at_int,M) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
tff(fact_7532_le__mult__nat__floor,axiom,
    ! [A: real,B: 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,A))),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,B)))),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))))) ).

% le_mult_nat_floor
tff(fact_7533_le__mult__nat__floor,axiom,
    ! [A: rat,B: 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,A))),aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,B)))),aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))))) ).

% le_mult_nat_floor
tff(fact_7534_split__nat,axiom,
    ! [P: fun_nat_bool,I: int] :
      ( pp(aa_nat_bool(P,aa_int_nat(nat2,I)))
    <=> ( ! [N3: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N3) )
           => pp(aa_nat_bool(P,N3)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
         => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).

% split_nat
tff(fact_7535_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_7536_nat__floor__neg,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_7537_floor__eq3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = N ) ) ) ).

% floor_eq3
tff(fact_7538_le__nat__floor,axiom,
    ! [X: nat,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A)))) ) ).

% le_nat_floor
tff(fact_7539_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_7540_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),M))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),aa_nat_int(semiri1314217659103216013at_int,M))) ) ) ).

% nat_less_iff
tff(fact_7541_frac__eq,axiom,
    ! [X: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,X) = X )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% frac_eq
tff(fact_7542_frac__eq,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,X) = X )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),one_one_rat)) ) ) ).

% frac_eq
tff(fact_7543_floor__eq4,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = N ) ) ) ).

% floor_eq4
tff(fact_7544_frac__add,axiom,
    ! [X: real,Y: 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,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real) ) ) ) ).

% frac_add
tff(fact_7545_frac__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
       => ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
       => ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat) ) ) ) ).

% frac_add
tff(fact_7546_diff__nat__eq__if,axiom,
    ! [Z6: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z6),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) = aa_int_nat(nat2,Z) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z6),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) = 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),Z6)),zero_zero_int)),zero_zero_nat),aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z6))) ) ) ) ).

% diff_nat_eq_if
tff(fact_7547_of__int__of__nat,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_rat(ring_1_of_int_rat,K) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_rat(ring_1_of_int_rat,K) = aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,K)) ) ) ) ).

% of_int_of_nat
tff(fact_7548_of__int__of__nat,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,K)) ) ) ) ).

% of_int_of_nat
tff(fact_7549_of__int__of__nat,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_int(ring_1_of_int_int,K) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_int(ring_1_of_int_int,K) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,K)) ) ) ) ).

% of_int_of_nat
tff(fact_7550_of__int__of__nat,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_real(ring_1_of_int_real,K) = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_real(ring_1_of_int_real,K) = aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,K)) ) ) ) ).

% of_int_of_nat
tff(fact_7551_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_7552_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_7553_incr__lemma,axiom,
    ! [D: int,Z: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),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,X),Z))),one_one_int)),D)))) ) ).

% incr_lemma
tff(fact_7554_decr__lemma,axiom,
    ! [D: int,X: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),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,X),Z))),one_one_int)),D))),Z)) ) ).

% decr_lemma
tff(fact_7555_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q4: 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,Q4),L)),R2) )
         => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R2))) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_7556_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_7557_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(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),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,Q3),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_7558_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(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K3) ) )
        | ? [L4: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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(aa_int2565445551602696444nt_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_7559_floor__add,axiom,
    ! [X: real,Y: 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,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = 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,X)),aa_real_int(archim6058952711729229775r_real,Y))),one_one_int) ) ) ) ).

% floor_add
tff(fact_7560_floor__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
       => ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
       => ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = 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,X)),aa_rat_int(archim3151403230148437115or_rat,Y))),one_one_int) ) ) ) ).

% floor_add
tff(fact_7561_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_7562_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)))) ) ) ) ).

% arctan_add
tff(fact_7563_powr__real__of__int,axiom,
    ! [X: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( 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,X),aa_int_real(ring_1_of_int_real,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),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,X),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,X),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,N)))) ) ) ) ) ).

% powr_real_of_int
tff(fact_7564_case__prod__Pair__iden,axiom,
    ! [P2: produc5207004643709237345nteger] : produc9069023536360486177nteger(produc4282985567423050387nteger,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_7565_case__prod__Pair__iden,axiom,
    ! [P2: produc1616345218596728535nteger] : produc5888382909295257357nteger(produc7384524951051686153nteger,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_7566_case__prod__Pair__iden,axiom,
    ! [P2: produc4441460161354545860nt_int] : produc7428976581010258791nt_int(produc1560874099174258678nt_int,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_7567_case__prod__Pair__iden,axiom,
    ! [P2: produc6768802789813714928nt_int] : produc3712466714231214015nt_int(produc4852315488734385442nt_int,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_7568_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_int_int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(product_Pair_int_int),P2) = P2 ).

% case_prod_Pair_iden
tff(fact_7569_the__elem__eq,axiom,
    ! [X: produc3843707927480180839at_nat] : the_el221668144340439132at_nat(insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) = X ).

% the_elem_eq
tff(fact_7570_the__elem__eq,axiom,
    ! [X: product_prod_nat_nat] : the_el2281957884133575798at_nat(insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) = X ).

% the_elem_eq
tff(fact_7571_the__elem__eq,axiom,
    ! [X: real] : the_elem_real(insert_real(X,bot_bot_set_real)) = X ).

% the_elem_eq
tff(fact_7572_the__elem__eq,axiom,
    ! [X: nat] : aa_set_nat_nat(the_elem_nat,insert_nat(X,bot_bot_set_nat)) = X ).

% the_elem_eq
tff(fact_7573_the__elem__eq,axiom,
    ! [X: int] : the_elem_int(insert_int(X,bot_bot_set_int)) = X ).

% the_elem_eq
tff(fact_7574_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),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,X),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_7575_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),X))
     => ( 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,X),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,X))) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_7576_sum__diff1_H__aux,axiom,
    ! [F3: set_complex,I6: set_complex,F: fun_complex_complex,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_ej(I6),F))),F3))
       => ( ( pp(aa_set_complex_bool(member_complex(I),I6))
           => ( groups808145749697022017omplex(F,minus_811609699411566653omplex(I6,insert_complex(I,bot_bot_set_complex))) = aa_complex_complex(minus_minus_complex(groups808145749697022017omplex(F,I6)),aa_complex_complex(F,I)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(I),I6))
           => ( groups808145749697022017omplex(F,minus_811609699411566653omplex(I6,insert_complex(I,bot_bot_set_complex))) = groups808145749697022017omplex(F,I6) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_7577_sum__diff1_H__aux,axiom,
    ! [F3: set_real,I6: set_real,F: fun_real_complex,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,F3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_ed(I6),F))),F3))
       => ( ( pp(aa_set_real_bool(member_real(I),I6))
           => ( groups5683813829254066239omplex(F,minus_minus_set_real(I6,insert_real(I,bot_bot_set_real))) = aa_complex_complex(minus_minus_complex(groups5683813829254066239omplex(F,I6)),aa_real_complex(F,I)) ) )
          & ( ~ pp(aa_set_real_bool(member_real(I),I6))
           => ( groups5683813829254066239omplex(F,minus_minus_set_real(I6,insert_real(I,bot_bot_set_real))) = groups5683813829254066239omplex(F,I6) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_7578_sum__diff1_H__aux,axiom,
    ! [F3: set_complex,I6: set_complex,F: fun_complex_real,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_er(I6),F))),F3))
       => ( ( pp(aa_set_complex_bool(member_complex(I),I6))
           => ( groups5737402329758386879x_real(F,minus_811609699411566653omplex(I6,insert_complex(I,bot_bot_set_complex))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5737402329758386879x_real(F,I6)),aa_complex_real(F,I)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(I),I6))
           => ( groups5737402329758386879x_real(F,minus_811609699411566653omplex(I6,insert_complex(I,bot_bot_set_complex))) = groups5737402329758386879x_real(F,I6) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_7579_sum__diff1_H__aux,axiom,
    ! [F3: set_real,I6: set_real,F: fun_real_real,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,F3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_el(I6),F))),F3))
       => ( ( pp(aa_set_real_bool(member_real(I),I6))
           => ( groups97945582718554045l_real(F,minus_minus_set_real(I6,insert_real(I,bot_bot_set_real))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups97945582718554045l_real(F,I6)),aa_real_real(F,I)) ) )
          & ( ~ pp(aa_set_real_bool(member_real(I),I6))
           => ( groups97945582718554045l_real(F,minus_minus_set_real(I6,insert_real(I,bot_bot_set_real))) = groups97945582718554045l_real(F,I6) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_7580_sum__diff1_H__aux,axiom,
    ! [F3: set_complex,I6: set_complex,F: fun_complex_rat,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(collect_complex(aa_fun2759627087638585025x_bool(aTP_Lamm_rx(I6),F))),F3))
       => ( ( pp(aa_set_complex_bool(member_complex(I),I6))
           => ( groups2276542476275365739ex_rat(F,minus_811609699411566653omplex(I6,insert_complex(I,bot_bot_set_complex))) = aa_rat_rat(minus_minus_rat(groups2276542476275365739ex_rat(F,I6)),aa_complex_rat(F,I)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(I),I6))
           => ( groups2276542476275365739ex_rat(F,minus_811609699411566653omplex(I6,insert_complex(I,bot_bot_set_complex))) = groups2276542476275365739ex_rat(F,I6) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_7581_sum__diff1_H__aux,axiom,
    ! [F3: set_real,I6: set_real,F: fun_real_rat,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,F3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_et(I6),F))),F3))
       => ( ( pp(aa_set_real_bool(member_real(I),I6))
           => ( groups3269169158384524137al_rat(F,minus_minus_set_real(I6,insert_real(I,bot_bot_set_real))) = aa_rat_rat(minus_minus_rat(groups3269169158384524137al_rat(F,I6)),aa_real_rat(F,I)) ) )
          & ( ~ pp(aa_set_real_bool(member_real(I),I6))
           => ( groups3269169158384524137al_rat(F,minus_minus_set_real(I6,insert_real(I,bot_bot_set_real))) = groups3269169158384524137al_rat(F,I6) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_7582_sum__diff1_H__aux,axiom,
    ! [F3: set_complex,I6: set_complex,F: fun_complex_int,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(collect_complex(aa_fun4637897056364002669x_bool(aTP_Lamm_ry(I6),F))),F3))
       => ( ( pp(aa_set_complex_bool(member_complex(I),I6))
           => ( groups2909182065852811199ex_int(F,minus_811609699411566653omplex(I6,insert_complex(I,bot_bot_set_complex))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups2909182065852811199ex_int(F,I6)),aa_complex_int(F,I)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(I),I6))
           => ( groups2909182065852811199ex_int(F,minus_811609699411566653omplex(I6,insert_complex(I,bot_bot_set_complex))) = groups2909182065852811199ex_int(F,I6) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_7583_sum__diff1_H__aux,axiom,
    ! [F3: set_real,I6: set_real,F: fun_real_int,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,F3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_fun2745684120339038577l_bool(aTP_Lamm_rz(I6),F))),F3))
       => ( ( pp(aa_set_real_bool(member_real(I),I6))
           => ( groups3901808747961969597al_int(F,minus_minus_set_real(I6,insert_real(I,bot_bot_set_real))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3901808747961969597al_int(F,I6)),aa_real_int(F,I)) ) )
          & ( ~ pp(aa_set_real_bool(member_real(I),I6))
           => ( groups3901808747961969597al_int(F,minus_minus_set_real(I6,insert_real(I,bot_bot_set_real))) = groups3901808747961969597al_int(F,I6) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_7584_sum__diff1_H__aux,axiom,
    ! [F3: set_nat,I6: set_nat,F: fun_nat_complex,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_ef(I6),F))),F3))
       => ( ( pp(aa_set_nat_bool(member_nat(I),I6))
           => ( groups8515261248781899619omplex(F,minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat))) = aa_complex_complex(minus_minus_complex(groups8515261248781899619omplex(F,I6)),aa_nat_complex(F,I)) ) )
          & ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
           => ( groups8515261248781899619omplex(F,minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat))) = groups8515261248781899619omplex(F,I6) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_7585_sum__diff1_H__aux,axiom,
    ! [F3: set_nat,I6: set_nat,F: fun_nat_real,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_en(I6),F))),F3))
       => ( ( pp(aa_set_nat_bool(member_nat(I),I6))
           => ( groups8560362682196896993t_real(F,minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8560362682196896993t_real(F,I6)),aa_nat_real(F,I)) ) )
          & ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
           => ( groups8560362682196896993t_real(F,minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat))) = groups8560362682196896993t_real(F,I6) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_7586_inverse__zero,axiom,
    aa_complex_complex(invers8013647133539491842omplex,zero_zero_complex) = zero_zero_complex ).

% inverse_zero
tff(fact_7587_inverse__zero,axiom,
    aa_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

% inverse_zero
tff(fact_7588_inverse__zero,axiom,
    aa_rat_rat(inverse_inverse_rat,zero_zero_rat) = zero_zero_rat ).

% inverse_zero
tff(fact_7589_inverse__nonzero__iff__nonzero,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_7590_inverse__nonzero__iff__nonzero,axiom,
    ! [A: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_7591_inverse__nonzero__iff__nonzero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_7592_exp__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% exp_less_cancel_iff
tff(fact_7593_exp__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y))) ) ).

% exp_less_mono
tff(fact_7594_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_7595_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_7596_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(inverse_inverse_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_7597_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_7598_inverse__less__iff__less,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% inverse_less_iff_less
tff(fact_7599_inverse__less__iff__less,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% inverse_less_iff_less
tff(fact_7600_inverse__less__iff__less__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_7601_inverse__less__iff__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_7602_inverse__negative__iff__negative,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% inverse_negative_iff_negative
tff(fact_7603_inverse__negative__iff__negative,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% inverse_negative_iff_negative
tff(fact_7604_inverse__positive__iff__positive,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% inverse_positive_iff_positive
tff(fact_7605_inverse__positive__iff__positive,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% inverse_positive_iff_positive
tff(fact_7606_exp__zero,axiom,
    aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).

% exp_zero
tff(fact_7607_exp__zero,axiom,
    aa_real_real(exp_real,zero_zero_real) = one_one_real ).

% exp_zero
tff(fact_7608_sum_Oempty_H,axiom,
    ! [P2: fun_real_complex] : groups5683813829254066239omplex(P2,bot_bot_set_real) = zero_zero_complex ).

% sum.empty'
tff(fact_7609_sum_Oempty_H,axiom,
    ! [P2: fun_real_real] : groups97945582718554045l_real(P2,bot_bot_set_real) = zero_zero_real ).

% sum.empty'
tff(fact_7610_sum_Oempty_H,axiom,
    ! [P2: fun_real_rat] : groups3269169158384524137al_rat(P2,bot_bot_set_real) = zero_zero_rat ).

% sum.empty'
tff(fact_7611_sum_Oempty_H,axiom,
    ! [P2: fun_real_nat] : groups3904299218471019873al_nat(P2,bot_bot_set_real) = zero_zero_nat ).

% sum.empty'
tff(fact_7612_sum_Oempty_H,axiom,
    ! [P2: fun_real_int] : groups3901808747961969597al_int(P2,bot_bot_set_real) = zero_zero_int ).

% sum.empty'
tff(fact_7613_sum_Oempty_H,axiom,
    ! [P2: fun_nat_complex] : groups8515261248781899619omplex(P2,bot_bot_set_nat) = zero_zero_complex ).

% sum.empty'
tff(fact_7614_sum_Oempty_H,axiom,
    ! [P2: fun_nat_real] : groups8560362682196896993t_real(P2,bot_bot_set_nat) = zero_zero_real ).

% sum.empty'
tff(fact_7615_sum_Oempty_H,axiom,
    ! [P2: fun_nat_rat] : groups1351286907653491341at_rat(P2,bot_bot_set_nat) = zero_zero_rat ).

% sum.empty'
tff(fact_7616_sum_Oempty_H,axiom,
    ! [P2: fun_nat_nat] : aa_set_nat_nat(groups1986416967739987077at_nat(P2),bot_bot_set_nat) = zero_zero_nat ).

% sum.empty'
tff(fact_7617_sum_Oempty_H,axiom,
    ! [P2: fun_nat_int] : groups1983926497230936801at_int(P2,bot_bot_set_nat) = zero_zero_int ).

% sum.empty'
tff(fact_7618_sum_Oeq__sum,axiom,
    ! [I6: set_nat,P2: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( aa_set_nat_nat(groups1986416967739987077at_nat(P2),I6) = aa_set_nat_nat(groups3542108847815614940at_nat(P2),I6) ) ) ).

% sum.eq_sum
tff(fact_7619_sum_Oeq__sum,axiom,
    ! [I6: set_complex,P2: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( groups808145749697022017omplex(P2,I6) = groups7754918857620584856omplex(P2,I6) ) ) ).

% sum.eq_sum
tff(fact_7620_sum_Oeq__sum,axiom,
    ! [I6: set_nat,P2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( groups8560362682196896993t_real(P2,I6) = groups6591440286371151544t_real(P2,I6) ) ) ).

% sum.eq_sum
tff(fact_7621_inverse__le__iff__le,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% inverse_le_iff_le
tff(fact_7622_inverse__le__iff__le,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% inverse_le_iff_le
tff(fact_7623_inverse__le__iff__le__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_7624_inverse__le__iff__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_7625_right__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,A)) = one_one_complex ) ) ).

% right_inverse
tff(fact_7626_right__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(inverse_inverse_real,A)) = one_one_real ) ) ).

% right_inverse
tff(fact_7627_right__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(inverse_inverse_rat,A)) = one_one_rat ) ) ).

% right_inverse
tff(fact_7628_left__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),A) = one_one_complex ) ) ).

% left_inverse
tff(fact_7629_left__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).

% left_inverse
tff(fact_7630_left__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).

% left_inverse
tff(fact_7631_one__less__exp__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(exp_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% one_less_exp_iff
tff(fact_7632_exp__less__one__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% exp_less_one_iff
tff(fact_7633_sum_Oinsert_H,axiom,
    ! [I6: set_real,P2: fun_real_complex,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_ed(I6),P2))))
     => ( ( pp(aa_set_real_bool(member_real(I),I6))
         => ( groups5683813829254066239omplex(P2,insert_real(I,I6)) = groups5683813829254066239omplex(P2,I6) ) )
        & ( ~ pp(aa_set_real_bool(member_real(I),I6))
         => ( groups5683813829254066239omplex(P2,insert_real(I,I6)) = aa_complex_complex(plus_plus_complex(aa_real_complex(P2,I)),groups5683813829254066239omplex(P2,I6)) ) ) ) ) ).

% sum.insert'
tff(fact_7634_sum_Oinsert_H,axiom,
    ! [I6: set_nat,P2: fun_nat_complex,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_ef(I6),P2))))
     => ( ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( groups8515261248781899619omplex(P2,insert_nat(I,I6)) = groups8515261248781899619omplex(P2,I6) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
         => ( groups8515261248781899619omplex(P2,insert_nat(I,I6)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(P2,I)),groups8515261248781899619omplex(P2,I6)) ) ) ) ) ).

% sum.insert'
tff(fact_7635_sum_Oinsert_H,axiom,
    ! [I6: set_int,P2: fun_int_complex,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_eh(I6),P2))))
     => ( ( pp(aa_set_int_bool(member_int(I),I6))
         => ( groups267424677133301183omplex(P2,insert_int(I,I6)) = groups267424677133301183omplex(P2,I6) ) )
        & ( ~ pp(aa_set_int_bool(member_int(I),I6))
         => ( groups267424677133301183omplex(P2,insert_int(I,I6)) = aa_complex_complex(plus_plus_complex(aa_int_complex(P2,I)),groups267424677133301183omplex(P2,I6)) ) ) ) ) ).

% sum.insert'
tff(fact_7636_sum_Oinsert_H,axiom,
    ! [I6: set_complex,P2: fun_complex_complex,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_ej(I6),P2))))
     => ( ( pp(aa_set_complex_bool(member_complex(I),I6))
         => ( groups808145749697022017omplex(P2,insert_complex(I,I6)) = groups808145749697022017omplex(P2,I6) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(I),I6))
         => ( groups808145749697022017omplex(P2,insert_complex(I,I6)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(P2,I)),groups808145749697022017omplex(P2,I6)) ) ) ) ) ).

% sum.insert'
tff(fact_7637_sum_Oinsert_H,axiom,
    ! [I6: set_real,P2: fun_real_real,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_el(I6),P2))))
     => ( ( pp(aa_set_real_bool(member_real(I),I6))
         => ( groups97945582718554045l_real(P2,insert_real(I,I6)) = groups97945582718554045l_real(P2,I6) ) )
        & ( ~ pp(aa_set_real_bool(member_real(I),I6))
         => ( groups97945582718554045l_real(P2,insert_real(I,I6)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(P2,I)),groups97945582718554045l_real(P2,I6)) ) ) ) ) ).

% sum.insert'
tff(fact_7638_sum_Oinsert_H,axiom,
    ! [I6: set_nat,P2: fun_nat_real,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_en(I6),P2))))
     => ( ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( groups8560362682196896993t_real(P2,insert_nat(I,I6)) = groups8560362682196896993t_real(P2,I6) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
         => ( groups8560362682196896993t_real(P2,insert_nat(I,I6)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(P2,I)),groups8560362682196896993t_real(P2,I6)) ) ) ) ) ).

% sum.insert'
tff(fact_7639_sum_Oinsert_H,axiom,
    ! [I6: set_int,P2: fun_int_real,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_ep(I6),P2))))
     => ( ( pp(aa_set_int_bool(member_int(I),I6))
         => ( groups1523912220035142973t_real(P2,insert_int(I,I6)) = groups1523912220035142973t_real(P2,I6) ) )
        & ( ~ pp(aa_set_int_bool(member_int(I),I6))
         => ( groups1523912220035142973t_real(P2,insert_int(I,I6)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(P2,I)),groups1523912220035142973t_real(P2,I6)) ) ) ) ) ).

% sum.insert'
tff(fact_7640_sum_Oinsert_H,axiom,
    ! [I6: set_complex,P2: fun_complex_real,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_er(I6),P2))))
     => ( ( pp(aa_set_complex_bool(member_complex(I),I6))
         => ( groups5737402329758386879x_real(P2,insert_complex(I,I6)) = groups5737402329758386879x_real(P2,I6) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(I),I6))
         => ( groups5737402329758386879x_real(P2,insert_complex(I,I6)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(P2,I)),groups5737402329758386879x_real(P2,I6)) ) ) ) ) ).

% sum.insert'
tff(fact_7641_sum_Oinsert_H,axiom,
    ! [I6: set_real,P2: fun_real_rat,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_et(I6),P2))))
     => ( ( pp(aa_set_real_bool(member_real(I),I6))
         => ( groups3269169158384524137al_rat(P2,insert_real(I,I6)) = groups3269169158384524137al_rat(P2,I6) ) )
        & ( ~ pp(aa_set_real_bool(member_real(I),I6))
         => ( groups3269169158384524137al_rat(P2,insert_real(I,I6)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(P2,I)),groups3269169158384524137al_rat(P2,I6)) ) ) ) ) ).

% sum.insert'
tff(fact_7642_sum_Oinsert_H,axiom,
    ! [I6: set_nat,P2: fun_nat_rat,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_ev(I6),P2))))
     => ( ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( groups1351286907653491341at_rat(P2,insert_nat(I,I6)) = groups1351286907653491341at_rat(P2,I6) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
         => ( groups1351286907653491341at_rat(P2,insert_nat(I,I6)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(P2,I)),groups1351286907653491341at_rat(P2,I6)) ) ) ) ) ).

% sum.insert'
tff(fact_7643_nonzero__norm__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(real_V7735802525324610683m_real,A)) ) ) ).

% nonzero_norm_inverse
tff(fact_7644_nonzero__norm__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = aa_real_real(inverse_inverse_real,aa_complex_real(real_V1022390504157884413omplex,A)) ) ) ).

% nonzero_norm_inverse
tff(fact_7645_field__class_Ofield__inverse__zero,axiom,
    aa_complex_complex(invers8013647133539491842omplex,zero_zero_complex) = zero_zero_complex ).

% field_class.field_inverse_zero
tff(fact_7646_field__class_Ofield__inverse__zero,axiom,
    aa_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

% field_class.field_inverse_zero
tff(fact_7647_field__class_Ofield__inverse__zero,axiom,
    aa_rat_rat(inverse_inverse_rat,zero_zero_rat) = zero_zero_rat ).

% field_class.field_inverse_zero
tff(fact_7648_inverse__zero__imp__zero,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = zero_zero_complex )
     => ( A = zero_zero_complex ) ) ).

% inverse_zero_imp_zero
tff(fact_7649_inverse__zero__imp__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = zero_zero_real )
     => ( A = zero_zero_real ) ) ).

% inverse_zero_imp_zero
tff(fact_7650_inverse__zero__imp__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = zero_zero_rat )
     => ( A = zero_zero_rat ) ) ).

% inverse_zero_imp_zero
tff(fact_7651_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(invers8013647133539491842omplex,B) )
     => ( ( A != zero_zero_complex )
       => ( ( B != zero_zero_complex )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_7652_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B) )
     => ( ( A != zero_zero_real )
       => ( ( B != zero_zero_real )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_7653_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(inverse_inverse_rat,B) )
     => ( ( A != zero_zero_rat )
       => ( ( B != zero_zero_rat )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_7654_nonzero__inverse__inverse__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_7655_nonzero__inverse__inverse__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,aa_real_real(inverse_inverse_real,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_7656_nonzero__inverse__inverse__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(inverse_inverse_rat,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_7657_nonzero__imp__inverse__nonzero,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) != zero_zero_complex ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_7658_nonzero__imp__inverse__nonzero,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,A) != zero_zero_real ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_7659_nonzero__imp__inverse__nonzero,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) != zero_zero_rat ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_7660_exp__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% exp_less_cancel
tff(fact_7661_exp__not__eq__zero,axiom,
    ! [X: complex] : aa_complex_complex(exp_complex,X) != zero_zero_complex ).

% exp_not_eq_zero
tff(fact_7662_exp__not__eq__zero,axiom,
    ! [X: real] : aa_real_real(exp_real,X) != zero_zero_real ).

% exp_not_eq_zero
tff(fact_7663_sum_Onon__neutral_H,axiom,
    ! [G: fun_real_complex,I6: set_real] : groups5683813829254066239omplex(G,collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_sa(G),I6))) = groups5683813829254066239omplex(G,I6) ).

% sum.non_neutral'
tff(fact_7664_sum_Onon__neutral_H,axiom,
    ! [G: fun_nat_complex,I6: set_nat] : groups8515261248781899619omplex(G,collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_sb(G),I6))) = groups8515261248781899619omplex(G,I6) ).

% sum.non_neutral'
tff(fact_7665_sum_Onon__neutral_H,axiom,
    ! [G: fun_int_complex,I6: set_int] : groups267424677133301183omplex(G,collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_sc(G),I6))) = groups267424677133301183omplex(G,I6) ).

% sum.non_neutral'
tff(fact_7666_sum_Onon__neutral_H,axiom,
    ! [G: fun_real_real,I6: set_real] : groups97945582718554045l_real(G,collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_sd(G),I6))) = groups97945582718554045l_real(G,I6) ).

% sum.non_neutral'
tff(fact_7667_sum_Onon__neutral_H,axiom,
    ! [G: fun_nat_real,I6: set_nat] : groups8560362682196896993t_real(G,collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_se(G),I6))) = groups8560362682196896993t_real(G,I6) ).

% sum.non_neutral'
tff(fact_7668_sum_Onon__neutral_H,axiom,
    ! [G: fun_int_real,I6: set_int] : groups1523912220035142973t_real(G,collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_sf(G),I6))) = groups1523912220035142973t_real(G,I6) ).

% sum.non_neutral'
tff(fact_7669_sum_Onon__neutral_H,axiom,
    ! [G: fun_real_rat,I6: set_real] : groups3269169158384524137al_rat(G,collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_sg(G),I6))) = groups3269169158384524137al_rat(G,I6) ).

% sum.non_neutral'
tff(fact_7670_sum_Onon__neutral_H,axiom,
    ! [G: fun_nat_rat,I6: set_nat] : groups1351286907653491341at_rat(G,collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_sh(G),I6))) = groups1351286907653491341at_rat(G,I6) ).

% sum.non_neutral'
tff(fact_7671_sum_Onon__neutral_H,axiom,
    ! [G: fun_int_rat,I6: set_int] : groups2350640619554545897nt_rat(G,collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_si(G),I6))) = groups2350640619554545897nt_rat(G,I6) ).

% sum.non_neutral'
tff(fact_7672_sum_Onon__neutral_H,axiom,
    ! [G: fun_real_nat,I6: set_real] : groups3904299218471019873al_nat(G,collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_sj(G),I6))) = groups3904299218471019873al_nat(G,I6) ).

% sum.non_neutral'
tff(fact_7673_norm__inverse__le__norm,axiom,
    ! [R2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),aa_real_real(real_V7735802525324610683m_real,X)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(inverse_inverse_real,X))),aa_real_real(inverse_inverse_real,R2))) ) ) ).

% norm_inverse_le_norm
tff(fact_7674_norm__inverse__le__norm,axiom,
    ! [R2: real,X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),aa_complex_real(real_V1022390504157884413omplex,X)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(invers8013647133539491842omplex,X))),aa_real_real(inverse_inverse_real,R2))) ) ) ).

% norm_inverse_le_norm
tff(fact_7675_inverse__less__imp__less,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% inverse_less_imp_less
tff(fact_7676_inverse__less__imp__less,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% inverse_less_imp_less
tff(fact_7677_less__imp__inverse__less,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A))) ) ) ).

% less_imp_inverse_less
tff(fact_7678_less__imp__inverse__less,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A))) ) ) ).

% less_imp_inverse_less
tff(fact_7679_inverse__less__imp__less__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% inverse_less_imp_less_neg
tff(fact_7680_inverse__less__imp__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% inverse_less_imp_less_neg
tff(fact_7681_less__imp__inverse__less__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A))) ) ) ).

% less_imp_inverse_less_neg
tff(fact_7682_less__imp__inverse__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A))) ) ) ).

% less_imp_inverse_less_neg
tff(fact_7683_inverse__negative__imp__negative,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real))
     => ( ( A != zero_zero_real )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).

% inverse_negative_imp_negative
tff(fact_7684_inverse__negative__imp__negative,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat))
     => ( ( A != zero_zero_rat )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ).

% inverse_negative_imp_negative
tff(fact_7685_inverse__positive__imp__positive,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A)))
     => ( ( A != zero_zero_real )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ).

% inverse_positive_imp_positive
tff(fact_7686_inverse__positive__imp__positive,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A)))
     => ( ( A != zero_zero_rat )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ).

% inverse_positive_imp_positive
tff(fact_7687_negative__imp__inverse__negative,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)) ) ).

% negative_imp_inverse_negative
tff(fact_7688_negative__imp__inverse__negative,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)) ) ).

% negative_imp_inverse_negative
tff(fact_7689_positive__imp__inverse__positive,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))) ) ).

% positive_imp_inverse_positive
tff(fact_7690_positive__imp__inverse__positive,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))) ) ).

% positive_imp_inverse_positive
tff(fact_7691_nonzero__inverse__mult__distrib,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,B)),aa_complex_complex(invers8013647133539491842omplex,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_7692_nonzero__inverse__mult__distrib,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_7693_nonzero__inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_7694_nonzero__inverse__minus__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(invers8013647133539491842omplex,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_7695_nonzero__inverse__minus__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_7696_nonzero__inverse__minus__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_7697_not__exp__less__zero,axiom,
    ! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),zero_zero_real)) ).

% not_exp_less_zero
tff(fact_7698_exp__gt__zero,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(exp_real,X))) ).

% exp_gt_zero
tff(fact_7699_exp__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ? [X4: real] : aa_real_real(exp_real,X4) = Y ) ).

% exp_total
tff(fact_7700_nonzero__abs__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(abs_abs_real,A)) ) ) ).

% nonzero_abs_inverse
tff(fact_7701_nonzero__abs__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(inverse_inverse_rat,A)) = aa_rat_rat(inverse_inverse_rat,aa_rat_rat(abs_abs_rat,A)) ) ) ).

% nonzero_abs_inverse
tff(fact_7702_inverse__le__imp__le,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% inverse_le_imp_le
tff(fact_7703_inverse__le__imp__le,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% inverse_le_imp_le
tff(fact_7704_le__imp__inverse__le,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A))) ) ) ).

% le_imp_inverse_le
tff(fact_7705_le__imp__inverse__le,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A))) ) ) ).

% le_imp_inverse_le
tff(fact_7706_inverse__le__imp__le__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% inverse_le_imp_le_neg
tff(fact_7707_inverse__le__imp__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% inverse_le_imp_le_neg
tff(fact_7708_le__imp__inverse__le__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A))) ) ) ).

% le_imp_inverse_le_neg
tff(fact_7709_le__imp__inverse__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A))) ) ) ).

% le_imp_inverse_le_neg
tff(fact_7710_inverse__le__1__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,X)),one_one_real))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

% inverse_le_1_iff
tff(fact_7711_inverse__le__1__iff,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,X)),one_one_rat))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ) ).

% inverse_le_1_iff
tff(fact_7712_one__less__inverse,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(inverse_inverse_real,A))) ) ) ).

% one_less_inverse
tff(fact_7713_one__less__inverse,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,A))) ) ) ).

% one_less_inverse
tff(fact_7714_one__less__inverse__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(inverse_inverse_real,X)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% one_less_inverse_iff
tff(fact_7715_one__less__inverse__iff,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,X)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),one_one_rat)) ) ) ).

% one_less_inverse_iff
tff(fact_7716_inverse__add,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),aa_complex_complex(invers8013647133539491842omplex,A))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% inverse_add
tff(fact_7717_inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(inverse_inverse_real,A))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% inverse_add
tff(fact_7718_inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(inverse_inverse_rat,A))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% inverse_add
tff(fact_7719_division__ring__inverse__add,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(plus_plus_complex(A),B))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_7720_division__ring__inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_7721_division__ring__inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_7722_field__class_Ofield__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),A) = one_one_complex ) ) ).

% field_class.field_inverse
tff(fact_7723_field__class_Ofield__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).

% field_class.field_inverse
tff(fact_7724_field__class_Ofield__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).

% field_class.field_inverse
tff(fact_7725_division__ring__inverse__diff,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(minus_minus_complex(B),A))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_7726_division__ring__inverse__diff,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_7727_division__ring__inverse__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(minus_minus_rat(B),A))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_7728_nonzero__inverse__eq__divide,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_7729_nonzero__inverse__eq__divide,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_7730_nonzero__inverse__eq__divide,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_7731_sum_Odistrib__triv_H,axiom,
    ! [I6: set_nat,G: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( groups8560362682196896993t_real(aa_fun1839206346889086768t_real(aTP_Lamm_sk(G),H),I6) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8560362682196896993t_real(G,I6)),groups8560362682196896993t_real(H,I6)) ) ) ).

% sum.distrib_triv'
tff(fact_7732_sum_Odistrib__triv_H,axiom,
    ! [I6: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( groups1523912220035142973t_real(aa_fun3252798400008884712t_real(aTP_Lamm_sl(G),H),I6) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups1523912220035142973t_real(G,I6)),groups1523912220035142973t_real(H,I6)) ) ) ).

% sum.distrib_triv'
tff(fact_7733_sum_Odistrib__triv_H,axiom,
    ! [I6: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( groups5737402329758386879x_real(aa_fun8933909628229591020x_real(aTP_Lamm_sm(G),H),I6) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5737402329758386879x_real(G,I6)),groups5737402329758386879x_real(H,I6)) ) ) ).

% sum.distrib_triv'
tff(fact_7734_sum_Odistrib__triv_H,axiom,
    ! [I6: set_nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( groups1351286907653491341at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_sn(G),H),I6) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups1351286907653491341at_rat(G,I6)),groups1351286907653491341at_rat(H,I6)) ) ) ).

% sum.distrib_triv'
tff(fact_7735_sum_Odistrib__triv_H,axiom,
    ! [I6: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( groups2350640619554545897nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_so(G),H),I6) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2350640619554545897nt_rat(G,I6)),groups2350640619554545897nt_rat(H,I6)) ) ) ).

% sum.distrib_triv'
tff(fact_7736_sum_Odistrib__triv_H,axiom,
    ! [I6: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( groups2276542476275365739ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_sp(G),H),I6) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2276542476275365739ex_rat(G,I6)),groups2276542476275365739ex_rat(H,I6)) ) ) ).

% sum.distrib_triv'
tff(fact_7737_sum_Odistrib__triv_H,axiom,
    ! [I6: set_nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( aa_set_nat_nat(groups1986416967739987077at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_sq(G),H)),I6) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups1986416967739987077at_nat(G),I6)),aa_set_nat_nat(groups1986416967739987077at_nat(H),I6)) ) ) ).

% sum.distrib_triv'
tff(fact_7738_sum_Odistrib__triv_H,axiom,
    ! [I6: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( groups2985770679641041633nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_sr(G),H),I6) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups2985770679641041633nt_nat(G,I6)),groups2985770679641041633nt_nat(H,I6)) ) ) ).

% sum.distrib_triv'
tff(fact_7739_sum_Odistrib__triv_H,axiom,
    ! [I6: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( groups2911672536361861475ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_ss(G),H),I6) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups2911672536361861475ex_nat(G,I6)),groups2911672536361861475ex_nat(H,I6)) ) ) ).

% sum.distrib_triv'
tff(fact_7740_sum_Odistrib__triv_H,axiom,
    ! [I6: set_nat,G: fun_nat_int,H: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( groups1983926497230936801at_int(aa_fun5565043122120104496at_int(aTP_Lamm_st(G),H),I6) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups1983926497230936801at_int(G,I6)),groups1983926497230936801at_int(H,I6)) ) ) ).

% sum.distrib_triv'
tff(fact_7741_exp__gt__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(exp_real,X))) ) ).

% exp_gt_one
tff(fact_7742_sum_Omono__neutral__cong__right_H,axiom,
    ! [S2: set_real,T2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
           => ( aa_real_complex(G,X4) = zero_zero_complex ) )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),S2))
             => ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
         => ( groups5683813829254066239omplex(G,T2) = groups5683813829254066239omplex(H,S2) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_7743_sum_Omono__neutral__cong__right_H,axiom,
    ! [S2: set_real,T2: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
           => ( aa_real_real(G,X4) = zero_zero_real ) )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),S2))
             => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
         => ( groups97945582718554045l_real(G,T2) = groups97945582718554045l_real(H,S2) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_7744_sum_Omono__neutral__cong__right_H,axiom,
    ! [S2: set_real,T2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
           => ( aa_real_rat(G,X4) = zero_zero_rat ) )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),S2))
             => ( aa_real_rat(G,X4) = aa_real_rat(H,X4) ) )
         => ( groups3269169158384524137al_rat(G,T2) = groups3269169158384524137al_rat(H,S2) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_7745_sum_Omono__neutral__cong__right_H,axiom,
    ! [S2: set_real,T2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
           => ( aa_real_nat(G,X4) = zero_zero_nat ) )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),S2))
             => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
         => ( groups3904299218471019873al_nat(G,T2) = groups3904299218471019873al_nat(H,S2) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_7746_sum_Omono__neutral__cong__right_H,axiom,
    ! [S2: set_real,T2: set_real,G: fun_real_int,H: fun_real_int] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
           => ( aa_real_int(G,X4) = zero_zero_int ) )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),S2))
             => ( aa_real_int(G,X4) = aa_real_int(H,X4) ) )
         => ( groups3901808747961969597al_int(G,T2) = groups3901808747961969597al_int(H,S2) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_7747_sum_Omono__neutral__cong__right_H,axiom,
    ! [S2: set_nat,T2: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
           => ( aa_nat_complex(G,X4) = zero_zero_complex ) )
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),S2))
             => ( aa_nat_complex(G,X4) = aa_nat_complex(H,X4) ) )
         => ( groups8515261248781899619omplex(G,T2) = groups8515261248781899619omplex(H,S2) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_7748_sum_Omono__neutral__cong__right_H,axiom,
    ! [S2: set_nat,T2: set_nat,G: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
           => ( aa_nat_real(G,X4) = zero_zero_real ) )
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),S2))
             => ( aa_nat_real(G,X4) = aa_nat_real(H,X4) ) )
         => ( groups8560362682196896993t_real(G,T2) = groups8560362682196896993t_real(H,S2) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_7749_sum_Omono__neutral__cong__right_H,axiom,
    ! [S2: set_nat,T2: set_nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
           => ( aa_nat_rat(G,X4) = zero_zero_rat ) )
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),S2))
             => ( aa_nat_rat(G,X4) = aa_nat_rat(H,X4) ) )
         => ( groups1351286907653491341at_rat(G,T2) = groups1351286907653491341at_rat(H,S2) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_7750_sum_Omono__neutral__cong__right_H,axiom,
    ! [S2: set_nat,T2: set_nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
           => ( aa_nat_nat(G,X4) = zero_zero_nat ) )
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),S2))
             => ( aa_nat_nat(G,X4) = aa_nat_nat(H,X4) ) )
         => ( aa_set_nat_nat(groups1986416967739987077at_nat(G),T2) = aa_set_nat_nat(groups1986416967739987077at_nat(H),S2) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_7751_sum_Omono__neutral__cong__right_H,axiom,
    ! [S2: set_nat,T2: set_nat,G: fun_nat_int,H: fun_nat_int] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
           => ( aa_nat_int(G,X4) = zero_zero_int ) )
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),S2))
             => ( aa_nat_int(G,X4) = aa_nat_int(H,X4) ) )
         => ( groups1983926497230936801at_int(G,T2) = groups1983926497230936801at_int(H,S2) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_7752_sum_Omono__neutral__cong__left_H,axiom,
    ! [S2: set_real,T2: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),minus_minus_set_real(T2,S2)))
           => ( aa_real_complex(H,I2) = zero_zero_complex ) )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),S2))
             => ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
         => ( groups5683813829254066239omplex(G,S2) = groups5683813829254066239omplex(H,T2) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_7753_sum_Omono__neutral__cong__left_H,axiom,
    ! [S2: set_real,T2: set_real,H: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),minus_minus_set_real(T2,S2)))
           => ( aa_real_real(H,I2) = zero_zero_real ) )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),S2))
             => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
         => ( groups97945582718554045l_real(G,S2) = groups97945582718554045l_real(H,T2) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_7754_sum_Omono__neutral__cong__left_H,axiom,
    ! [S2: set_real,T2: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),minus_minus_set_real(T2,S2)))
           => ( aa_real_rat(H,I2) = zero_zero_rat ) )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),S2))
             => ( aa_real_rat(G,X4) = aa_real_rat(H,X4) ) )
         => ( groups3269169158384524137al_rat(G,S2) = groups3269169158384524137al_rat(H,T2) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_7755_sum_Omono__neutral__cong__left_H,axiom,
    ! [S2: set_real,T2: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),minus_minus_set_real(T2,S2)))
           => ( aa_real_nat(H,I2) = zero_zero_nat ) )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),S2))
             => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
         => ( groups3904299218471019873al_nat(G,S2) = groups3904299218471019873al_nat(H,T2) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_7756_sum_Omono__neutral__cong__left_H,axiom,
    ! [S2: set_real,T2: set_real,H: fun_real_int,G: fun_real_int] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),minus_minus_set_real(T2,S2)))
           => ( aa_real_int(H,I2) = zero_zero_int ) )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),S2))
             => ( aa_real_int(G,X4) = aa_real_int(H,X4) ) )
         => ( groups3901808747961969597al_int(G,S2) = groups3901808747961969597al_int(H,T2) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_7757_sum_Omono__neutral__cong__left_H,axiom,
    ! [S2: set_nat,T2: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),minus_minus_set_nat(T2,S2)))
           => ( aa_nat_complex(H,I2) = zero_zero_complex ) )
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),S2))
             => ( aa_nat_complex(G,X4) = aa_nat_complex(H,X4) ) )
         => ( groups8515261248781899619omplex(G,S2) = groups8515261248781899619omplex(H,T2) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_7758_sum_Omono__neutral__cong__left_H,axiom,
    ! [S2: set_nat,T2: set_nat,H: fun_nat_real,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),minus_minus_set_nat(T2,S2)))
           => ( aa_nat_real(H,I2) = zero_zero_real ) )
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),S2))
             => ( aa_nat_real(G,X4) = aa_nat_real(H,X4) ) )
         => ( groups8560362682196896993t_real(G,S2) = groups8560362682196896993t_real(H,T2) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_7759_sum_Omono__neutral__cong__left_H,axiom,
    ! [S2: set_nat,T2: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),minus_minus_set_nat(T2,S2)))
           => ( aa_nat_rat(H,I2) = zero_zero_rat ) )
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),S2))
             => ( aa_nat_rat(G,X4) = aa_nat_rat(H,X4) ) )
         => ( groups1351286907653491341at_rat(G,S2) = groups1351286907653491341at_rat(H,T2) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_7760_sum_Omono__neutral__cong__left_H,axiom,
    ! [S2: set_nat,T2: set_nat,H: fun_nat_nat,G: fun_nat_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),minus_minus_set_nat(T2,S2)))
           => ( aa_nat_nat(H,I2) = zero_zero_nat ) )
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),S2))
             => ( aa_nat_nat(G,X4) = aa_nat_nat(H,X4) ) )
         => ( aa_set_nat_nat(groups1986416967739987077at_nat(G),S2) = aa_set_nat_nat(groups1986416967739987077at_nat(H),T2) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_7761_sum_Omono__neutral__cong__left_H,axiom,
    ! [S2: set_nat,T2: set_nat,H: fun_nat_int,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),minus_minus_set_nat(T2,S2)))
           => ( aa_nat_int(H,I2) = zero_zero_int ) )
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),S2))
             => ( aa_nat_int(G,X4) = aa_nat_int(H,X4) ) )
         => ( groups1983926497230936801at_int(G,S2) = groups1983926497230936801at_int(H,T2) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_7762_sum_Omono__neutral__right_H,axiom,
    ! [S2: set_nat,T2: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
           => ( aa_nat_complex(G,X4) = zero_zero_complex ) )
       => ( groups8515261248781899619omplex(G,T2) = groups8515261248781899619omplex(G,S2) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_7763_sum_Omono__neutral__right_H,axiom,
    ! [S2: set_nat,T2: set_nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
           => ( aa_nat_real(G,X4) = zero_zero_real ) )
       => ( groups8560362682196896993t_real(G,T2) = groups8560362682196896993t_real(G,S2) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_7764_sum_Omono__neutral__right_H,axiom,
    ! [S2: set_nat,T2: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
           => ( aa_nat_rat(G,X4) = zero_zero_rat ) )
       => ( groups1351286907653491341at_rat(G,T2) = groups1351286907653491341at_rat(G,S2) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_7765_sum_Omono__neutral__right_H,axiom,
    ! [S2: set_nat,T2: set_nat,G: fun_nat_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
           => ( aa_nat_nat(G,X4) = zero_zero_nat ) )
       => ( aa_set_nat_nat(groups1986416967739987077at_nat(G),T2) = aa_set_nat_nat(groups1986416967739987077at_nat(G),S2) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_7766_sum_Omono__neutral__right_H,axiom,
    ! [S2: set_nat,T2: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
           => ( aa_nat_int(G,X4) = zero_zero_int ) )
       => ( groups1983926497230936801at_int(G,T2) = groups1983926497230936801at_int(G,S2) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_7767_sum_Omono__neutral__right_H,axiom,
    ! [S2: set_int,T2: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
           => ( aa_int_complex(G,X4) = zero_zero_complex ) )
       => ( groups267424677133301183omplex(G,T2) = groups267424677133301183omplex(G,S2) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_7768_sum_Omono__neutral__right_H,axiom,
    ! [S2: set_int,T2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
           => ( aa_int_real(G,X4) = zero_zero_real ) )
       => ( groups1523912220035142973t_real(G,T2) = groups1523912220035142973t_real(G,S2) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_7769_sum_Omono__neutral__right_H,axiom,
    ! [S2: set_int,T2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
           => ( aa_int_rat(G,X4) = zero_zero_rat ) )
       => ( groups2350640619554545897nt_rat(G,T2) = groups2350640619554545897nt_rat(G,S2) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_7770_sum_Omono__neutral__right_H,axiom,
    ! [S2: set_int,T2: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
           => ( aa_int_nat(G,X4) = zero_zero_nat ) )
       => ( groups2985770679641041633nt_nat(G,T2) = groups2985770679641041633nt_nat(G,S2) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_7771_sum_Omono__neutral__right_H,axiom,
    ! [S2: set_int,T2: set_int,G: fun_int_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
           => ( aa_int_int(G,X4) = zero_zero_int ) )
       => ( groups2983280209131991357nt_int(G,T2) = groups2983280209131991357nt_int(G,S2) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_7772_sum_Omono__neutral__left_H,axiom,
    ! [S2: set_nat,T2: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
           => ( aa_nat_complex(G,X4) = zero_zero_complex ) )
       => ( groups8515261248781899619omplex(G,S2) = groups8515261248781899619omplex(G,T2) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_7773_sum_Omono__neutral__left_H,axiom,
    ! [S2: set_nat,T2: set_nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
           => ( aa_nat_real(G,X4) = zero_zero_real ) )
       => ( groups8560362682196896993t_real(G,S2) = groups8560362682196896993t_real(G,T2) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_7774_sum_Omono__neutral__left_H,axiom,
    ! [S2: set_nat,T2: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
           => ( aa_nat_rat(G,X4) = zero_zero_rat ) )
       => ( groups1351286907653491341at_rat(G,S2) = groups1351286907653491341at_rat(G,T2) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_7775_sum_Omono__neutral__left_H,axiom,
    ! [S2: set_nat,T2: set_nat,G: fun_nat_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
           => ( aa_nat_nat(G,X4) = zero_zero_nat ) )
       => ( aa_set_nat_nat(groups1986416967739987077at_nat(G),S2) = aa_set_nat_nat(groups1986416967739987077at_nat(G),T2) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_7776_sum_Omono__neutral__left_H,axiom,
    ! [S2: set_nat,T2: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
           => ( aa_nat_int(G,X4) = zero_zero_int ) )
       => ( groups1983926497230936801at_int(G,S2) = groups1983926497230936801at_int(G,T2) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_7777_sum_Omono__neutral__left_H,axiom,
    ! [S2: set_int,T2: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
           => ( aa_int_complex(G,X4) = zero_zero_complex ) )
       => ( groups267424677133301183omplex(G,S2) = groups267424677133301183omplex(G,T2) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_7778_sum_Omono__neutral__left_H,axiom,
    ! [S2: set_int,T2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
           => ( aa_int_real(G,X4) = zero_zero_real ) )
       => ( groups1523912220035142973t_real(G,S2) = groups1523912220035142973t_real(G,T2) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_7779_sum_Omono__neutral__left_H,axiom,
    ! [S2: set_int,T2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
           => ( aa_int_rat(G,X4) = zero_zero_rat ) )
       => ( groups2350640619554545897nt_rat(G,S2) = groups2350640619554545897nt_rat(G,T2) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_7780_sum_Omono__neutral__left_H,axiom,
    ! [S2: set_int,T2: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
           => ( aa_int_nat(G,X4) = zero_zero_nat ) )
       => ( groups2985770679641041633nt_nat(G,S2) = groups2985770679641041633nt_nat(G,T2) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_7781_sum_Omono__neutral__left_H,axiom,
    ! [S2: set_int,T2: set_int,G: fun_int_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
           => ( aa_int_int(G,X4) = zero_zero_int ) )
       => ( groups2983280209131991357nt_int(G,S2) = groups2983280209131991357nt_int(G,T2) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_7782_inverse__le__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ) ).

% inverse_le_iff
tff(fact_7783_inverse__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ) ).

% inverse_le_iff
tff(fact_7784_inverse__less__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ) ).

% inverse_less_iff
tff(fact_7785_inverse__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ) ).

% inverse_less_iff
tff(fact_7786_one__le__inverse,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(inverse_inverse_real,A))) ) ) ).

% one_le_inverse
tff(fact_7787_one__le__inverse,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,A))) ) ) ).

% one_le_inverse
tff(fact_7788_inverse__less__1__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,X)),one_one_real))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).

% inverse_less_1_iff
tff(fact_7789_inverse__less__1__iff,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,X)),one_one_rat))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X)) ) ) ).

% inverse_less_1_iff
tff(fact_7790_one__le__inverse__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(inverse_inverse_real,X)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ).

% one_le_inverse_iff
tff(fact_7791_one__le__inverse__iff,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,X)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),one_one_rat)) ) ) ).

% one_le_inverse_iff
tff(fact_7792_inverse__diff__inverse,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(minus_minus_complex(A),B))),aa_complex_complex(invers8013647133539491842omplex,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_7793_inverse__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = 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_real_real(inverse_inverse_real,A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))),aa_real_real(inverse_inverse_real,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_7794_inverse__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(minus_minus_rat(A),B))),aa_rat_rat(inverse_inverse_rat,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_7795_reals__Archimedean,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))),X)) ) ).

% reals_Archimedean
tff(fact_7796_reals__Archimedean,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)))),X)) ) ).

% reals_Archimedean
tff(fact_7797_sum_Odistrib_H,axiom,
    ! [I6: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_ed(I6),G))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_ed(I6),H))))
       => ( groups5683813829254066239omplex(aa_fun2233226938935547116omplex(aTP_Lamm_su(G),H),I6) = aa_complex_complex(plus_plus_complex(groups5683813829254066239omplex(G,I6)),groups5683813829254066239omplex(H,I6)) ) ) ) ).

% sum.distrib'
tff(fact_7798_sum_Odistrib_H,axiom,
    ! [I6: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_ef(I6),G))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_ef(I6),H))))
       => ( groups8515261248781899619omplex(aa_fun2955221450537884980omplex(aTP_Lamm_sv(G),H),I6) = aa_complex_complex(plus_plus_complex(groups8515261248781899619omplex(G,I6)),groups8515261248781899619omplex(H,I6)) ) ) ) ).

% sum.distrib'
tff(fact_7799_sum_Odistrib_H,axiom,
    ! [I6: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_eh(I6),G))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_eh(I6),H))))
       => ( groups267424677133301183omplex(aa_fun3088484340408535532omplex(aTP_Lamm_sw(G),H),I6) = aa_complex_complex(plus_plus_complex(groups267424677133301183omplex(G,I6)),groups267424677133301183omplex(H,I6)) ) ) ) ).

% sum.distrib'
tff(fact_7800_sum_Odistrib_H,axiom,
    ! [I6: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_ej(I6),G))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_ej(I6),H))))
       => ( groups808145749697022017omplex(aa_fun3187328805393345264omplex(aTP_Lamm_sx(G),H),I6) = aa_complex_complex(plus_plus_complex(groups808145749697022017omplex(G,I6)),groups808145749697022017omplex(H,I6)) ) ) ) ).

% sum.distrib'
tff(fact_7801_sum_Odistrib_H,axiom,
    ! [I6: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_el(I6),G))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_el(I6),H))))
       => ( groups97945582718554045l_real(aa_fun6946558556880649704l_real(aTP_Lamm_sy(G),H),I6) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups97945582718554045l_real(G,I6)),groups97945582718554045l_real(H,I6)) ) ) ) ).

% sum.distrib'
tff(fact_7802_sum_Odistrib_H,axiom,
    ! [I6: set_nat,G: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_en(I6),G))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_en(I6),H))))
       => ( groups8560362682196896993t_real(aa_fun1839206346889086768t_real(aTP_Lamm_sk(G),H),I6) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8560362682196896993t_real(G,I6)),groups8560362682196896993t_real(H,I6)) ) ) ) ).

% sum.distrib'
tff(fact_7803_sum_Odistrib_H,axiom,
    ! [I6: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_ep(I6),G))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_ep(I6),H))))
       => ( groups1523912220035142973t_real(aa_fun3252798400008884712t_real(aTP_Lamm_sl(G),H),I6) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups1523912220035142973t_real(G,I6)),groups1523912220035142973t_real(H,I6)) ) ) ) ).

% sum.distrib'
tff(fact_7804_sum_Odistrib_H,axiom,
    ! [I6: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_er(I6),G))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_er(I6),H))))
       => ( groups5737402329758386879x_real(aa_fun8933909628229591020x_real(aTP_Lamm_sm(G),H),I6) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5737402329758386879x_real(G,I6)),groups5737402329758386879x_real(H,I6)) ) ) ) ).

% sum.distrib'
tff(fact_7805_sum_Odistrib_H,axiom,
    ! [I6: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_et(I6),G))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_et(I6),H))))
       => ( groups3269169158384524137al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_sz(G),H),I6) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3269169158384524137al_rat(G,I6)),groups3269169158384524137al_rat(H,I6)) ) ) ) ).

% sum.distrib'
tff(fact_7806_sum_Odistrib_H,axiom,
    ! [I6: set_nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_ev(I6),G))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_ev(I6),H))))
       => ( groups1351286907653491341at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_sn(G),H),I6) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups1351286907653491341at_rat(G,I6)),groups1351286907653491341at_rat(H,I6)) ) ) ) ).

% sum.distrib'
tff(fact_7807_forall__pos__mono__1,axiom,
    ! [P: fun_real_bool,E2: real] :
      ( ! [D4: real,E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D4),E))
         => ( pp(aa_real_bool(P,D4))
           => pp(aa_real_bool(P,E)) ) )
     => ( ! [N2: nat] : pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(P,E2)) ) ) ) ).

% forall_pos_mono_1
tff(fact_7808_forall__pos__mono,axiom,
    ! [P: fun_real_bool,E2: real] :
      ( ! [D4: real,E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D4),E))
         => ( pp(aa_real_bool(P,D4))
           => pp(aa_real_bool(P,E)) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero_nat )
           => pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2)))) )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(P,E2)) ) ) ) ).

% forall_pos_mono
tff(fact_7809_real__arch__inverse,axiom,
    ! [E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
    <=> ? [N3: nat] :
          ( ( N3 != zero_zero_nat )
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N3))))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N3))),E2)) ) ) ).

% real_arch_inverse
tff(fact_7810_sum_OG__def,axiom,
    ! [I6: set_real,P2: fun_real_complex] :
      ( ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_ed(I6),P2))))
       => ( groups5683813829254066239omplex(P2,I6) = groups5754745047067104278omplex(P2,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_ed(I6),P2))) ) )
      & ( ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_ed(I6),P2))))
       => ( groups5683813829254066239omplex(P2,I6) = zero_zero_complex ) ) ) ).

% sum.G_def
tff(fact_7811_sum_OG__def,axiom,
    ! [I6: set_nat,P2: fun_nat_complex] :
      ( ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_ef(I6),P2))))
       => ( groups8515261248781899619omplex(P2,I6) = groups2073611262835488442omplex(P2,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_ef(I6),P2))) ) )
      & ( ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_ef(I6),P2))))
       => ( groups8515261248781899619omplex(P2,I6) = zero_zero_complex ) ) ) ).

% sum.G_def
tff(fact_7812_sum_OG__def,axiom,
    ! [I6: set_int,P2: fun_int_complex] :
      ( ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_eh(I6),P2))))
       => ( groups267424677133301183omplex(P2,I6) = groups3049146728041665814omplex(P2,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_eh(I6),P2))) ) )
      & ( ~ pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_eh(I6),P2))))
       => ( groups267424677133301183omplex(P2,I6) = zero_zero_complex ) ) ) ).

% sum.G_def
tff(fact_7813_sum_OG__def,axiom,
    ! [I6: set_real,P2: fun_real_real] :
      ( ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_el(I6),P2))))
       => ( groups97945582718554045l_real(P2,I6) = groups8097168146408367636l_real(P2,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_el(I6),P2))) ) )
      & ( ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_el(I6),P2))))
       => ( groups97945582718554045l_real(P2,I6) = zero_zero_real ) ) ) ).

% sum.G_def
tff(fact_7814_sum_OG__def,axiom,
    ! [I6: set_int,P2: fun_int_real] :
      ( ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_ep(I6),P2))))
       => ( groups1523912220035142973t_real(P2,I6) = groups8778361861064173332t_real(P2,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_ep(I6),P2))) ) )
      & ( ~ pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_ep(I6),P2))))
       => ( groups1523912220035142973t_real(P2,I6) = zero_zero_real ) ) ) ).

% sum.G_def
tff(fact_7815_sum_OG__def,axiom,
    ! [I6: set_complex,P2: fun_complex_real] :
      ( ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_er(I6),P2))))
       => ( groups5737402329758386879x_real(P2,I6) = groups5808333547571424918x_real(P2,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_er(I6),P2))) ) )
      & ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_er(I6),P2))))
       => ( groups5737402329758386879x_real(P2,I6) = zero_zero_real ) ) ) ).

% sum.G_def
tff(fact_7816_sum_OG__def,axiom,
    ! [I6: set_real,P2: fun_real_rat] :
      ( ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_et(I6),P2))))
       => ( groups3269169158384524137al_rat(P2,I6) = groups1300246762558778688al_rat(P2,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_et(I6),P2))) ) )
      & ( ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_et(I6),P2))))
       => ( groups3269169158384524137al_rat(P2,I6) = zero_zero_rat ) ) ) ).

% sum.G_def
tff(fact_7817_sum_OG__def,axiom,
    ! [I6: set_nat,P2: fun_nat_rat] :
      ( ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_ev(I6),P2))))
       => ( groups1351286907653491341at_rat(P2,I6) = groups2906978787729119204at_rat(P2,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_ev(I6),P2))) ) )
      & ( ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_ev(I6),P2))))
       => ( groups1351286907653491341at_rat(P2,I6) = zero_zero_rat ) ) ) ).

% sum.G_def
tff(fact_7818_sum_OG__def,axiom,
    ! [I6: set_int,P2: fun_int_rat] :
      ( ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun258058769140594373t_bool(aTP_Lamm_ta(I6),P2))))
       => ( groups2350640619554545897nt_rat(P2,I6) = groups3906332499630173760nt_rat(P2,collect_int(aa_fun258058769140594373t_bool(aTP_Lamm_ta(I6),P2))) ) )
      & ( ~ pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun258058769140594373t_bool(aTP_Lamm_ta(I6),P2))))
       => ( groups2350640619554545897nt_rat(P2,I6) = zero_zero_rat ) ) ) ).

% sum.G_def
tff(fact_7819_sum_OG__def,axiom,
    ! [I6: set_complex,P2: fun_complex_rat] :
      ( ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2759627087638585025x_bool(aTP_Lamm_rx(I6),P2))))
       => ( groups2276542476275365739ex_rat(P2,I6) = groups5058264527183730370ex_rat(P2,collect_complex(aa_fun2759627087638585025x_bool(aTP_Lamm_rx(I6),P2))) ) )
      & ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2759627087638585025x_bool(aTP_Lamm_rx(I6),P2))))
       => ( groups2276542476275365739ex_rat(P2,I6) = zero_zero_rat ) ) ) ).

% sum.G_def
tff(fact_7820_ex__inverse__of__nat__less,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2))),X)) ) ) ).

% ex_inverse_of_nat_less
tff(fact_7821_ex__inverse__of__nat__less,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ? [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,N2))),X)) ) ) ).

% ex_inverse_of_nat_less
tff(fact_7822_power__diff__conv__inverse,axiom,
    ! [X: complex,M: nat,N: nat] :
      ( ( X != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_complex(power_power_complex(X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,X)),M)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_7823_power__diff__conv__inverse,axiom,
    ! [X: real,M: nat,N: nat] :
      ( ( X != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,X)),M)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_7824_power__diff__conv__inverse,axiom,
    ! [X: rat,M: nat,N: nat] :
      ( ( X != zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_rat(power_power_rat(X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),N)),aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,X)),M)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_7825_exp__divide__power__eq,axiom,
    ! [N: nat,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,aa_complex_complex(divide1717551699836669952omplex(X),aa_nat_complex(semiri8010041392384452111omplex,N)))),N) = aa_complex_complex(exp_complex,X) ) ) ).

% exp_divide_power_eq
tff(fact_7826_exp__divide__power__eq,axiom,
    ! [N: nat,X: real] :
      ( 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,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_nat_real(semiri5074537144036343181t_real,N)))),N) = aa_real_real(exp_real,X) ) ) ).

% exp_divide_power_eq
tff(fact_7827_log__inverse,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(inverse_inverse_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)) ) ) ) ) ).

% log_inverse
tff(fact_7828_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => 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,X),aa_real_real(inverse_inverse_real,X)))) ) ).

% plus_inverse_ge_2
tff(fact_7829_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,X) ) ) ).

% real_inv_sqrt_pow2
tff(fact_7830_sum__diff1_H,axiom,
    ! [I6: set_complex,F: fun_complex_complex,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_ej(I6),F))))
     => ( ( pp(aa_set_complex_bool(member_complex(I),I6))
         => ( groups808145749697022017omplex(F,minus_811609699411566653omplex(I6,insert_complex(I,bot_bot_set_complex))) = aa_complex_complex(minus_minus_complex(groups808145749697022017omplex(F,I6)),aa_complex_complex(F,I)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(I),I6))
         => ( groups808145749697022017omplex(F,minus_811609699411566653omplex(I6,insert_complex(I,bot_bot_set_complex))) = groups808145749697022017omplex(F,I6) ) ) ) ) ).

% sum_diff1'
tff(fact_7831_sum__diff1_H,axiom,
    ! [I6: set_real,F: fun_real_complex,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_ed(I6),F))))
     => ( ( pp(aa_set_real_bool(member_real(I),I6))
         => ( groups5683813829254066239omplex(F,minus_minus_set_real(I6,insert_real(I,bot_bot_set_real))) = aa_complex_complex(minus_minus_complex(groups5683813829254066239omplex(F,I6)),aa_real_complex(F,I)) ) )
        & ( ~ pp(aa_set_real_bool(member_real(I),I6))
         => ( groups5683813829254066239omplex(F,minus_minus_set_real(I6,insert_real(I,bot_bot_set_real))) = groups5683813829254066239omplex(F,I6) ) ) ) ) ).

% sum_diff1'
tff(fact_7832_sum__diff1_H,axiom,
    ! [I6: set_int,F: fun_int_complex,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_eh(I6),F))))
     => ( ( pp(aa_set_int_bool(member_int(I),I6))
         => ( groups267424677133301183omplex(F,minus_minus_set_int(I6,insert_int(I,bot_bot_set_int))) = aa_complex_complex(minus_minus_complex(groups267424677133301183omplex(F,I6)),aa_int_complex(F,I)) ) )
        & ( ~ pp(aa_set_int_bool(member_int(I),I6))
         => ( groups267424677133301183omplex(F,minus_minus_set_int(I6,insert_int(I,bot_bot_set_int))) = groups267424677133301183omplex(F,I6) ) ) ) ) ).

% sum_diff1'
tff(fact_7833_sum__diff1_H,axiom,
    ! [I6: set_complex,F: fun_complex_real,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_er(I6),F))))
     => ( ( pp(aa_set_complex_bool(member_complex(I),I6))
         => ( groups5737402329758386879x_real(F,minus_811609699411566653omplex(I6,insert_complex(I,bot_bot_set_complex))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5737402329758386879x_real(F,I6)),aa_complex_real(F,I)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(I),I6))
         => ( groups5737402329758386879x_real(F,minus_811609699411566653omplex(I6,insert_complex(I,bot_bot_set_complex))) = groups5737402329758386879x_real(F,I6) ) ) ) ) ).

% sum_diff1'
tff(fact_7834_sum__diff1_H,axiom,
    ! [I6: set_real,F: fun_real_real,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_el(I6),F))))
     => ( ( pp(aa_set_real_bool(member_real(I),I6))
         => ( groups97945582718554045l_real(F,minus_minus_set_real(I6,insert_real(I,bot_bot_set_real))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups97945582718554045l_real(F,I6)),aa_real_real(F,I)) ) )
        & ( ~ pp(aa_set_real_bool(member_real(I),I6))
         => ( groups97945582718554045l_real(F,minus_minus_set_real(I6,insert_real(I,bot_bot_set_real))) = groups97945582718554045l_real(F,I6) ) ) ) ) ).

% sum_diff1'
tff(fact_7835_sum__diff1_H,axiom,
    ! [I6: set_int,F: fun_int_real,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_ep(I6),F))))
     => ( ( pp(aa_set_int_bool(member_int(I),I6))
         => ( groups1523912220035142973t_real(F,minus_minus_set_int(I6,insert_int(I,bot_bot_set_int))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups1523912220035142973t_real(F,I6)),aa_int_real(F,I)) ) )
        & ( ~ pp(aa_set_int_bool(member_int(I),I6))
         => ( groups1523912220035142973t_real(F,minus_minus_set_int(I6,insert_int(I,bot_bot_set_int))) = groups1523912220035142973t_real(F,I6) ) ) ) ) ).

% sum_diff1'
tff(fact_7836_sum__diff1_H,axiom,
    ! [I6: set_complex,F: fun_complex_rat,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2759627087638585025x_bool(aTP_Lamm_rx(I6),F))))
     => ( ( pp(aa_set_complex_bool(member_complex(I),I6))
         => ( groups2276542476275365739ex_rat(F,minus_811609699411566653omplex(I6,insert_complex(I,bot_bot_set_complex))) = aa_rat_rat(minus_minus_rat(groups2276542476275365739ex_rat(F,I6)),aa_complex_rat(F,I)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(I),I6))
         => ( groups2276542476275365739ex_rat(F,minus_811609699411566653omplex(I6,insert_complex(I,bot_bot_set_complex))) = groups2276542476275365739ex_rat(F,I6) ) ) ) ) ).

% sum_diff1'
tff(fact_7837_sum__diff1_H,axiom,
    ! [I6: set_real,F: fun_real_rat,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_et(I6),F))))
     => ( ( pp(aa_set_real_bool(member_real(I),I6))
         => ( groups3269169158384524137al_rat(F,minus_minus_set_real(I6,insert_real(I,bot_bot_set_real))) = aa_rat_rat(minus_minus_rat(groups3269169158384524137al_rat(F,I6)),aa_real_rat(F,I)) ) )
        & ( ~ pp(aa_set_real_bool(member_real(I),I6))
         => ( groups3269169158384524137al_rat(F,minus_minus_set_real(I6,insert_real(I,bot_bot_set_real))) = groups3269169158384524137al_rat(F,I6) ) ) ) ) ).

% sum_diff1'
tff(fact_7838_sum__diff1_H,axiom,
    ! [I6: set_int,F: fun_int_rat,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun258058769140594373t_bool(aTP_Lamm_ta(I6),F))))
     => ( ( pp(aa_set_int_bool(member_int(I),I6))
         => ( groups2350640619554545897nt_rat(F,minus_minus_set_int(I6,insert_int(I,bot_bot_set_int))) = aa_rat_rat(minus_minus_rat(groups2350640619554545897nt_rat(F,I6)),aa_int_rat(F,I)) ) )
        & ( ~ pp(aa_set_int_bool(member_int(I),I6))
         => ( groups2350640619554545897nt_rat(F,minus_minus_set_int(I6,insert_int(I,bot_bot_set_int))) = groups2350640619554545897nt_rat(F,I6) ) ) ) ) ).

% sum_diff1'
tff(fact_7839_sum__diff1_H,axiom,
    ! [I6: set_complex,F: fun_complex_int,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun4637897056364002669x_bool(aTP_Lamm_ry(I6),F))))
     => ( ( pp(aa_set_complex_bool(member_complex(I),I6))
         => ( groups2909182065852811199ex_int(F,minus_811609699411566653omplex(I6,insert_complex(I,bot_bot_set_complex))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups2909182065852811199ex_int(F,I6)),aa_complex_int(F,I)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(I),I6))
         => ( groups2909182065852811199ex_int(F,minus_811609699411566653omplex(I6,insert_complex(I,bot_bot_set_complex))) = groups2909182065852811199ex_int(F,I6) ) ) ) ) ).

% sum_diff1'
tff(fact_7840_Maclaurin__exp__lt,axiom,
    ! [X: real,N: nat] :
      ( ( X != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T6)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(exp_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tb,X),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,T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_7841_log__base__10__eq1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( 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))))),X) = 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,X)) ) ) ).

% log_base_10_eq1
tff(fact_7842_powr__def,axiom,
    ! [X: real,A: real] :
      ( ( ( X = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A) = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A) = aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(ln_ln_real,X))) ) ) ) ).

% powr_def
tff(fact_7843_log__base__10__eq2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( 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))))),X) = 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,X)) ) ) ).

% log_base_10_eq2
tff(fact_7844_cosh__zero__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cosh_complex,X) = zero_zero_complex )
    <=> ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% cosh_zero_iff
tff(fact_7845_cosh__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(cosh_real,X) = zero_zero_real )
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% cosh_zero_iff
tff(fact_7846_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_7847_fact__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).

% fact_0
tff(fact_7848_fact__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).

% fact_0
tff(fact_7849_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_7850_fact__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).

% fact_0
tff(fact_7851_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_7852_cosh__0,axiom,
    aa_complex_complex(cosh_complex,zero_zero_complex) = one_one_complex ).

% cosh_0
tff(fact_7853_cosh__0,axiom,
    aa_real_real(cosh_real,zero_zero_real) = one_one_real ).

% cosh_0
tff(fact_7854_ln__inj__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( ( aa_real_real(ln_ln_real,X) = aa_real_real(ln_ln_real,Y) )
        <=> ( X = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_7855_ln__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ).

% ln_less_cancel_iff
tff(fact_7856_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_7857_fact__Suc__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).

% fact_Suc_0
tff(fact_7858_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_7859_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_7860_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_7861_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ).

% ln_le_cancel_iff
tff(fact_7862_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( aa_real_real(ln_ln_real,X) = zero_zero_real )
      <=> ( X = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_7863_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).

% ln_gt_zero_iff
tff(fact_7864_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% ln_less_zero_iff
tff(fact_7865_exp__ln__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,X)) = X )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% exp_ln_iff
tff(fact_7866_exp__ln,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(exp_real,aa_real_real(ln_ln_real,X)) = X ) ) ).

% exp_ln
tff(fact_7867_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ).

% ln_le_zero_iff
tff(fact_7868_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

% ln_ge_zero_iff
tff(fact_7869_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_complex(semiri5044797733671781792omplex,N) != zero_zero_complex ).

% fact_nonzero
tff(fact_7870_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,N) != zero_zero_rat ).

% fact_nonzero
tff(fact_7871_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) != zero_zero_int ).

% fact_nonzero
tff(fact_7872_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,N) != zero_zero_real ).

% fact_nonzero
tff(fact_7873_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) != zero_zero_nat ).

% fact_nonzero
tff(fact_7874_ln__less__self,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),X)) ) ).

% ln_less_self
tff(fact_7875_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N))) ).

% fact_ge_zero
tff(fact_7876_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).

% fact_ge_zero
tff(fact_7877_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,N))) ).

% fact_ge_zero
tff(fact_7878_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_zero
tff(fact_7879_fact__not__neg,axiom,
    ! [N: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,N)),zero_zero_rat)) ).

% fact_not_neg
tff(fact_7880_fact__not__neg,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,N)),zero_zero_int)) ).

% fact_not_neg
tff(fact_7881_fact__not__neg,axiom,
    ! [N: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,N)),zero_zero_real)) ).

% fact_not_neg
tff(fact_7882_fact__not__neg,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),zero_zero_nat)) ).

% fact_not_neg
tff(fact_7883_fact__gt__zero,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N))) ).

% fact_gt_zero
tff(fact_7884_fact__gt__zero,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).

% fact_gt_zero
tff(fact_7885_fact__gt__zero,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,N))) ).

% fact_gt_zero
tff(fact_7886_fact__gt__zero,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_gt_zero
tff(fact_7887_fact__ge__1,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,N))) ).

% fact_ge_1
tff(fact_7888_fact__ge__1,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).

% fact_ge_1
tff(fact_7889_fact__ge__1,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(semiri2265585572941072030t_real,N))) ).

% fact_ge_1
tff(fact_7890_fact__ge__1,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_1
tff(fact_7891_fact__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,N))) ) ).

% fact_mono
tff(fact_7892_fact__mono,axiom,
    ! [M: nat,N: nat] :
      ( 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(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,N))) ) ).

% fact_mono
tff(fact_7893_fact__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,M)),aa_nat_real(semiri2265585572941072030t_real,N))) ) ).

% fact_mono
tff(fact_7894_fact__mono,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
tff(fact_7895_cosh__real__pos,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cosh_real,X))) ).

% cosh_real_pos
tff(fact_7896_fact__dvd,axiom,
    ! [N: nat,M: nat] :
      ( 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(semiri1406184849735516958ct_int,N)),aa_nat_int(semiri1406184849735516958ct_int,M))) ) ).

% fact_dvd
tff(fact_7897_fact__dvd,axiom,
    ! [N: nat,M: nat] :
      ( 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(semiri3624122377584611663nteger,N)),aa_nat_Code_integer(semiri3624122377584611663nteger,M))) ) ).

% fact_dvd
tff(fact_7898_fact__dvd,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(dvd_dvd_real,aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri2265585572941072030t_real,M))) ) ).

% fact_dvd
tff(fact_7899_fact__dvd,axiom,
    ! [N: nat,M: 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(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,M))) ) ).

% fact_dvd
tff(fact_7900_ln__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),X)) ) ).

% ln_bound
tff(fact_7901_ln__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X))) ) ).

% ln_gt_zero
tff(fact_7902_ln__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),zero_zero_real)) ) ) ).

% ln_less_zero
tff(fact_7903_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_7904_fact__less__mono,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_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,N))) ) ) ).

% fact_less_mono
tff(fact_7905_fact__less__mono,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_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,N))) ) ) ).

% fact_less_mono
tff(fact_7906_fact__less__mono,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_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,M)),aa_nat_real(semiri2265585572941072030t_real,N))) ) ) ).

% fact_less_mono
tff(fact_7907_fact__less__mono,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
tff(fact_7908_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_7909_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_7910_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y))) ) ) ).

% cosh_real_strict_mono
tff(fact_7911_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(modulo_modulo_int(aa_nat_int(semiri1406184849735516958ct_int,N)),aa_nat_int(semiri1406184849735516958ct_int,M)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_7912_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( modulo364778990260209775nteger(aa_nat_Code_integer(semiri3624122377584611663nteger,N),aa_nat_Code_integer(semiri3624122377584611663nteger,M)) = zero_z3403309356797280102nteger ) ) ).

% fact_mod
tff(fact_7913_fact__mod,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(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,M)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_7914_fact__le__power,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,N)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).

% fact_le_power
tff(fact_7915_fact__le__power,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,N)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).

% fact_le_power
tff(fact_7916_fact__le__power,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).

% fact_le_power
tff(fact_7917_fact__le__power,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).

% fact_le_power
tff(fact_7918_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_7919_cosh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(inverse_inverse_real,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_7920_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_mult
tff(fact_7921_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( aa_real_real(ln_ln_real,X) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real) )
       => ( X = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_7922_ln__div,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_div
tff(fact_7923_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(ln_ln_real,X)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,Y)),X)) ) ) ).

% ln_ge_iff
tff(fact_7924_ln__inverse,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(ln_ln_real,aa_real_real(inverse_inverse_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,X)) ) ) ).

% ln_inverse
tff(fact_7925_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_Code_integer(semiri3624122377584611663nteger,N))) ) ).

% choose_dvd
tff(fact_7926_choose__dvd,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(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_rat(semiri773545260158071498ct_rat,N))) ) ).

% choose_dvd
tff(fact_7927_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_int(semiri1406184849735516958ct_int,N))) ) ).

% choose_dvd
tff(fact_7928_choose__dvd,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(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_real(semiri2265585572941072030t_real,N))) ) ).

% choose_dvd
tff(fact_7929_choose__dvd,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(dvd_dvd_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(semiri1408675320244567234ct_nat,N))) ) ).

% choose_dvd
tff(fact_7930_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_7931_ln__le__minus__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real))) ) ).

% ln_le_minus_one
tff(fact_7932_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => 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,X)),aa_real_real(ln_ln_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),Y))) ) ) ).

% ln_diff_le
tff(fact_7933_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => 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),X))),X)) ) ).

% ln_add_one_self_le_self2
tff(fact_7934_ln__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(ln_ln_real,X)) ) ) ).

% ln_realpow
tff(fact_7935_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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),X))),aa_real_real(uminus_uminus_real,X))) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_7936_ln__powr__bound,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),A))) ) ) ).

% ln_powr_bound
tff(fact_7937_ln__powr__bound2,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => 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,X)),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),A)),X))) ) ) ).

% ln_powr_bound2
tff(fact_7938_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
         => ( ( B != one_one_real )
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
             => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_real_real(ln_ln_real,A))),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_7939_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_complex(semiri5044797733671781792omplex,M) = one_one_complex ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_complex(semiri5044797733671781792omplex,M) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7940_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_rat(semiri773545260158071498ct_rat,M) = one_one_rat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_rat(semiri773545260158071498ct_rat,M) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7941_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_int(semiri1406184849735516958ct_int,M) = one_one_int ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_int(semiri1406184849735516958ct_int,M) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7942_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_real(semiri2265585572941072030t_real,M) = one_one_real ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_real(semiri2265585572941072030t_real,M) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7943_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = one_one_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7944_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_complex(semiri5044797733671781792omplex,N) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7945_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7946_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_int(semiri1406184849735516958ct_int,N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7947_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_real(semiri2265585572941072030t_real,N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7948_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7949_binomial__fact,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)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,N)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))) ) ) ).

% binomial_fact
tff(fact_7950_binomial__fact,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)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,N)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))) ) ) ).

% binomial_fact
tff(fact_7951_binomial__fact,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)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri2265585572941072030t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))) ) ) ).

% binomial_fact
tff(fact_7952_fact__binomial,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,N)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ) ).

% fact_binomial
tff(fact_7953_fact__binomial,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,N)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ) ).

% fact_binomial
tff(fact_7954_fact__binomial,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ) ).

% fact_binomial
tff(fact_7955_ln__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,X)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_7956_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_na5723442673449079972x_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun5746040925223427547t_real(aTP_Lamm_tc(X),Diff),set_ord_lessThan_nat(N)) = aa_complex_real(aa_nat480766339200631653x_real(Diff,zero_zero_nat),zero_zero_complex) ) ) ) ).

% Maclaurin_zero
tff(fact_7957_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_na6724076748692969250l_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_td(X),Diff),set_ord_lessThan_nat(N)) = aa_real_real(aa_nat_fun_real_real(Diff,zero_zero_nat),zero_zero_real) ) ) ) ).

% Maclaurin_zero
tff(fact_7958_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_nat_fun_rat_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun507281639146497029t_real(aTP_Lamm_te(X),Diff),set_ord_lessThan_nat(N)) = aa_rat_real(aa_nat_fun_rat_real(Diff,zero_zero_nat),zero_zero_rat) ) ) ) ).

% Maclaurin_zero
tff(fact_7959_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_nat_fun_nat_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun7039663905532832253t_real(aTP_Lamm_tf(X),Diff),set_ord_lessThan_nat(N)) = aa_nat_real(aa_nat_fun_nat_real(Diff,zero_zero_nat),zero_zero_nat) ) ) ) ).

% Maclaurin_zero
tff(fact_7960_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_nat_fun_int_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun241848180832411225t_real(aTP_Lamm_tg(X),Diff),set_ord_lessThan_nat(N)) = aa_int_real(aa_nat_fun_int_real(Diff,zero_zero_nat),zero_zero_int) ) ) ) ).

% Maclaurin_zero
tff(fact_7961_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))
     => ? [B7: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_th(H),J),set_ord_lessThan_nat(N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B7),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_7962_sin__coeff__def,axiom,
    ! [X2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X2))
       => ( aa_nat_real(sin_coeff,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))),X2))
       => ( aa_nat_real(sin_coeff,X2) = 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,X2),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X2)) ) ) ) ).

% sin_coeff_def
tff(fact_7963_gbinomial__code,axiom,
    ! [K: nat,A: complex] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_complex(gbinomial_complex(A),K) = one_one_complex ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_ti(A),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_7964_gbinomial__code,axiom,
    ! [K: nat,A: rat] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_rat(gbinomial_rat(A),K) = one_one_rat ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(divide_divide_rat(aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_tj(A),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_7965_gbinomial__code,axiom,
    ! [K: nat,A: real] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = one_one_real ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_tk(A),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_7966_ln__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(ln_ln_real,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_tl,X)) ) ) ) ).

% ln_series
tff(fact_7967_tanh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,X)) = 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,X),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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_7968_sin__x__sin__y,axiom,
    ! [X: complex,Y: complex] : pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_tn(X),Y)),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(sin_complex,Y)))) ).

% sin_x_sin_y
tff(fact_7969_sin__x__sin__y,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_tp(X),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y)))) ).

% sin_x_sin_y
tff(fact_7970_scaleR__cancel__right,axiom,
    ! [A: real,X: complex,B: real] :
      ( ( aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),X) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,B),X) )
    <=> ( ( A = B )
        | ( X = zero_zero_complex ) ) ) ).

% scaleR_cancel_right
tff(fact_7971_scaleR__cancel__right,axiom,
    ! [A: real,X: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),X) )
    <=> ( ( A = B )
        | ( X = zero_zero_real ) ) ) ).

% scaleR_cancel_right
tff(fact_7972_scaleR__zero__right,axiom,
    ! [A: real] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),zero_zero_complex) = zero_zero_complex ).

% scaleR_zero_right
tff(fact_7973_scaleR__zero__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),zero_zero_real) = zero_zero_real ).

% scaleR_zero_right
tff(fact_7974_sin__zero,axiom,
    aa_complex_complex(sin_complex,zero_zero_complex) = zero_zero_complex ).

% sin_zero
tff(fact_7975_sin__zero,axiom,
    aa_real_real(sin_real,zero_zero_real) = zero_zero_real ).

% sin_zero
tff(fact_7976_tanh__0,axiom,
    aa_complex_complex(tanh_complex,zero_zero_complex) = zero_zero_complex ).

% tanh_0
tff(fact_7977_tanh__0,axiom,
    aa_real_real(tanh_real,zero_zero_real) = zero_zero_real ).

% tanh_0
tff(fact_7978_tanh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% tanh_real_less_iff
tff(fact_7979_suminf__zero,axiom,
    suminf_complex(aTP_Lamm_qg) = zero_zero_complex ).

% suminf_zero
tff(fact_7980_suminf__zero,axiom,
    suminf_real(aTP_Lamm_ft) = zero_zero_real ).

% suminf_zero
tff(fact_7981_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_ck) = zero_zero_nat ).

% suminf_zero
tff(fact_7982_suminf__zero,axiom,
    suminf_int(aTP_Lamm_qh) = zero_zero_int ).

% suminf_zero
tff(fact_7983_scaleR__eq__0__iff,axiom,
    ! [A: real,X: complex] :
      ( ( aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),X) = zero_zero_complex )
    <=> ( ( A = zero_zero_real )
        | ( X = zero_zero_complex ) ) ) ).

% scaleR_eq_0_iff
tff(fact_7984_scaleR__eq__0__iff,axiom,
    ! [A: real,X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( X = zero_zero_real ) ) ) ).

% scaleR_eq_0_iff
tff(fact_7985_scaleR__zero__left,axiom,
    ! [X: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,zero_zero_real),X) = zero_zero_complex ).

% scaleR_zero_left
tff(fact_7986_scaleR__zero__left,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,zero_zero_real),X) = zero_zero_real ).

% scaleR_zero_left
tff(fact_7987_tanh__real__pos__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(tanh_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% tanh_real_pos_iff
tff(fact_7988_tanh__real__neg__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% tanh_real_neg_iff
tff(fact_7989_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_7990_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_gv(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_7991_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_gx(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_7992_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_7993_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_7994_scaleR__right__imp__eq,axiom,
    ! [X: complex,A: real,B: real] :
      ( ( X != zero_zero_complex )
     => ( ( aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),X) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,B),X) )
       => ( A = B ) ) ) ).

% scaleR_right_imp_eq
tff(fact_7995_scaleR__right__imp__eq,axiom,
    ! [X: real,A: real,B: real] :
      ( ( X != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),X) )
       => ( A = B ) ) ) ).

% scaleR_right_imp_eq
tff(fact_7996_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_7997_tanh__real__lt__1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X)),one_one_real)) ).

% tanh_real_lt_1
tff(fact_7998_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_7999_scaleR__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),C))) ) ) ).

% scaleR_right_mono_neg
tff(fact_8000_scaleR__right__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),X))) ) ) ).

% scaleR_right_mono
tff(fact_8001_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_8002_vector__fraction__eq__iff,axiom,
    ! [U: real,V: real,A: complex,X: complex] :
      ( ( aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),V)),A) = X )
    <=> ( ( ( V = zero_zero_real )
         => ( X = zero_zero_complex ) )
        & ( ( V != zero_zero_real )
         => ( aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,U),A) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,V),X) ) ) ) ) ).

% vector_fraction_eq_iff
tff(fact_8003_vector__fraction__eq__iff,axiom,
    ! [U: real,V: real,A: real,X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),V)),A) = X )
    <=> ( ( ( V = zero_zero_real )
         => ( X = zero_zero_real ) )
        & ( ( V != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,U),A) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,V),X) ) ) ) ) ).

% vector_fraction_eq_iff
tff(fact_8004_eq__vector__fraction__iff,axiom,
    ! [X: complex,U: real,V: real,A: complex] :
      ( ( X = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),V)),A) )
    <=> ( ( ( V = zero_zero_real )
         => ( X = zero_zero_complex ) )
        & ( ( V != zero_zero_real )
         => ( aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,V),X) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,U),A) ) ) ) ) ).

% eq_vector_fraction_iff
tff(fact_8005_eq__vector__fraction__iff,axiom,
    ! [X: real,U: real,V: real,A: real] :
      ( ( X = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),V)),A) )
    <=> ( ( ( V = zero_zero_real )
         => ( X = zero_zero_real ) )
        & ( ( V != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,V),X) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,U),A) ) ) ) ) ).

% eq_vector_fraction_iff
tff(fact_8006_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: 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(real_V1485227260804924795R_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D))) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_8007_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: 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(real_V1485227260804924795R_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),E2)),D)))
    <=> 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(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E2)),C)),D)) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_8008_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_complex(F,N2) = zero_zero_complex ) )
       => ( suminf_complex(F) = groups2073611262835488442omplex(F,N4) ) ) ) ).

% suminf_finite
tff(fact_8009_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => ( suminf_int(F) = groups3539618377306564664at_int(F,N4) ) ) ) ).

% suminf_finite
tff(fact_8010_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => ( suminf_nat(F) = aa_set_nat_nat(groups3542108847815614940at_nat(F),N4) ) ) ) ).

% suminf_finite
tff(fact_8011_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => ( suminf_real(F) = groups6591440286371151544t_real(F,N4) ) ) ) ).

% suminf_finite
tff(fact_8012_tanh__real__gt__neg1,axiom,
    ! [X: 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,X))) ).

% tanh_real_gt_neg1
tff(fact_8013_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_8014_zero__le__scaleR__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( A = zero_zero_real ) ) ) ).

% zero_le_scaleR_iff
tff(fact_8015_scaleR__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( A = zero_zero_real ) ) ) ).

% scaleR_le_0_iff
tff(fact_8016_scaleR__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),B))) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_8017_scaleR__nonpos__nonneg,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),zero_zero_real)) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_8018_scaleR__nonneg__nonpos,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),zero_zero_real)) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_8019_scaleR__nonneg__nonneg,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X))) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_8020_split__scaleR__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),B))) ) ).

% split_scaleR_pos_le
tff(fact_8021_split__scaleR__neg__le,axiom,
    ! [A: real,X: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),zero_zero_real)) ) ).

% split_scaleR_neg_le
tff(fact_8022_scaleR__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),D))) ) ) ) ) ).

% scaleR_mono'
tff(fact_8023_scaleR__mono,axiom,
    ! [A: real,B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),Y))) ) ) ) ) ).

% scaleR_mono
tff(fact_8024_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_8025_scaleR__left__le__one__le,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),X)) ) ) ).

% scaleR_left_le_one_le
tff(fact_8026_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_8027_neg__less__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A))) ) ) ).

% neg_less_divideR_eq
tff(fact_8028_neg__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),B)) ) ) ).

% neg_divideR_less_eq
tff(fact_8029_pos__less__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),B)) ) ) ).

% pos_less_divideR_eq
tff(fact_8030_pos__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A))) ) ) ).

% pos_divideR_less_eq
tff(fact_8031_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: real,X: complex] :
      ( ( A != zero_zero_real )
     => ( ( X != zero_zero_complex )
       => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),X)) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(inverse_inverse_real,A)),aa_complex_complex(invers8013647133539491842omplex,X)) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_8032_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: real,X: real] :
      ( ( A != zero_zero_real )
     => ( ( X != zero_zero_real )
       => ( aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,X)) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_8033_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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,X))) ) ) ).

% sin_gt_zero_02
tff(fact_8034_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_8035_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa2: nat,Xb2: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,Xa2),Xb2),Xc) = Y )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb2),Xa2))
         => ( Y = Xc ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb2),Xa2))
         => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa2),one_one_nat)),Xb2),aa_nat_nat(aa_nat_fun_nat_nat(X,Xa2),Xc)) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_8036_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [B: nat,A: nat,F: fun_nat_fun_nat_nat,Acc: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc) = Acc ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc)) ) ) ) ).

% fold_atLeastAtMost_nat.simps
tff(fact_8037_pos__le__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_8038_pos__minus__divideR__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A))) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_8039_neg__le__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A))) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_8040_neg__minus__divideR__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_8041_neg__minus__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_8042_neg__less__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A))) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_8043_pos__minus__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A))) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_8044_pos__less__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_8045_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_8046_suminf__geometric,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real))
     => ( suminf_real(aa_real_fun_nat_real(power_power_real,C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),C)) ) ) ).

% suminf_geometric
tff(fact_8047_suminf__geometric,axiom,
    ! [C: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real))
     => ( suminf_complex(power_power_complex(C)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C)) ) ) ).

% suminf_geometric
tff(fact_8048_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,A: nat,B: nat] : groups2073611262835488442omplex(F,set_or1269000886237332187st_nat(A,B)) = aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_tq(F),A),B),zero_zero_complex) ).

% sum_atLeastAtMost_code
tff(fact_8049_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(A,B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_tr(F),A,B),zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_8050_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(A,B)) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_ts(F),A,B),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_8051_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(A,B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_tt(F),A),B),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_8052_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B: nat] : groups6591440286371151544t_real(F,set_or1269000886237332187st_nat(A,B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_tu(F),A),B),zero_zero_real) ).

% sum_atLeastAtMost_code
tff(fact_8053_tanh__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cosh_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cosh_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(tanh_complex,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tanh_complex,X)),aa_complex_complex(tanh_complex,Y))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tanh_complex,X)),aa_complex_complex(tanh_complex,Y)))) ) ) ) ).

% tanh_add
tff(fact_8054_tanh__add,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cosh_real,X) != zero_zero_real )
     => ( ( aa_real_real(cosh_real,Y) != zero_zero_real )
       => ( aa_real_real(tanh_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y)))) ) ) ) ).

% tanh_add
tff(fact_8055_cosh__converges,axiom,
    ! [X: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_tv,X)),aa_real_real(cosh_real,X))) ).

% cosh_converges
tff(fact_8056_cosh__converges,axiom,
    ! [X: complex] : pp(aa_complex_bool(sums_complex(aTP_Lamm_tw(X)),aa_complex_complex(cosh_complex,X))) ).

% cosh_converges
tff(fact_8057_sums__cos__x__plus__y,axiom,
    ! [X: complex,Y: complex] : pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_ty(X),Y)),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y)))) ).

% sums_cos_x_plus_y
tff(fact_8058_sums__cos__x__plus__y,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_ua(X),Y)),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)))) ).

% sums_cos_x_plus_y
tff(fact_8059_cos__x__cos__y,axiom,
    ! [X: complex,Y: complex] : pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_uc(X),Y)),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y)))) ).

% cos_x_cos_y
tff(fact_8060_cos__x__cos__y,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_ue(X),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y)))) ).

% cos_x_cos_y
tff(fact_8061_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X: 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),X))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),X))
            & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_uf,X),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,T6),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,X),N))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_8062_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [T6: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),X))
          & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_uf,X),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,T6),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,X),N))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_8063_sinh__converges,axiom,
    ! [X: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_ug,X)),aa_real_real(sinh_real,X))) ).

% sinh_converges
tff(fact_8064_sinh__converges,axiom,
    ! [X: complex] : pp(aa_complex_bool(sums_complex(aTP_Lamm_uh(X)),aa_complex_complex(sinh_complex,X))) ).

% sinh_converges
tff(fact_8065_sinh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X)),aa_real_real(sinh_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% sinh_real_less_iff
tff(fact_8066_sinh__real__neg__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% sinh_real_neg_iff
tff(fact_8067_sinh__real__pos__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sinh_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% sinh_real_pos_iff
tff(fact_8068_sinh__0,axiom,
    aa_complex_complex(sinh_complex,zero_zero_complex) = zero_zero_complex ).

% sinh_0
tff(fact_8069_sinh__0,axiom,
    aa_real_real(sinh_real,zero_zero_real) = zero_zero_real ).

% sinh_0
tff(fact_8070_cos__zero,axiom,
    aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_8071_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_8072_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),pi))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_8073_cos__monotone__0__pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).

% cos_monotone_0_pi
tff(fact_8074_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y2))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),pi))
      & ( aa_real_real(sin_real,Y2) = aa_real_real(sin_real,X) )
      & ( aa_real_real(cos_real,Y2) = aa_real_real(cos_real,X) ) ) ).

% sincos_principal_value
tff(fact_8075_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,X))) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_8076_pi__gt__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),pi)) ).

% pi_gt_zero
tff(fact_8077_pi__not__less__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),zero_zero_real)) ).

% pi_not_less_zero
tff(fact_8078_sin__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),pi))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).

% sin_gt_zero
tff(fact_8079_sinh__less__cosh__real,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X))) ).

% sinh_less_cosh_real
tff(fact_8080_cos__one__sin__zero,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cos_complex,X) = one_one_complex )
     => ( aa_complex_complex(sin_complex,X) = zero_zero_complex ) ) ).

% cos_one_sin_zero
tff(fact_8081_cos__one__sin__zero,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
     => ( aa_real_real(sin_real,X) = zero_zero_real ) ) ).

% cos_one_sin_zero
tff(fact_8082_cos__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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,X))) ) ) ).

% cos_gt_zero
tff(fact_8083_sin__eq__0__pi,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),pi))
       => ( ( aa_real_real(sin_real,X) = zero_zero_real )
         => ( X = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
tff(fact_8084_sin__zero__pi__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),pi))
     => ( ( aa_real_real(sin_real,X) = zero_zero_real )
      <=> ( X = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
tff(fact_8085_cos__gt__zero__pi,axiom,
    ! [X: 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))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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,X))) ) ) ).

% cos_gt_zero_pi
tff(fact_8086_sin__zero__norm__cos__one,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(cos_real,X)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_8087_sin__zero__norm__cos__one,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(sin_complex,X) = zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(cos_complex,X)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_8088_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_8089_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_8090_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_8091_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_8092_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
             => ( ( X = aa_real_real(cos_real,T6) )
               => ( Y != aa_real_real(sin_real,T6) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_8093_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_8094_sin__gt__zero2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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,X))) ) ) ).

% sin_gt_zero2
tff(fact_8095_sin__lt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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,X)),zero_zero_real)) ) ) ).

% sin_lt_zero
tff(fact_8096_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_8097_arctan__ubound,axiom,
    ! [Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y)),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_8098_cos__double__less__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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))),X))),one_one_real)) ) ) ).

% cos_double_less_one
tff(fact_8099_sinh__zero__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(sinh_complex,X) = zero_zero_complex )
    <=> pp(aa_set_complex_bool(member_complex(aa_complex_complex(exp_complex,X)),insert_complex(one_one_complex,insert_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex),bot_bot_set_complex)))) ) ).

% sinh_zero_iff
tff(fact_8100_sinh__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sinh_real,X) = zero_zero_real )
    <=> pp(aa_set_real_bool(member_real(aa_real_real(exp_real,X)),insert_real(one_one_real,insert_real(aa_real_real(uminus_uminus_real,one_one_real),bot_bot_set_real)))) ) ).

% sinh_zero_iff
tff(fact_8101_sin__le__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,pi),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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,X)),zero_zero_real)) ) ) ).

% sin_le_zero
tff(fact_8102_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_8103_sin__less__zero,axiom,
    ! [X: 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)))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real)) ) ) ).

% sin_less_zero
tff(fact_8104_sin__mono__less__eq,axiom,
    ! [X: real,Y: 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))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),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))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),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,X)),aa_real_real(sin_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_8105_sin__monotone__2pi,axiom,
    ! [Y: real,X: 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))))),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),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,Y)),aa_real_real(sin_real,X))) ) ) ) ).

% sin_monotone_2pi
tff(fact_8106_arctan__bounded,axiom,
    ! [Y: 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,Y)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y)),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_8107_arctan__lbound,axiom,
    ! [Y: 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,Y))) ).

% arctan_lbound
tff(fact_8108_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_8109_sinh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(inverse_inverse_real,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_8110_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X: 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,X),zero_zero_real))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),zero_zero_real))
            & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ui,X),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,T6),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,X),N))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_8111_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),X))
            & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ui,X),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,T6),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,X),N))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_8112_tan__double,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) != zero_zero_complex )
       => ( aa_complex_complex(tan_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(tan_complex,X))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(tan_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_8113_tan__double,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_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))),X)) != zero_zero_real )
       => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) = 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))),aa_real_real(tan_real,X))),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(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_8114_sin__tan,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),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,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(tan_real,X)),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,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_8115_cos__tan,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),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,X) = 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,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_8116_tan__zero,axiom,
    aa_complex_complex(tan_complex,zero_zero_complex) = zero_zero_complex ).

% tan_zero
tff(fact_8117_tan__zero,axiom,
    aa_real_real(tan_real,zero_zero_real) = zero_zero_real ).

% tan_zero
tff(fact_8118_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_8119_tan__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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,X))) ) ) ).

% tan_gt_zero
tff(fact_8120_lemma__tan__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ? [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,Y),aa_real_real(tan_real,X4))) ) ) ).

% lemma_tan_total
tff(fact_8121_lemma__tan__total1,axiom,
    ! [Y: 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) = Y ) ) ).

% lemma_tan_total1
tff(fact_8122_tan__mono__lt__eq,axiom,
    ! [X: real,Y: 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))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),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,X)),aa_real_real(tan_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_8123_tan__monotone_H,axiom,
    ! [Y: real,X: 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))))),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),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))))),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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,Y),X))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X))) ) ) ) ) ) ).

% tan_monotone'
tff(fact_8124_tan__monotone,axiom,
    ! [Y: real,X: 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))))),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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,Y)),aa_real_real(tan_real,X))) ) ) ) ).

% tan_monotone
tff(fact_8125_tan__total,axiom,
    ! [Y: 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) = Y )
      & ! [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))))),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)))))
            & ( aa_real_real(tan_real,Y3) = Y ) )
         => ( Y3 = X4 ) ) ) ).

% tan_total
tff(fact_8126_add__tan__eq,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(X),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y))) ) ) ) ).

% add_tan_eq
tff(fact_8127_add__tan__eq,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).

% add_tan_eq
tff(fact_8128_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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,X))) ) ) ).

% tan_pos_pi2_le
tff(fact_8129_tan__total__pos,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ? [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) = Y ) ) ) ).

% tan_total_pos
tff(fact_8130_tan__less__zero,axiom,
    ! [X: 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)))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,X)),zero_zero_real)) ) ) ).

% tan_less_zero
tff(fact_8131_tan__mono__le__eq,axiom,
    ! [X: real,Y: 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))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),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,X)),aa_real_real(tan_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_8132_tan__mono__le,axiom,
    ! [X: real,Y: 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))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),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,X)),aa_real_real(tan_real,Y))) ) ) ) ).

% tan_mono_le
tff(fact_8133_tan__bound__pi2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),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,X))),one_one_real)) ) ).

% tan_bound_pi2
tff(fact_8134_arctan,axiom,
    ! [Y: 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,Y)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y)),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,Y)) = Y ) ) ).

% arctan
tff(fact_8135_arctan__tan,axiom,
    ! [X: 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))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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,X)) = X ) ) ) ).

% arctan_tan
tff(fact_8136_arctan__unique,axiom,
    ! [X: real,Y: 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))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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,X) = Y )
         => ( aa_real_real(arctan,Y) = X ) ) ) ) ).

% arctan_unique
tff(fact_8137_tan__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).

% tan_add
tff(fact_8138_tan__add,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_add
tff(fact_8139_tan__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(X),Y)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(minus_minus_complex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_8140_tan__diff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_8141_lemma__tan__add1,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_8142_lemma__tan__add1,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_8143_tan__total__pi4,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ? [Z3: 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)))))),Z3))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),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,Z3) = X ) ) ) ).

% tan_total_pi4
tff(fact_8144_tan__sec,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(tan_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(cos_complex,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% tan_sec
tff(fact_8145_tan__sec,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( 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,X)),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(inverse_inverse_real,aa_real_real(cos_real,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% tan_sec
tff(fact_8146_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),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,T6)),aa_real_real(sin_real,T6)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_8147_cot__less__zero,axiom,
    ! [X: 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)))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cot_real,X)),zero_zero_real)) ) ) ).

% cot_less_zero
tff(fact_8148_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,set_ord_lessThan_nat(K))),suminf_real(F))) ) ) ).

% sum_pos_lt_pair
tff(fact_8149_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_uj,Y)) ).

% arctan_def
tff(fact_8150_cot__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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,X))) ) ) ).

% cot_gt_zero
tff(fact_8151_cot__zero,axiom,
    aa_complex_complex(cot_complex,zero_zero_complex) = zero_zero_complex ).

% cot_zero
tff(fact_8152_cot__zero,axiom,
    aa_real_real(cot_real,zero_zero_real) = zero_zero_real ).

% cot_zero
tff(fact_8153_summable__single,axiom,
    ! [I: nat,F: fun_nat_complex] : summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_gc(I),F)) ).

% summable_single
tff(fact_8154_summable__single,axiom,
    ! [I: nat,F: fun_nat_real] : summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_qi(I),F)) ).

% summable_single
tff(fact_8155_summable__single,axiom,
    ! [I: nat,F: fun_nat_nat] : summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qj(I),F)) ).

% summable_single
tff(fact_8156_summable__single,axiom,
    ! [I: nat,F: fun_nat_int] : summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_qk(I),F)) ).

% summable_single
tff(fact_8157_summable__zero,axiom,
    summable_complex(aTP_Lamm_qg) ).

% summable_zero
tff(fact_8158_summable__zero,axiom,
    summable_real(aTP_Lamm_ft) ).

% summable_zero
tff(fact_8159_summable__zero,axiom,
    summable_nat(aTP_Lamm_ck) ).

% summable_zero
tff(fact_8160_summable__zero,axiom,
    summable_int(aTP_Lamm_qh) ).

% summable_zero
tff(fact_8161_summable__cmult__iff,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ql(C),F))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_cmult_iff
tff(fact_8162_summable__cmult__iff,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_qm(C),F))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_8163_summable__divide__iff,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_uk(F),C))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_divide_iff
tff(fact_8164_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ul(F),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_8165_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qw(A2),F)) ) ).

% summable_If_finite_set
tff(fact_8166_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_qz(A2),F)) ) ).

% summable_If_finite_set
tff(fact_8167_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qy(A2),F)) ) ).

% summable_If_finite_set
tff(fact_8168_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_qx(A2),F)) ) ).

% summable_If_finite_set
tff(fact_8169_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qs(P),F)) ) ).

% summable_If_finite
tff(fact_8170_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_qv(P),F)) ) ).

% summable_If_finite
tff(fact_8171_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qu(P),F)) ) ).

% summable_If_finite
tff(fact_8172_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_qt(P),F)) ) ).

% summable_If_finite
tff(fact_8173_summable__geometric__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_real(power_power_real,C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)) ) ).

% summable_geometric_iff
tff(fact_8174_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( summable_complex(power_power_complex(C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)) ) ).

% summable_geometric_iff
tff(fact_8175_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N4: nat,F: fun_nat_real] :
      ( summable_real(G)
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N4),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2))),aa_nat_real(G,N2))) )
       => summable_real(F) ) ) ).

% summable_comparison_test'
tff(fact_8176_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N4: nat,F: fun_nat_complex] :
      ( summable_real(G)
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N4),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))),aa_nat_real(G,N2))) )
       => summable_complex(F) ) ) ).

% summable_comparison_test'
tff(fact_8177_summable__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2))),aa_nat_real(G,N2))) )
     => ( summable_real(G)
       => summable_real(F) ) ) ).

% summable_comparison_test
tff(fact_8178_summable__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))),aa_nat_real(G,N2))) )
     => ( summable_real(G)
       => summable_complex(F) ) ) ).

% summable_comparison_test
tff(fact_8179_summable__const__iff,axiom,
    ! [C: complex] :
      ( summable_complex(aTP_Lamm_um(C))
    <=> ( C = zero_zero_complex ) ) ).

% summable_const_iff
tff(fact_8180_summable__const__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_un,C))
    <=> ( C = zero_zero_real ) ) ).

% summable_const_iff
tff(fact_8181_powser__insidea,axiom,
    ! [F: fun_nat_real,X: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_nr(F),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,X)))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_uo(F),Z)) ) ) ).

% powser_insidea
tff(fact_8182_powser__insidea,axiom,
    ! [F: fun_nat_complex,X: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ni(F),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,X)))
       => summable_real(aa_com20334259897932645t_real(aTP_Lamm_up(F),Z)) ) ) ).

% powser_insidea
tff(fact_8183_suminf__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G,N2)))
     => ( summable_real(F)
       => ( summable_real(G)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(F)),suminf_real(G))) ) ) ) ).

% suminf_le
tff(fact_8184_suminf__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(G,N2)))
     => ( summable_nat(F)
       => ( summable_nat(G)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,suminf_nat(F)),suminf_nat(G))) ) ) ) ).

% suminf_le
tff(fact_8185_suminf__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(G,N2)))
     => ( summable_int(F)
       => ( summable_int(G)
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,suminf_int(F)),suminf_int(G))) ) ) ) ).

% suminf_le
tff(fact_8186_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_complex(F,N2) = zero_zero_complex ) )
       => summable_complex(F) ) ) ).

% summable_finite
tff(fact_8187_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => summable_real(F) ) ) ).

% summable_finite
tff(fact_8188_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => summable_nat(F) ) ) ).

% summable_finite
tff(fact_8189_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => summable_int(F) ) ) ).

% summable_finite
tff(fact_8190_summable__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ql(C),F))
     => ( ( C != zero_zero_complex )
       => summable_complex(F) ) ) ).

% summable_mult_D
tff(fact_8191_summable__mult__D,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_qm(C),F))
     => ( ( C != zero_zero_real )
       => summable_real(F) ) ) ).

% summable_mult_D
tff(fact_8192_summable__zero__power,axiom,
    summable_real(aa_real_fun_nat_real(power_power_real,zero_zero_real)) ).

% summable_zero_power
tff(fact_8193_summable__zero__power,axiom,
    summable_int(power_power_int(zero_zero_int)) ).

% summable_zero_power
tff(fact_8194_summable__zero__power,axiom,
    summable_complex(power_power_complex(zero_zero_complex)) ).

% summable_zero_power
tff(fact_8195_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => ( ( suminf_real(F) = zero_zero_real )
        <=> ! [N3: nat] : aa_nat_real(F,N3) = zero_zero_real ) ) ) ).

% suminf_eq_zero_iff
tff(fact_8196_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => ( ( suminf_nat(F) = zero_zero_nat )
        <=> ! [N3: nat] : aa_nat_nat(F,N3) = zero_zero_nat ) ) ) ).

% suminf_eq_zero_iff
tff(fact_8197_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => ( ( suminf_int(F) = zero_zero_int )
        <=> ! [N3: nat] : aa_nat_int(F,N3) = zero_zero_int ) ) ) ).

% suminf_eq_zero_iff
tff(fact_8198_suminf__nonneg,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),suminf_real(F))) ) ) ).

% suminf_nonneg
tff(fact_8199_suminf__nonneg,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),suminf_nat(F))) ) ) ).

% suminf_nonneg
tff(fact_8200_suminf__nonneg,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),suminf_int(F))) ) ) ).

% suminf_nonneg
tff(fact_8201_suminf__pos,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,N2)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F))) ) ) ).

% suminf_pos
tff(fact_8202_suminf__pos,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F))) ) ) ).

% suminf_pos
tff(fact_8203_suminf__pos,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,N2)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F))) ) ) ).

% suminf_pos
tff(fact_8204_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_gv(F)) ).

% summable_0_powser
tff(fact_8205_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_gx(F)) ).

% summable_0_powser
tff(fact_8206_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_gv(F)) ).

% summable_zero_power'
tff(fact_8207_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_gx(F)) ).

% summable_zero_power'
tff(fact_8208_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_uq(F)) ).

% summable_zero_power'
tff(fact_8209_summable__norm__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))),aa_nat_real(G,N2))) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_ur(F)) ) ) ).

% summable_norm_comparison_test
tff(fact_8210_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N2))),aa_nat_real(G,N2))) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_rm(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_8211_suminf__pos__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F)))
        <=> ? [I4: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I4))) ) ) ) ).

% suminf_pos_iff
tff(fact_8212_suminf__pos__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F)))
        <=> ? [I4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I4))) ) ) ) ).

% suminf_pos_iff
tff(fact_8213_suminf__pos__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F)))
        <=> ? [I4: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I4))) ) ) ) ).

% suminf_pos_iff
tff(fact_8214_suminf__pos2,axiom,
    ! [F: fun_nat_real,I: nat] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F))) ) ) ) ).

% suminf_pos2
tff(fact_8215_suminf__pos2,axiom,
    ! [F: fun_nat_nat,I: nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F))) ) ) ) ).

% suminf_pos2
tff(fact_8216_suminf__pos2,axiom,
    ! [F: fun_nat_int,I: nat] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F))) ) ) ) ).

% suminf_pos2
tff(fact_8217_suminf__le__const,axiom,
    ! [F: fun_nat_int,X: int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,set_ord_lessThan_nat(N2))),X))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,suminf_int(F)),X)) ) ) ).

% suminf_le_const
tff(fact_8218_suminf__le__const,axiom,
    ! [F: fun_nat_nat,X: nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),set_ord_lessThan_nat(N2))),X))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,suminf_nat(F)),X)) ) ) ).

% suminf_le_const
tff(fact_8219_suminf__le__const,axiom,
    ! [F: fun_nat_real,X: real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(F,set_ord_lessThan_nat(N2))),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(F)),X)) ) ) ).

% suminf_le_const
tff(fact_8220_powser__inside,axiom,
    ! [F: fun_nat_real,X: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_nr(F),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,X)))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_nr(F),Z)) ) ) ).

% powser_inside
tff(fact_8221_powser__inside,axiom,
    ! [F: fun_nat_complex,X: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ni(F),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,X)))
       => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ni(F),Z)) ) ) ).

% powser_inside
tff(fact_8222_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_int,X: int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,set_ord_lessThan_nat(N2))),X))
       => summable_int(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8223_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_nat,X: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),set_ord_lessThan_nat(N2))),X))
       => summable_nat(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8224_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_real,X: real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(F,set_ord_lessThan_nat(N2))),X))
       => summable_real(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8225_bounded__imp__summable,axiom,
    ! [A: fun_nat_int,B2: int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(A,N2)))
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(A,set_ord_atMost_nat(N2))),B2))
       => summable_int(A) ) ) ).

% bounded_imp_summable
tff(fact_8226_bounded__imp__summable,axiom,
    ! [A: fun_nat_nat,B2: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(A,N2)))
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),set_ord_atMost_nat(N2))),B2))
       => summable_nat(A) ) ) ).

% bounded_imp_summable
tff(fact_8227_bounded__imp__summable,axiom,
    ! [A: fun_nat_real,B2: real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(A,set_ord_atMost_nat(N2))),B2))
       => summable_real(A) ) ) ).

% bounded_imp_summable
tff(fact_8228_summable__geometric,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real))
     => summable_real(aa_real_fun_nat_real(power_power_real,C)) ) ).

% summable_geometric
tff(fact_8229_summable__geometric,axiom,
    ! [C: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real))
     => summable_complex(power_power_complex(C)) ) ).

% summable_geometric
tff(fact_8230_complete__algebra__summable__geometric,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),one_one_real))
     => summable_real(aa_real_fun_nat_real(power_power_real,X)) ) ).

% complete_algebra_summable_geometric
tff(fact_8231_complete__algebra__summable__geometric,axiom,
    ! [X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),one_one_real))
     => summable_complex(power_power_complex(X)) ) ).

% complete_algebra_summable_geometric
tff(fact_8232_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_ln(F)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),aa_nat_real(F,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_8233_the__elem__def,axiom,
    ! [X6: set_Pr958786334691620121nt_int] : the_el8326832613380209454nt_int(X6) = the_Pr4378521158711661632nt_int(aTP_Lamm_us(X6)) ).

% the_elem_def
tff(fact_8234_the__elem__def,axiom,
    ! [X6: set_Pr4329608150637261639at_nat] : the_el221668144340439132at_nat(X6) = the_Pr4604535484834969198at_nat(aTP_Lamm_ut(X6)) ).

% the_elem_def
tff(fact_8235_the__elem__def,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] : the_el2281957884133575798at_nat(X6) = the_Pr7557018466319803784at_nat(aTP_Lamm_uu(X6)) ).

% the_elem_def
tff(fact_8236_the__elem__def,axiom,
    ! [X6: set_real] : the_elem_real(X6) = the_real(aTP_Lamm_uv(X6)) ).

% the_elem_def
tff(fact_8237_the__elem__def,axiom,
    ! [X6: set_nat] : aa_set_nat_nat(the_elem_nat,X6) = the_nat(aa_set3744802623701456604t_bool(aTP_Lamm_uw,X6)) ).

% the_elem_def
tff(fact_8238_the__elem__def,axiom,
    ! [X6: set_int] : the_elem_int(X6) = the_int(aTP_Lamm_ux(X6)) ).

% the_elem_def
tff(fact_8239_sum__le__suminf,axiom,
    ! [F: fun_nat_int,I6: set_nat] :
      ( summable_int(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I6))
       => ( ! [N2: nat] :
              ( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I6)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,I6)),suminf_int(F))) ) ) ) ).

% sum_le_suminf
tff(fact_8240_sum__le__suminf,axiom,
    ! [F: fun_nat_nat,I6: set_nat] :
      ( summable_nat(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I6))
       => ( ! [N2: nat] :
              ( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I6)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),I6)),suminf_nat(F))) ) ) ) ).

% sum_le_suminf
tff(fact_8241_sum__le__suminf,axiom,
    ! [F: fun_nat_real,I6: set_nat] :
      ( summable_real(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I6))
       => ( ! [N2: nat] :
              ( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I6)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(F,I6)),suminf_real(F))) ) ) ) ).

% sum_le_suminf
tff(fact_8242_sum__less__suminf,axiom,
    ! [F: fun_nat_int,N: nat] :
      ( summable_int(F)
     => ( ! [M5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M5))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,M5))) )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups3539618377306564664at_int(F,set_ord_lessThan_nat(N))),suminf_int(F))) ) ) ).

% sum_less_suminf
tff(fact_8243_sum__less__suminf,axiom,
    ! [F: fun_nat_nat,N: nat] :
      ( summable_nat(F)
     => ( ! [M5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,M5))) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),set_ord_lessThan_nat(N))),suminf_nat(F))) ) ) ).

% sum_less_suminf
tff(fact_8244_sum__less__suminf,axiom,
    ! [F: fun_nat_real,N: nat] :
      ( summable_real(F)
     => ( ! [M5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,M5))) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups6591440286371151544t_real(F,set_ord_lessThan_nat(N))),suminf_real(F))) ) ) ).

% sum_less_suminf
tff(fact_8245_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ni(F),Z))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_ni(F),Z)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(F,zero_zero_nat)),aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_uy(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_8246_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_nr(F),Z))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_nr(F),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,zero_zero_nat)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_uz(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_8247_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ni(F),Z))
     => ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_uy(F),Z))),Z) = aa_complex_complex(minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_ni(F),Z))),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_8248_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_nr(F),Z))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_uz(F),Z))),Z) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_nr(F),Z))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_8249_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_complex,E2: real] :
      ( summable_complex(F)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
       => ~ ! [N8: nat] :
              ~ ! [M2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),M2))
                 => ! [N5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,set_or1269000886237332187st_nat(M2,N5)))),E2)) ) ) ) ).

% summable_partial_sum_bound
tff(fact_8250_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_real,E2: real] :
      ( summable_real(F)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
       => ~ ! [N8: nat] :
              ~ ! [M2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),M2))
                 => ! [N5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,groups6591440286371151544t_real(F,set_or1269000886237332187st_nat(M2,N5)))),E2)) ) ) ) ).

% summable_partial_sum_bound
tff(fact_8251_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
     => ( summable_real(F)
       => ? [N8: nat] :
          ! [N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_va(F),N5)))),R2)) ) ) ) ).

% suminf_exist_split
tff(fact_8252_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
     => ( summable_complex(F)
       => ? [N8: nat] :
          ! [N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_vb(F),N5)))),R2)) ) ) ) ).

% suminf_exist_split
tff(fact_8253_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_nr(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_8254_Abel__lemma,axiom,
    ! [R2: real,R0: real,A: fun_nat_complex,M4: 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_real,R2),R0))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(A,N2))),aa_nat_real(aa_real_fun_nat_real(power_power_real,R0),N2))),M4))
         => summable_real(aa_fun7054779078796324018t_real(aTP_Lamm_vc(R2),A)) ) ) ) ).

% Abel_lemma
tff(fact_8255_summable__ratio__test,axiom,
    ! [C: real,N4: nat,F: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),one_one_real))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N4),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,aa_nat_nat(suc,N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2))))) )
       => summable_real(F) ) ) ).

% summable_ratio_test
tff(fact_8256_summable__ratio__test,axiom,
    ! [C: real,N4: nat,F: fun_nat_complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),one_one_real))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N4),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,aa_nat_nat(suc,N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))))) )
       => summable_complex(F) ) ) ).

% summable_ratio_test
tff(fact_8257_sum__less__suminf2,axiom,
    ! [F: fun_nat_int,N: nat,I: nat] :
      ( summable_int(F)
     => ( ! [M5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M5))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,M5))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups3539618377306564664at_int(F,set_ord_lessThan_nat(N))),suminf_int(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8258_sum__less__suminf2,axiom,
    ! [F: fun_nat_nat,N: nat,I: nat] :
      ( summable_nat(F)
     => ( ! [M5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,M5))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),set_ord_lessThan_nat(N))),suminf_nat(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8259_sum__less__suminf2,axiom,
    ! [F: fun_nat_real,N: nat,I: nat] :
      ( summable_real(F)
     => ( ! [M5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,M5))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups6591440286371151544t_real(F,set_ord_lessThan_nat(N))),suminf_real(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8260_diffs__equiv,axiom,
    ! [C: fun_nat_complex,X: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_vd(C),X))
     => pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_ve(C),X)),suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_vd(C),X)))) ) ).

% diffs_equiv
tff(fact_8261_diffs__equiv,axiom,
    ! [C: fun_nat_real,X: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_vf(C),X))
     => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_vg(C),X)),suminf_real(aa_real_fun_nat_real(aTP_Lamm_vf(C),X)))) ) ).

% diffs_equiv
tff(fact_8262_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),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,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,Y)),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_8263_pochhammer__code,axiom,
    ! [N: nat,A: complex] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_vh(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_complex) ) ) ) ).

% pochhammer_code
tff(fact_8264_pochhammer__code,axiom,
    ! [N: nat,A: rat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_vi(A),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_8265_pochhammer__code,axiom,
    ! [N: nat,A: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(A),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(A),N) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_vj(A),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_8266_pochhammer__code,axiom,
    ! [N: nat,A: real] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_vk(A),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_8267_pochhammer__code,axiom,
    ! [N: nat,A: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_vl(A),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_8268_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_8269_Cauchy__iff2,axiom,
    ! [X6: fun_nat_real] :
      ( topolo4055970368930404560y_real(X6)
    <=> ! [J3: nat] :
        ? [M9: nat] :
        ! [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),M3))
         => ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),N3))
             => 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(X6,M3)),aa_nat_real(X6,N3)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3))))) ) ) ) ).

% Cauchy_iff2
tff(fact_8270_or_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,zero_zero_nat),A) = A ).

% or.left_neutral
tff(fact_8271_or_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se1409905431419307370or_int(zero_zero_int),A) = A ).

% or.left_neutral
tff(fact_8272_or_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),zero_zero_nat) = A ).

% or.right_neutral
tff(fact_8273_or_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se1409905431419307370or_int(A),zero_zero_int) = A ).

% or.right_neutral
tff(fact_8274_The__split__eq,axiom,
    ! [X: fun_Co6545587727236015215e_term,Y: produc8923325533196201883nteger] : the_Pr1822819638310571034nteger(produc401098500374025047r_bool(aa_Pro4785828671252635594r_bool(aTP_Lamm_vm(X),Y))) = aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,X),Y) ).

% The_split_eq
tff(fact_8275_The__split__eq,axiom,
    ! [X: fun_Pr272337284026993017e_term,Y: produc8923325533196201883nteger] : the_Pr8744451031814498960nteger(produc1236900166980094561r_bool(aa_Pro8070820463477474004r_bool(aTP_Lamm_vn(X),Y))) = aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,X),Y) ).

% The_split_eq
tff(fact_8276_The__split__eq,axiom,
    ! [X: fun_Pr9164807099015279978e_term,Y: product_prod_int_int] : the_Pr8879353902343807613nt_int(produc625993674133193396t_bool(aa_Pro7791619889881332805t_bool(aTP_Lamm_vo(X),Y))) = aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,X),Y) ).

% The_split_eq
tff(fact_8277_The__split__eq,axiom,
    ! [X: fun_in6429348457993719614e_term,Y: product_prod_int_int] : the_Pr1349890353615962025nt_int(produc6627716702706253960t_bool(aa_Pro4937060880834667033t_bool(aTP_Lamm_vp(X),Y))) = aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,X),Y) ).

% The_split_eq
tff(fact_8278_The__split__eq,axiom,
    ! [X: nat,Y: nat] : the_Pr7557018466319803784at_nat(produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_vq(X),Y))) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y) ).

% The_split_eq
tff(fact_8279_The__split__eq,axiom,
    ! [X: int,Y: int] : the_Pr4378521158711661632nt_int(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_vr(X),Y))) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y) ).

% The_split_eq
tff(fact_8280_pochhammer__0,axiom,
    ! [A: complex] : aa_nat_complex(comm_s2602460028002588243omplex(A),zero_zero_nat) = one_one_complex ).

% pochhammer_0
tff(fact_8281_pochhammer__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_8282_pochhammer__0,axiom,
    ! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).

% pochhammer_0
tff(fact_8283_pochhammer__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),zero_zero_nat) = one_one_nat ).

% pochhammer_0
tff(fact_8284_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_8285_or__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% or_nat_numerals(4)
tff(fact_8286_or__nat__numerals_I2_J,axiom,
    ! [Y: 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,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_nat_numerals(2)
tff(fact_8287_or__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% or_nat_numerals(3)
tff(fact_8288_or__nat__numerals_I1_J,axiom,
    ! [Y: 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,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_8289_or__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% or_eq_0_iff
tff(fact_8290_or__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(bit_se1409905431419307370or_int(A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & ( B = zero_zero_int ) ) ) ).

% or_eq_0_iff
tff(fact_8291_bit_Odisj__zero__right,axiom,
    ! [X: int] : aa_int_int(bit_se1409905431419307370or_int(X),zero_zero_int) = X ).

% bit.disj_zero_right
tff(fact_8292_pochhammer__pos,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,X),N))) ) ).

% pochhammer_pos
tff(fact_8293_pochhammer__pos,axiom,
    ! [X: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),N))) ) ).

% pochhammer_pos
tff(fact_8294_pochhammer__pos,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
     => 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,X),N))) ) ).

% pochhammer_pos
tff(fact_8295_pochhammer__pos,axiom,
    ! [X: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X),N))) ) ).

% pochhammer_pos
tff(fact_8296_pochhammer__neq__0__mono,axiom,
    ! [A: complex,M: nat,N: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),M) != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_8297_pochhammer__neq__0__mono,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),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,A),N) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_8298_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),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(A),N) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_8299_pochhammer__eq__0__mono,axiom,
    ! [A: complex,N: nat,M: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),M) = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_8300_pochhammer__eq__0__mono,axiom,
    ! [A: real,N: nat,M: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),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,A),M) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_8301_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N: nat,M: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),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(A),M) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_8302_pochhammer__nonneg,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => 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,X),N))) ) ).

% pochhammer_nonneg
tff(fact_8303_pochhammer__nonneg,axiom,
    ! [X: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),N))) ) ).

% pochhammer_nonneg
tff(fact_8304_pochhammer__nonneg,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
     => 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,X),N))) ) ).

% pochhammer_nonneg
tff(fact_8305_pochhammer__nonneg,axiom,
    ! [X: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X),N))) ) ).

% pochhammer_nonneg
tff(fact_8306_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),N) = zero_zero_complex ) ) ) ).

% pochhammer_0_left
tff(fact_8307_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_8308_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_8309_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_8310_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N) = zero_zero_int ) ) ) ).

% pochhammer_0_left
tff(fact_8311_bit_Ocomplement__unique,axiom,
    ! [A: code_integer,X: code_integer,Y: code_integer] :
      ( ( bit_se3949692690581998587nteger(A,X) = zero_z3403309356797280102nteger )
     => ( ( bit_se1080825931792720795nteger(A,X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
       => ( ( bit_se3949692690581998587nteger(A,Y) = zero_z3403309356797280102nteger )
         => ( ( bit_se1080825931792720795nteger(A,Y) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
           => ( X = Y ) ) ) ) ) ).

% bit.complement_unique
tff(fact_8312_bit_Ocomplement__unique,axiom,
    ! [A: int,X: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(A),X) = zero_zero_int )
     => ( ( aa_int_int(bit_se1409905431419307370or_int(A),X) = aa_int_int(uminus_uminus_int,one_one_int) )
       => ( ( aa_int_int(bit_se725231765392027082nd_int(A),Y) = zero_zero_int )
         => ( ( aa_int_int(bit_se1409905431419307370or_int(A),Y) = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( X = Y ) ) ) ) ) ).

% bit.complement_unique
tff(fact_8313_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = zero_zero_complex )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
          & ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_8314_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = zero_zero_rat )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
          & ( A = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_8315_pochhammer__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = zero_zero_real )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
          & ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_8316_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) = zero_zero_complex )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8317_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) = zero_zero_rat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8318_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) = zero_z3403309356797280102nteger )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8319_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) = zero_zero_int )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8320_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) = zero_zero_real )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8321_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8322_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8323_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8324_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8325_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8326_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8327_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8328_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8329_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8330_pochhammer__of__nat__eq__0__lemma_H,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(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8331_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y))) ) ) ) ).

% arcsin_less_arcsin
tff(fact_8332_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ).

% arcsin_less_mono
tff(fact_8333_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(Z),N) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),M)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_8334_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(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),M)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_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_8335_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_8336_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(comm_s4660882817536571857er_int(Z),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),M)),aa_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_8337_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_8338_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => ( aa_real_real(cos_real,aa_real_real(arcsin,X)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_8339_Cauchy__iff,axiom,
    ! [X6: fun_nat_complex] :
      ( topolo6517432010174082258omplex(X6)
    <=> ! [E3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E3))
         => ? [M9: nat] :
            ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),M3))
             => ! [N3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),N3))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(X6,M3)),aa_nat_complex(X6,N3)))),E3)) ) ) ) ) ).

% Cauchy_iff
tff(fact_8340_Cauchy__iff,axiom,
    ! [X6: fun_nat_real] :
      ( topolo4055970368930404560y_real(X6)
    <=> ! [E3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E3))
         => ? [M9: nat] :
            ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),M3))
             => ! [N3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),N3))
                 => 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,aa_nat_real(X6,M3)),aa_nat_real(X6,N3)))),E3)) ) ) ) ) ).

% Cauchy_iff
tff(fact_8341_CauchyI,axiom,
    ! [X6: fun_nat_complex] :
      ( ! [E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => ? [M10: nat] :
            ! [M5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M10),M5))
             => ! [N2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M10),N2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(X6,M5)),aa_nat_complex(X6,N2)))),E)) ) ) )
     => topolo6517432010174082258omplex(X6) ) ).

% CauchyI
tff(fact_8342_CauchyI,axiom,
    ! [X6: fun_nat_real] :
      ( ! [E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => ? [M10: nat] :
            ! [M5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M10),M5))
             => ! [N2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M10),N2))
                 => 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,aa_nat_real(X6,M5)),aa_nat_real(X6,N2)))),E)) ) ) )
     => topolo4055970368930404560y_real(X6) ) ).

% CauchyI
tff(fact_8343_CauchyD,axiom,
    ! [X6: fun_nat_complex,E2: real] :
      ( topolo6517432010174082258omplex(X6)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
       => ? [M8: nat] :
          ! [M2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M8),M2))
           => ! [N5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M8),N5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(X6,M2)),aa_nat_complex(X6,N5)))),E2)) ) ) ) ) ).

% CauchyD
tff(fact_8344_CauchyD,axiom,
    ! [X6: fun_nat_real,E2: real] :
      ( topolo4055970368930404560y_real(X6)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
       => ? [M8: nat] :
          ! [M2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M8),M2))
           => ! [N5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M8),N5))
               => 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,aa_nat_real(X6,M2)),aa_nat_real(X6,N5)))),E2)) ) ) ) ) ).

% CauchyD
tff(fact_8345_floor__real__def,axiom,
    ! [X: real] : aa_real_int(archim6058952711729229775r_real,X) = the_int(aa_real_fun_int_bool(aTP_Lamm_vs,X)) ).

% floor_real_def
tff(fact_8346_termdiff__converges,axiom,
    ! [X: real,K4: real,C: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),K4))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X4)),K4))
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_nr(C),X4)) )
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_vt(X),C)) ) ) ).

% termdiff_converges
tff(fact_8347_termdiff__converges,axiom,
    ! [X: complex,K4: real,C: fun_nat_complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),K4))
     => ( ! [X4: complex] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X4)),K4))
           => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ni(C),X4)) )
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vu(X),C)) ) ) ).

% termdiff_converges
tff(fact_8348_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),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_8349_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),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_8350_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,N))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = groups6464643781859351333omplex(aTP_Lamm_vv(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_8351_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,N: nat] : aa_rat_rat(aa_rat_fun_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(aa_rat_fun_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_vw(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_8352_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_vx,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_8353_floor__rat__def,axiom,
    ! [X: rat] : aa_rat_int(archim3151403230148437115or_rat,X) = the_int(aTP_Lamm_vy(X)) ).

% floor_rat_def
tff(fact_8354_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_8355_VEBT_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Node(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_list_VEBT_VEBT(size_size_VEBT_VEBT,X13)),aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size(3)
tff(fact_8356_is__singleton__the__elem,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(is_sin2937591304547752795at_nat,A2))
    <=> ( A2 = insert9069300056098147895at_nat(the_el221668144340439132at_nat(A2),bot_bo228742789529271731at_nat) ) ) ).

% is_singleton_the_elem
tff(fact_8357_is__singleton__the__elem,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(is_sin2850979758926227957at_nat,A2))
    <=> ( A2 = insert8211810215607154385at_nat(the_el2281957884133575798at_nat(A2),bot_bo2099793752762293965at_nat) ) ) ).

% is_singleton_the_elem
tff(fact_8358_is__singleton__the__elem,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(is_singleton_real,A2))
    <=> ( A2 = insert_real(the_elem_real(A2),bot_bot_set_real) ) ) ).

% is_singleton_the_elem
tff(fact_8359_is__singleton__the__elem,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(is_singleton_nat,A2))
    <=> ( A2 = insert_nat(aa_set_nat_nat(the_elem_nat,A2),bot_bot_set_nat) ) ) ).

% is_singleton_the_elem
tff(fact_8360_is__singleton__the__elem,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(is_singleton_int,A2))
    <=> ( A2 = insert_int(the_elem_int(A2),bot_bot_set_int) ) ) ).

% is_singleton_the_elem
tff(fact_8361_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(K),L)),zero_zero_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(ord_less_int,L),zero_zero_int)) ) ) ).

% or_negative_int_iff
tff(fact_8362_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,set_or4665077453230672383an_nat(L,U))) ).

% finite_atLeastLessThan
tff(fact_8363_atLeastLessThan__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(I),set_or3540276404033026485et_nat(L,U)))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,L),I))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_8364_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_8365_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_8366_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_8367_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_8368_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_8369_atLeastLessThan__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( pp(aa_set_int_bool(member_int(I),aa_int_set_int(aa_int8639817059373041269et_int(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_8370_prod__zero__iff,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( groups6464643781859351333omplex(F,A2) = zero_zero_complex )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
            & ( aa_nat_complex(F,X3) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_8371_prod__zero__iff,axiom,
    ! [A2: set_int,F: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups7440179247065528705omplex(F,A2) = zero_zero_complex )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
            & ( aa_int_complex(F,X3) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_8372_prod__zero__iff,axiom,
    ! [A2: set_complex,F: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups3708469109370488835omplex(F,A2) = zero_zero_complex )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
            & ( aa_complex_complex(F,X3) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_8373_prod__zero__iff,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( groups129246275422532515t_real(F,A2) = zero_zero_real )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
            & ( aa_nat_real(F,X3) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_8374_prod__zero__iff,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups2316167850115554303t_real(F,A2) = zero_zero_real )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
            & ( aa_int_real(F,X3) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_8375_prod__zero__iff,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups766887009212190081x_real(F,A2) = zero_zero_real )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
            & ( aa_complex_real(F,X3) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_8376_prod__zero__iff,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( groups73079841787564623at_rat(F,A2) = zero_zero_rat )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
            & ( aa_nat_rat(F,X3) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_8377_prod__zero__iff,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups1072433553688619179nt_rat(F,A2) = zero_zero_rat )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
            & ( aa_int_rat(F,X3) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_8378_prod__zero__iff,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups225925009352817453ex_rat(F,A2) = zero_zero_rat )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
            & ( aa_complex_rat(F,X3) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_8379_prod__zero__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups1707563613775114915nt_nat(F,A2) = zero_zero_nat )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
            & ( aa_int_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_8380_atLeastLessThan__empty,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( set_or66887138388493659n_real(A,B) = bot_bot_set_real ) ) ).

% atLeastLessThan_empty
tff(fact_8381_atLeastLessThan__empty,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
     => ( set_or8585797421378605585et_int(A,B) = bot_bot_set_set_int ) ) ).

% atLeastLessThan_empty
tff(fact_8382_atLeastLessThan__empty,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( set_or4029947393144176647an_rat(A,B) = bot_bot_set_rat ) ) ).

% atLeastLessThan_empty
tff(fact_8383_atLeastLessThan__empty,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( set_or1222409239386451017an_num(A,B) = bot_bot_set_num ) ) ).

% atLeastLessThan_empty
tff(fact_8384_atLeastLessThan__empty,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( set_or4665077453230672383an_nat(A,B) = bot_bot_set_nat ) ) ).

% atLeastLessThan_empty
tff(fact_8385_atLeastLessThan__empty,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,A),B) = bot_bot_set_int ) ) ).

% atLeastLessThan_empty
tff(fact_8386_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_8387_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_8388_ivl__subset,axiom,
    ! [I: nat,J: nat,M: nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_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_8389_ivl__subset,axiom,
    ! [I: int,J: int,M: int,N: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,I),J)),aa_int_set_int(aa_int8639817059373041269et_int(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_8390_atLeastLessThan__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real = set_or66887138388493659n_real(A,B) )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_8391_atLeastLessThan__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat = set_or4029947393144176647an_rat(A,B) )
    <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_8392_atLeastLessThan__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or1222409239386451017an_num(A,B) )
    <=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_8393_atLeastLessThan__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = set_or4665077453230672383an_nat(A,B) )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_8394_atLeastLessThan__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,A),B) )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_8395_atLeastLessThan__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( set_or66887138388493659n_real(A,B) = bot_bot_set_real )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_8396_atLeastLessThan__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = bot_bot_set_rat )
    <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_8397_atLeastLessThan__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or1222409239386451017an_num(A,B) = bot_bot_set_num )
    <=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_8398_atLeastLessThan__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( set_or4665077453230672383an_nat(A,B) = bot_bot_set_nat )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_8399_atLeastLessThan__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,A),B) = bot_bot_set_int )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_8400_infinite__Ico__iff,axiom,
    ! [A: real,B: real] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,set_or66887138388493659n_real(A,B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% infinite_Ico_iff
tff(fact_8401_infinite__Ico__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_set_rat_bool(finite_finite_rat,set_or4029947393144176647an_rat(A,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% infinite_Ico_iff
tff(fact_8402_prod_Oempty,axiom,
    ! [G: fun_real_complex] : groups713298508707869441omplex(G,bot_bot_set_real) = one_one_complex ).

% prod.empty
tff(fact_8403_prod_Oempty,axiom,
    ! [G: fun_real_real] : groups1681761925125756287l_real(G,bot_bot_set_real) = one_one_real ).

% prod.empty
tff(fact_8404_prod_Oempty,axiom,
    ! [G: fun_real_rat] : groups4061424788464935467al_rat(G,bot_bot_set_real) = one_one_rat ).

% prod.empty
tff(fact_8405_prod_Oempty,axiom,
    ! [G: fun_real_nat] : groups4696554848551431203al_nat(G,bot_bot_set_real) = one_one_nat ).

% prod.empty
tff(fact_8406_prod_Oempty,axiom,
    ! [G: fun_real_int] : groups4694064378042380927al_int(G,bot_bot_set_real) = one_one_int ).

% prod.empty
tff(fact_8407_prod_Oempty,axiom,
    ! [G: fun_nat_complex] : groups6464643781859351333omplex(G,bot_bot_set_nat) = one_one_complex ).

% prod.empty
tff(fact_8408_prod_Oempty,axiom,
    ! [G: fun_nat_real] : groups129246275422532515t_real(G,bot_bot_set_nat) = one_one_real ).

% prod.empty
tff(fact_8409_prod_Oempty,axiom,
    ! [G: fun_nat_rat] : groups73079841787564623at_rat(G,bot_bot_set_nat) = one_one_rat ).

% prod.empty
tff(fact_8410_prod_Oempty,axiom,
    ! [G: fun_nat_int] : groups705719431365010083at_int(G,bot_bot_set_nat) = one_one_int ).

% prod.empty
tff(fact_8411_prod_Oempty,axiom,
    ! [G: fun_int_complex] : groups7440179247065528705omplex(G,bot_bot_set_int) = one_one_complex ).

% prod.empty
tff(fact_8412_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_complex] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups6464643781859351333omplex(G,A2) = one_one_complex ) ) ).

% prod.infinite
tff(fact_8413_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups7440179247065528705omplex(G,A2) = one_one_complex ) ) ).

% prod.infinite
tff(fact_8414_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups3708469109370488835omplex(G,A2) = one_one_complex ) ) ).

% prod.infinite
tff(fact_8415_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_real] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups129246275422532515t_real(G,A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_8416_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups2316167850115554303t_real(G,A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_8417_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups766887009212190081x_real(G,A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_8418_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_rat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups73079841787564623at_rat(G,A2) = one_one_rat ) ) ).

% prod.infinite
tff(fact_8419_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups1072433553688619179nt_rat(G,A2) = one_one_rat ) ) ).

% prod.infinite
tff(fact_8420_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups225925009352817453ex_rat(G,A2) = one_one_rat ) ) ).

% prod.infinite
tff(fact_8421_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups1707563613775114915nt_nat(G,A2) = one_one_nat ) ) ).

% prod.infinite
tff(fact_8422_dvd__prod__eqI,axiom,
    ! [A2: set_real,A: real,B: nat,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => ( ( B = aa_real_nat(F,A) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),groups4696554848551431203al_nat(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_8423_dvd__prod__eqI,axiom,
    ! [A2: set_int,A: int,B: nat,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => ( ( B = aa_int_nat(F,A) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),groups1707563613775114915nt_nat(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_8424_dvd__prod__eqI,axiom,
    ! [A2: set_complex,A: complex,B: nat,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(A),A2))
       => ( ( B = aa_complex_nat(F,A) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),groups861055069439313189ex_nat(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_8425_dvd__prod__eqI,axiom,
    ! [A2: set_real,A: real,B: int,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => ( ( B = aa_real_int(F,A) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),groups4694064378042380927al_int(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_8426_dvd__prod__eqI,axiom,
    ! [A2: set_nat,A: nat,B: int,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => ( ( B = aa_nat_int(F,A) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),groups705719431365010083at_int(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_8427_dvd__prod__eqI,axiom,
    ! [A2: set_int,A: int,B: int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => ( ( B = aa_int_int(F,A) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),groups1705073143266064639nt_int(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_8428_dvd__prod__eqI,axiom,
    ! [A2: set_complex,A: complex,B: int,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(A),A2))
       => ( ( B = aa_complex_int(F,A) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),groups858564598930262913ex_int(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_8429_dvd__prod__eqI,axiom,
    ! [A2: set_real,A: real,B: code_integer,F: fun_re6978948838062957830nteger] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => ( ( B = aa_real_Code_integer(F,A) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),groups6225526099057966256nteger(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_8430_dvd__prod__eqI,axiom,
    ! [A2: set_nat,A: nat,B: code_integer,F: fun_nat_Code_integer] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => ( ( B = aa_nat_Code_integer(F,A) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),groups3455450783089532116nteger(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_8431_dvd__prod__eqI,axiom,
    ! [A2: set_int,A: int,B: code_integer,F: fun_int_Code_integer] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => ( ( B = aa_int_Code_integer(F,A) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),groups3827104343326376752nteger(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_8432_dvd__prodI,axiom,
    ! [A2: set_real,A: real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_real_nat(F,A)),groups4696554848551431203al_nat(F,A2))) ) ) ).

% dvd_prodI
tff(fact_8433_dvd__prodI,axiom,
    ! [A2: set_int,A: int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(F,A)),groups1707563613775114915nt_nat(F,A2))) ) ) ).

% dvd_prodI
tff(fact_8434_dvd__prodI,axiom,
    ! [A2: set_complex,A: complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(A),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_complex_nat(F,A)),groups861055069439313189ex_nat(F,A2))) ) ) ).

% dvd_prodI
tff(fact_8435_dvd__prodI,axiom,
    ! [A2: set_real,A: real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_real_int(F,A)),groups4694064378042380927al_int(F,A2))) ) ) ).

% dvd_prodI
tff(fact_8436_dvd__prodI,axiom,
    ! [A2: set_nat,A: nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(F,A)),groups705719431365010083at_int(F,A2))) ) ) ).

% dvd_prodI
tff(fact_8437_dvd__prodI,axiom,
    ! [A2: set_int,A: int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(F,A)),groups1705073143266064639nt_int(F,A2))) ) ) ).

% dvd_prodI
tff(fact_8438_dvd__prodI,axiom,
    ! [A2: set_complex,A: complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(A),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_complex_int(F,A)),groups858564598930262913ex_int(F,A2))) ) ) ).

% dvd_prodI
tff(fact_8439_dvd__prodI,axiom,
    ! [A2: set_real,A: real,F: fun_re6978948838062957830nteger] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_real_Code_integer(F,A)),groups6225526099057966256nteger(F,A2))) ) ) ).

% dvd_prodI
tff(fact_8440_dvd__prodI,axiom,
    ! [A2: set_nat,A: nat,F: fun_nat_Code_integer] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(F,A)),groups3455450783089532116nteger(F,A2))) ) ) ).

% dvd_prodI
tff(fact_8441_dvd__prodI,axiom,
    ! [A2: set_int,A: int,F: fun_int_Code_integer] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_int_Code_integer(F,A)),groups3827104343326376752nteger(F,A2))) ) ) ).

% dvd_prodI
tff(fact_8442_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_8443_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_8444_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_8445_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(aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,I),M),aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,I),N)) = aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,N),M) ) ) ).

% ivl_diff
tff(fact_8446_prod_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_vz(A),B),S2) = aa_real_complex(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_vz(A),B),S2) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_8447_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_wa(A),B),S2) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_wa(A),B),S2) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_8448_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_wb(A),B),S2) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_wb(A),B),S2) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_8449_prod_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_wc(A),B),S2) = aa_complex_complex(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_wc(A),B),S2) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_8450_prod_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_wd(A),B),S2) = aa_real_real(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_wd(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_8451_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_we(A),B),S2) = aa_nat_real(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_we(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_8452_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_wf(A),B),S2) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_wf(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_8453_prod_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_wg(A),B),S2) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_wg(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_8454_prod_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_wh(A),B),S2) = aa_real_rat(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_wh(A),B),S2) = one_one_rat ) ) ) ) ).

% prod.delta'
tff(fact_8455_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_wi(A),B),S2) = aa_nat_rat(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_wi(A),B),S2) = one_one_rat ) ) ) ) ).

% prod.delta'
tff(fact_8456_prod_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_wj(A),B),S2) = aa_real_complex(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_wj(A),B),S2) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_8457_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_wk(A),B),S2) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_wk(A),B),S2) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_8458_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_wl(A),B),S2) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_wl(A),B),S2) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_8459_prod_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_wm(A),B),S2) = aa_complex_complex(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_wm(A),B),S2) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_8460_prod_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_wn(A),B),S2) = aa_real_real(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_wn(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_8461_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_wo(A),B),S2) = aa_nat_real(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_wo(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_8462_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_wp(A),B),S2) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_wp(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_8463_prod_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_wq(A),B),S2) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_wq(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_8464_prod_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_wr(A),B),S2) = aa_real_rat(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_wr(A),B),S2) = one_one_rat ) ) ) ) ).

% prod.delta
tff(fact_8465_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ws(A),B),S2) = aa_nat_rat(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ws(A),B),S2) = one_one_rat ) ) ) ) ).

% prod.delta
tff(fact_8466_is__singletonI,axiom,
    ! [X: produc3843707927480180839at_nat] : pp(aa_set4148026973699692364t_bool(is_sin2937591304547752795at_nat,insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))) ).

% is_singletonI
tff(fact_8467_is__singletonI,axiom,
    ! [X: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(is_sin2850979758926227957at_nat,insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))) ).

% is_singletonI
tff(fact_8468_is__singletonI,axiom,
    ! [X: real] : pp(aa_set_real_bool(is_singleton_real,insert_real(X,bot_bot_set_real))) ).

% is_singletonI
tff(fact_8469_is__singletonI,axiom,
    ! [X: nat] : pp(aa_set_nat_bool(is_singleton_nat,insert_nat(X,bot_bot_set_nat))) ).

% is_singletonI
tff(fact_8470_is__singletonI,axiom,
    ! [X: int] : pp(aa_set_int_bool(is_singleton_int,insert_int(X,bot_bot_set_int))) ).

% is_singletonI
tff(fact_8471_prod_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X),A2))
       => ( groups713298508707869441omplex(G,insert_real(X,A2)) = aa_complex_complex(times_times_complex(aa_real_complex(G,X)),groups713298508707869441omplex(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8472_prod_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
       => ( groups6464643781859351333omplex(G,insert_nat(X,A2)) = aa_complex_complex(times_times_complex(aa_nat_complex(G,X)),groups6464643781859351333omplex(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8473_prod_Oinsert,axiom,
    ! [A2: set_int,X: int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ~ pp(aa_set_int_bool(member_int(X),A2))
       => ( groups7440179247065528705omplex(G,insert_int(X,A2)) = aa_complex_complex(times_times_complex(aa_int_complex(G,X)),groups7440179247065528705omplex(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8474_prod_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups3708469109370488835omplex(G,insert_complex(X,A2)) = aa_complex_complex(times_times_complex(aa_complex_complex(G,X)),groups3708469109370488835omplex(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8475_prod_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X),A2))
       => ( groups1681761925125756287l_real(G,insert_real(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,X)),groups1681761925125756287l_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8476_prod_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
       => ( groups129246275422532515t_real(G,insert_nat(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,X)),groups129246275422532515t_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8477_prod_Oinsert,axiom,
    ! [A2: set_int,X: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ~ pp(aa_set_int_bool(member_int(X),A2))
       => ( groups2316167850115554303t_real(G,insert_int(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,X)),groups2316167850115554303t_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8478_prod_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
       => ( groups766887009212190081x_real(G,insert_complex(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,X)),groups766887009212190081x_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8479_prod_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X),A2))
       => ( groups4061424788464935467al_rat(G,insert_real(X,A2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(G,X)),groups4061424788464935467al_rat(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8480_prod_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
       => ( groups73079841787564623at_rat(G,insert_nat(X,A2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,X)),groups73079841787564623at_rat(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8481_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_8482_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(plus_plus_complex(groups2073611262835488442omplex(G,set_or4665077453230672383an_nat(M,N))),aa_nat_complex(G,N)) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_8483_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(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,set_or4665077453230672383an_nat(M,N))),aa_nat_rat(G,N)) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_8484_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_8485_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_8486_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_8487_prod_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))
       => ( groups6464643781859351333omplex(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = one_one_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups6464643781859351333omplex(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,set_or1269000886237332187st_nat(M,N))),aa_nat_complex(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_8488_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_8489_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(aa_rat_fun_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_8490_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_8491_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))
       => ( aa_set_nat_nat(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))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(M,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_8492_prod_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))
       => ( groups6464643781859351333omplex(G,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = one_one_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups6464643781859351333omplex(G,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,set_or4665077453230672383an_nat(M,N))),aa_nat_complex(G,N)) ) ) ) ).

% prod.op_ivl_Suc
tff(fact_8493_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_8494_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(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,set_or4665077453230672383an_nat(M,N))),aa_nat_rat(G,N)) ) ) ) ).

% prod.op_ivl_Suc
tff(fact_8495_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_8496_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))
       => ( aa_set_nat_nat(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))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G),set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),set_or4665077453230672383an_nat(M,N))),aa_nat_nat(G,N)) ) ) ) ).

% prod.op_ivl_Suc
tff(fact_8497_sgn__rat__def,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( aa_rat_rat(sgn_sgn_rat,A) = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
           => ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat ) )
          & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
           => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ) ) ).

% sgn_rat_def
tff(fact_8498_abs__rat__def,axiom,
    ! [A: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ) ).

% abs_rat_def
tff(fact_8499_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P2: nat,G: fun_nat_complex] :
      ( 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_complex_complex(times_times_complex(groups6464643781859351333omplex(G,set_or4665077453230672383an_nat(M,N))),groups6464643781859351333omplex(G,set_or4665077453230672383an_nat(N,P2))) = groups6464643781859351333omplex(G,set_or4665077453230672383an_nat(M,P2)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_8500_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_8501_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(aa_rat_fun_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_8502_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_8503_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,aa_set_nat_nat(groups708209901874060359at_nat(G),set_or4665077453230672383an_nat(M,N))),aa_set_nat_nat(groups708209901874060359at_nat(G),set_or4665077453230672383an_nat(N,P2))) = aa_set_nat_nat(groups708209901874060359at_nat(G),set_or4665077453230672383an_nat(M,P2)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_8504_less__eq__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
        | ( X = Y ) ) ) ).

% less_eq_rat_def
tff(fact_8505_prod_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( ( A = C )
     => ( ( B = D )
       => ( ! [X4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),X4))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),D))
               => ( aa_nat_nat(G,X4) = aa_nat_nat(H,X4) ) ) )
         => ( aa_set_nat_nat(groups708209901874060359at_nat(G),set_or4665077453230672383an_nat(A,B)) = aa_set_nat_nat(groups708209901874060359at_nat(H),set_or4665077453230672383an_nat(C,D)) ) ) ) ) ).

% prod.ivl_cong
tff(fact_8506_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R2))
     => ~ ! [S4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),S4))
           => ! [T6: rat] :
                ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),T6))
               => ( R2 != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,S4),T6) ) ) ) ) ).

% obtain_pos_sum
tff(fact_8507_atLeastLessThan__inj_I2_J,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( set_or66887138388493659n_real(A,B) = set_or66887138388493659n_real(C,D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_8508_atLeastLessThan__inj_I2_J,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(C,D) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_8509_atLeastLessThan__inj_I2_J,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(C,D) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
       => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),D))
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_8510_atLeastLessThan__inj_I2_J,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( set_or4665077453230672383an_nat(A,B) = set_or4665077453230672383an_nat(C,D) )
     => ( 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,C),D))
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_8511_atLeastLessThan__inj_I2_J,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,A),B) = aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,C),D) )
     => ( 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,C),D))
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_8512_atLeastLessThan__inj_I1_J,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( set_or66887138388493659n_real(A,B) = set_or66887138388493659n_real(C,D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_8513_atLeastLessThan__inj_I1_J,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(C,D) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_8514_atLeastLessThan__inj_I1_J,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(C,D) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
       => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),D))
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_8515_atLeastLessThan__inj_I1_J,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( set_or4665077453230672383an_nat(A,B) = set_or4665077453230672383an_nat(C,D) )
     => ( 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,C),D))
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_8516_atLeastLessThan__inj_I1_J,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,A),B) = aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,C),D) )
     => ( 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,C),D))
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_8517_atLeastLessThan__eq__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( ( set_or66887138388493659n_real(A,B) = set_or66887138388493659n_real(C,D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_8518_atLeastLessThan__eq__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(C,D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_8519_atLeastLessThan__eq__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),D))
       => ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(C,D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_8520_atLeastLessThan__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: 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,C),D))
       => ( ( set_or4665077453230672383an_nat(A,B) = set_or4665077453230672383an_nat(C,D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_8521_atLeastLessThan__eq__iff,axiom,
    ! [A: int,B: int,C: int,D: 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,C),D))
       => ( ( aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,A),B) = aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,C),D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_8522_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_complex,N: nat] : groups6464643781859351333omplex(G,set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,set_or4665077453230672383an_nat(zero_zero_nat,N))),aa_nat_complex(G,N)) ).

% prod.atLeast0_lessThan_Suc
tff(fact_8523_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_8524_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(aa_rat_fun_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_8525_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_8526_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_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,aa_set_nat_nat(groups708209901874060359at_nat(G),set_or4665077453230672383an_nat(zero_zero_nat,N))),aa_nat_nat(G,N)) ).

% prod.atLeast0_lessThan_Suc
tff(fact_8527_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( groups6464643781859351333omplex(G,set_or4665077453230672383an_nat(M,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(G,M)),groups6464643781859351333omplex(G,set_or4665077453230672383an_nat(aa_nat_nat(suc,M),N))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_8528_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_8529_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(aa_rat_fun_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_8530_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_8531_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))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),set_or4665077453230672383an_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups708209901874060359at_nat(G),set_or4665077453230672383an_nat(aa_nat_nat(suc,M),N))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_8532_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( groups6464643781859351333omplex(G,set_or4665077453230672383an_nat(A,aa_nat_nat(suc,B))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,set_or4665077453230672383an_nat(A,B))),aa_nat_complex(G,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_8533_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( groups129246275422532515t_real(G,set_or4665077453230672383an_nat(A,aa_nat_nat(suc,B))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,set_or4665077453230672383an_nat(A,B))),aa_nat_real(G,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_8534_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( groups73079841787564623at_rat(G,set_or4665077453230672383an_nat(A,aa_nat_nat(suc,B))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,set_or4665077453230672383an_nat(A,B))),aa_nat_rat(G,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_8535_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( groups705719431365010083at_int(G,set_or4665077453230672383an_nat(A,aa_nat_nat(suc,B))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or4665077453230672383an_nat(A,B))),aa_nat_int(G,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_8536_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),set_or4665077453230672383an_nat(A,aa_nat_nat(suc,B))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),set_or4665077453230672383an_nat(A,B))),aa_nat_nat(G,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_8537_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_nat,G: fun_real_fun_nat_nat,R: fun_re9121486324101951821t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups4696554848551431203al_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_wt(B2),G),R),A2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_wu(A2),G),R)),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_8538_prod_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_nat,G: fun_int_fun_nat_nat,R: fun_int_fun_nat_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups1707563613775114915nt_nat(aa_fun5132377469993263754nt_nat(aa_fun348247319442030677nt_nat(aTP_Lamm_wv(B2),G),R),A2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun4133023758092209198at_nat(aa_fun7384697781603784697at_nat(aTP_Lamm_ww(A2),G),R)),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_8539_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B2: set_nat,G: fun_co636942507509095880at_nat,R: fun_co2841405903858895691t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups861055069439313189ex_nat(aa_fun9050545535516344074ex_nat(aa_fun2757644259618170323ex_nat(aTP_Lamm_wx(B2),G),R),A2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun996456248398975532at_nat(aa_fun8754082201773474805at_nat(aTP_Lamm_wy(A2),G),R)),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_8540_prod_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_real,G: fun_nat_fun_real_nat,R: fun_na6801867027684862029l_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_real_bool(finite_finite_real,B2))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun8064793419647352750at_nat(aa_fun581391774837177593at_nat(aTP_Lamm_wz(B2),G),R)),A2) = groups4696554848551431203al_nat(aa_fun389422839623581578al_nat(aa_fun9122955408058973909al_nat(aTP_Lamm_xa(A2),G),R),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_8541_prod_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_int,G: fun_nat_fun_int_nat,R: fun_nat_fun_int_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun7682780481535005230at_nat(aa_fun503657754853467129at_nat(aTP_Lamm_xc(B2),G),R)),A2) = groups1707563613775114915nt_nat(aa_fun8682134193436059786nt_nat(aa_fun2690579329546488917nt_nat(aTP_Lamm_xe(A2),G),R),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_8542_prod_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_complex,G: fun_na8457291176974909128ex_nat,R: fun_na5941084329111710283x_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun7456585239162712364at_nat(aa_fun5124956507111473653at_nat(aTP_Lamm_xf(B2),G),R)),A2) = groups861055069439313189ex_nat(aa_fun8860606018881707530ex_nat(aa_fun2725303033966715859ex_nat(aTP_Lamm_xh(A2),G),R),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_8543_prod_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_fun_nat_nat,R: fun_nat_fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun4644938397070871434at_nat(aa_fun7390876557192650417at_nat(aTP_Lamm_xj(B2),G),R)),A2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun4644938397070871434at_nat(aa_fun7390876557192650417at_nat(aTP_Lamm_xl(A2),G),R)),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_8544_prod_Oswap__restrict,axiom,
    ! [A2: set_list_nat,B2: set_nat,G: fun_li2771310203728655226at_nat,R: fun_li4440390755235668505t_bool] :
      ( finite8100373058378681591st_nat(A2)
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups2907647131375434839at_nat(aa_fun8445324095761362826at_nat(aa_fun504646108086548129at_nat(aTP_Lamm_xn(B2),G),R),A2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun1837141565787665786at_nat(aa_fun6292763642686614161at_nat(aTP_Lamm_xq(A2),G),R)),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_8545_prod_Oswap__restrict,axiom,
    ! [A2: set_set_nat,B2: set_nat,G: fun_se1179353343636554400at_nat,R: fun_se680526843285281907t_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups4248547760180025341at_nat(aa_fun7954195629112176906at_nat(aa_fun2162494588655638267at_nat(aTP_Lamm_xs(B2),G),R),A2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5592962576178777428at_nat(aa_fun3299576543290263109at_nat(aTP_Lamm_xv(A2),G),R)),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_8546_prod_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_list_nat,G: fun_na7671319193681171066at_nat,R: fun_na5614638261385347865t_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( finite8100373058378681591st_nat(B2)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5703926054570661498at_nat(aa_fun1685042437804453009at_nat(aTP_Lamm_xx(B2),G),R)),A2) = groups2907647131375434839at_nat(aa_fun4481408125089960074at_nat(aa_fun2629866330252507297at_nat(aTP_Lamm_ya(A2),G),R),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_8547_prod_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups6464643781859351333omplex(G,set_or1269000886237332187st_nat(M,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(G,N)),groups6464643781859351333omplex(G,set_or4665077453230672383an_nat(M,N))) ) ) ).

% prod.last_plus
tff(fact_8548_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_8549_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(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,N)),groups73079841787564623at_rat(G,set_or4665077453230672383an_nat(M,N))) ) ) ).

% prod.last_plus
tff(fact_8550_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_8551_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))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,N)),aa_set_nat_nat(groups708209901874060359at_nat(G),set_or4665077453230672383an_nat(M,N))) ) ) ).

% prod.last_plus
tff(fact_8552_prod__Suc__Suc__fact,axiom,
    ! [N: nat] : aa_set_nat_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_8553_prod__Suc__fact,axiom,
    ! [N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(zero_zero_nat,N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_fact
tff(fact_8554_prod_Onested__swap,axiom,
    ! [A: fun_nat_fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_yb(A)),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_yc(A),N)),set_or4665077453230672383an_nat(zero_zero_nat,N)) ).

% prod.nested_swap
tff(fact_8555_prod_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))
       => ( groups6464643781859351333omplex(G,set_or1269000886237332187st_nat(M,N)) = one_one_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups6464643781859351333omplex(G,set_or1269000886237332187st_nat(M,N)) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,set_or4665077453230672383an_nat(M,N))),aa_nat_complex(G,N)) ) ) ) ).

% prod.head_if
tff(fact_8556_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_8557_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(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,set_or4665077453230672383an_nat(M,N))),aa_nat_rat(G,N)) ) ) ) ).

% prod.head_if
tff(fact_8558_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_8559_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))
       => ( aa_set_nat_nat(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))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),set_or4665077453230672383an_nat(M,N))),aa_nat_nat(G,N)) ) ) ) ).

% prod.head_if
tff(fact_8560_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A2))
         => ( 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,A2)),groups1681761925125756287l_real(G,A2))) ) ).

% prod_mono
tff(fact_8561_prod__mono,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => ( 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,A2)),groups129246275422532515t_real(G,A2))) ) ).

% prod_mono
tff(fact_8562_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),A2))
         => ( 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,A2)),groups2316167850115554303t_real(G,A2))) ) ).

% prod_mono
tff(fact_8563_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A2))
         => ( 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,A2)),groups4061424788464935467al_rat(G,A2))) ) ).

% prod_mono
tff(fact_8564_prod__mono,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => ( 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,A2)),groups73079841787564623at_rat(G,A2))) ) ).

% prod_mono
tff(fact_8565_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),A2))
         => ( 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,A2)),groups1072433553688619179nt_rat(G,A2))) ) ).

% prod_mono
tff(fact_8566_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A2))
         => ( 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,A2)),groups4696554848551431203al_nat(G,A2))) ) ).

% prod_mono
tff(fact_8567_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2)))
            & 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,groups1707563613775114915nt_nat(F,A2)),groups1707563613775114915nt_nat(G,A2))) ) ).

% prod_mono
tff(fact_8568_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,I2)))
            & 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,groups4694064378042380927al_int(F,A2)),groups4694064378042380927al_int(G,A2))) ) ).

% prod_mono
tff(fact_8569_prod__mono,axiom,
    ! [A2: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,I2)))
            & 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,groups705719431365010083at_int(F,A2)),groups705719431365010083at_int(G,A2))) ) ).

% prod_mono
tff(fact_8570_prod__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => 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),aa_set_nat_nat(groups708209901874060359at_nat(F),A2))) ) ).

% prod_nonneg
tff(fact_8571_prod__pos,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => 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),aa_set_nat_nat(groups708209901874060359at_nat(F),A2))) ) ).

% prod_pos
tff(fact_8572_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => 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,A2))) ) ).

% prod_ge_1
tff(fact_8573_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => 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,A2))) ) ).

% prod_ge_1
tff(fact_8574_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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,A2))) ) ).

% prod_ge_1
tff(fact_8575_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => 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,A2))) ) ).

% prod_ge_1
tff(fact_8576_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => 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,A2))) ) ).

% prod_ge_1
tff(fact_8577_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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,A2))) ) ).

% prod_ge_1
tff(fact_8578_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => 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,A2))) ) ).

% prod_ge_1
tff(fact_8579_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_int_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),groups1707563613775114915nt_nat(F,A2))) ) ).

% prod_ge_1
tff(fact_8580_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_int] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),groups4694064378042380927al_int(F,A2))) ) ).

% prod_ge_1
tff(fact_8581_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),groups705719431365010083at_int(F,A2))) ) ).

% prod_ge_1
tff(fact_8582_fact__prod__Suc,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(zero_zero_nat,N))) ).

% fact_prod_Suc
tff(fact_8583_fact__prod__Suc,axiom,
    ! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,N) = aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(zero_zero_nat,N))) ).

% fact_prod_Suc
tff(fact_8584_fact__prod__Suc,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(zero_zero_nat,N))) ).

% fact_prod_Suc
tff(fact_8585_prod__zero,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ? [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),A2))
            & ( aa_nat_complex(F,X2) = zero_zero_complex ) )
       => ( groups6464643781859351333omplex(F,A2) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_8586_prod__zero,axiom,
    ! [A2: set_int,F: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ? [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
            & ( aa_int_complex(F,X2) = zero_zero_complex ) )
       => ( groups7440179247065528705omplex(F,A2) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_8587_prod__zero,axiom,
    ! [A2: set_complex,F: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ? [X2: complex] :
            ( pp(aa_set_complex_bool(member_complex(X2),A2))
            & ( aa_complex_complex(F,X2) = zero_zero_complex ) )
       => ( groups3708469109370488835omplex(F,A2) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_8588_prod__zero,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ? [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),A2))
            & ( aa_nat_real(F,X2) = zero_zero_real ) )
       => ( groups129246275422532515t_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_8589_prod__zero,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ? [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
            & ( aa_int_real(F,X2) = zero_zero_real ) )
       => ( groups2316167850115554303t_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_8590_prod__zero,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ? [X2: complex] :
            ( pp(aa_set_complex_bool(member_complex(X2),A2))
            & ( aa_complex_real(F,X2) = zero_zero_real ) )
       => ( groups766887009212190081x_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_8591_prod__zero,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ? [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),A2))
            & ( aa_nat_rat(F,X2) = zero_zero_rat ) )
       => ( groups73079841787564623at_rat(F,A2) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_8592_prod__zero,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ? [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
            & ( aa_int_rat(F,X2) = zero_zero_rat ) )
       => ( groups1072433553688619179nt_rat(F,A2) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_8593_prod__zero,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ? [X2: complex] :
            ( pp(aa_set_complex_bool(member_complex(X2),A2))
            & ( aa_complex_rat(F,X2) = zero_zero_rat ) )
       => ( groups225925009352817453ex_rat(F,A2) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_8594_prod__zero,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ? [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
            & ( aa_int_nat(F,X2) = zero_zero_nat ) )
       => ( groups1707563613775114915nt_nat(F,A2) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_8595_atLeastLessThan__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or4029947393144176647an_rat(A,B)),set_or4029947393144176647an_rat(C,D)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),D)) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_8596_atLeastLessThan__subset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_or1222409239386451017an_num(A,B)),set_or1222409239386451017an_num(C,D)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        | ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
          & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),D)) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_8597_atLeastLessThan__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_or4665077453230672383an_nat(A,B)),set_or4665077453230672383an_nat(C,D)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),D)) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_8598_atLeastLessThan__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,A),B)),aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,C),D)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),D)) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_8599_infinite__Ico,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ~ pp(aa_set_real_bool(finite_finite_real,set_or66887138388493659n_real(A,B))) ) ).

% infinite_Ico
tff(fact_8600_infinite__Ico,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ~ pp(aa_set_rat_bool(finite_finite_rat,set_or4029947393144176647an_rat(A,B))) ) ).

% infinite_Ico
tff(fact_8601_all__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
         => pp(aa_nat_bool(P,M3)) )
    <=> ! [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_8602_ex__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
          & pp(aa_nat_bool(P,M3)) )
    <=> ? [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_8603_fact__prod__rev,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(minus_minus_nat,N)),set_or4665077453230672383an_nat(zero_zero_nat,N))) ).

% fact_prod_rev
tff(fact_8604_fact__prod__rev,axiom,
    ! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,N) = aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(minus_minus_nat,N)),set_or4665077453230672383an_nat(zero_zero_nat,N))) ).

% fact_prod_rev
tff(fact_8605_fact__prod__rev,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(minus_minus_nat,N)),set_or4665077453230672383an_nat(zero_zero_nat,N))) ).

% fact_prod_rev
tff(fact_8606_pochhammer__prod,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = groups73079841787564623at_rat(aTP_Lamm_yd(A),set_or4665077453230672383an_nat(zero_zero_nat,N)) ).

% pochhammer_prod
tff(fact_8607_pochhammer__prod,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),N) = groups705719431365010083at_int(aTP_Lamm_ye(A),set_or4665077453230672383an_nat(zero_zero_nat,N)) ).

% pochhammer_prod
tff(fact_8608_pochhammer__prod,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_yf,A),set_or4665077453230672383an_nat(zero_zero_nat,N)) ).

% pochhammer_prod
tff(fact_8609_pochhammer__prod,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_yg,A)),set_or4665077453230672383an_nat(zero_zero_nat,N)) ).

% pochhammer_prod
tff(fact_8610_lessThan__atLeast0,axiom,
    ! [N: nat] : set_ord_lessThan_nat(N) = set_or4665077453230672383an_nat(zero_zero_nat,N) ).

% lessThan_atLeast0
tff(fact_8611_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_complex,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups713298508707869441omplex(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bm(A2),P))) = groups713298508707869441omplex(aa_fun6813637976755515871omplex(aTP_Lamm_yh(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8612_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_complex,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups6464643781859351333omplex(G,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bp(A2),P))) = groups6464643781859351333omplex(aa_fun8588439219086927967omplex(aTP_Lamm_yi(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8613_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_complex,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups7440179247065528705omplex(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bq(A2),P))) = groups7440179247065528705omplex(aa_fun3606824691168040799omplex(aTP_Lamm_yj(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8614_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_complex,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups3708469109370488835omplex(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_kp(A2),P))) = groups3708469109370488835omplex(aa_fun1034237174562290143omplex(aTP_Lamm_yk(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8615_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1681761925125756287l_real(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bm(A2),P))) = groups1681761925125756287l_real(aa_fun6348708079608492509l_real(aTP_Lamm_yl(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8616_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_real,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups129246275422532515t_real(G,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bp(A2),P))) = groups129246275422532515t_real(aa_fun1750223370215434333t_real(aTP_Lamm_ym(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8617_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_real,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups2316167850115554303t_real(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bq(A2),P))) = groups2316167850115554303t_real(aa_fun6974987029372589917t_real(aTP_Lamm_yn(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8618_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_real,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups766887009212190081x_real(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_kp(A2),P))) = groups766887009212190081x_real(aa_fun2902006202505214429x_real(aTP_Lamm_yo(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8619_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_rat,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups4061424788464935467al_rat(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bm(A2),P))) = groups4061424788464935467al_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_yp(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8620_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_rat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups73079841787564623at_rat(G,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bp(A2),P))) = groups73079841787564623at_rat(aa_fun5656162108953708681at_rat(aTP_Lamm_yq(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8621_atLeastLessThan0,axiom,
    ! [M: nat] : set_or4665077453230672383an_nat(M,zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_8622_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => ( 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,A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_8623_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( 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,A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_8624_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => ( 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,A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_8625_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => ( 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,A2)),one_one_rat)) ) ).

% prod_le_1
tff(fact_8626_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( 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,A2)),one_one_rat)) ) ).

% prod_le_1
tff(fact_8627_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => ( 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,A2)),one_one_rat)) ) ).

% prod_le_1
tff(fact_8628_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => ( 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,A2)),one_one_nat)) ) ).

% prod_le_1
tff(fact_8629_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => ( 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,aa_int_nat(F,X4)),one_one_nat)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1707563613775114915nt_nat(F,A2)),one_one_nat)) ) ).

% prod_le_1
tff(fact_8630_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_int] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,X4)))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,X4)),one_one_int)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups4694064378042380927al_int(F,A2)),one_one_int)) ) ).

% prod_le_1
tff(fact_8631_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( 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,aa_nat_int(F,X4)),one_one_int)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups705719431365010083at_int(F,A2)),one_one_int)) ) ).

% prod_le_1
tff(fact_8632_is__singletonI_H,axiom,
    ! [A2: set_set_nat] :
      ( ( A2 != bot_bot_set_set_nat )
     => ( ! [X4: set_nat,Y2: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
           => ( pp(aa_set_set_nat_bool(member_set_nat(Y2),A2))
             => ( X4 = Y2 ) ) )
       => pp(aa_set_set_nat_bool(is_singleton_set_nat,A2)) ) ) ).

% is_singletonI'
tff(fact_8633_is__singletonI_H,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( A2 != bot_bo2099793752762293965at_nat )
     => ( ! [X4: product_prod_nat_nat,Y2: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),A2))
           => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y2),A2))
             => ( X4 = Y2 ) ) )
       => pp(aa_set3135850010578026034t_bool(is_sin2850979758926227957at_nat,A2)) ) ) ).

% is_singletonI'
tff(fact_8634_is__singletonI_H,axiom,
    ! [A2: set_real] :
      ( ( A2 != bot_bot_set_real )
     => ( ! [X4: real,Y2: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
           => ( pp(aa_set_real_bool(member_real(Y2),A2))
             => ( X4 = Y2 ) ) )
       => pp(aa_set_real_bool(is_singleton_real,A2)) ) ) ).

% is_singletonI'
tff(fact_8635_is__singletonI_H,axiom,
    ! [A2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ! [X4: nat,Y2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
           => ( pp(aa_set_nat_bool(member_nat(Y2),A2))
             => ( X4 = Y2 ) ) )
       => pp(aa_set_nat_bool(is_singleton_nat,A2)) ) ) ).

% is_singletonI'
tff(fact_8636_is__singletonI_H,axiom,
    ! [A2: set_int] :
      ( ( A2 != bot_bot_set_int )
     => ( ! [X4: int,Y2: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => ( pp(aa_set_int_bool(member_int(Y2),A2))
             => ( X4 = Y2 ) ) )
       => pp(aa_set_int_bool(is_singleton_int,A2)) ) ) ).

% is_singletonI'
tff(fact_8637_prod_Orelated,axiom,
    ! [R: fun_co1378794544173401837x_bool,S2: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,one_one_complex),one_one_complex))
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X15),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y15),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(times_times_complex(X15),Y15)),aa_complex_complex(times_times_complex(X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S2))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => 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,groups6464643781859351333omplex(H,S2)),groups6464643781859351333omplex(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_8638_prod_Orelated,axiom,
    ! [R: fun_co1378794544173401837x_bool,S2: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,one_one_complex),one_one_complex))
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X15),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y15),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(times_times_complex(X15),Y15)),aa_complex_complex(times_times_complex(X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => 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,groups7440179247065528705omplex(H,S2)),groups7440179247065528705omplex(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_8639_prod_Orelated,axiom,
    ! [R: fun_co1378794544173401837x_bool,S2: set_complex,H: fun_complex_complex,G: fun_complex_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,one_one_complex),one_one_complex))
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X15),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y15),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(times_times_complex(X15),Y15)),aa_complex_complex(times_times_complex(X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(H,X4)),aa_complex_complex(G,X4))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,groups3708469109370488835omplex(H,S2)),groups3708469109370488835omplex(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_8640_prod_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S2: 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,Y15: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X15),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y15),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S2))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => 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,S2)),groups129246275422532515t_real(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_8641_prod_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S2: 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,Y15: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X15),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y15),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => 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,S2)),groups2316167850115554303t_real(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_8642_prod_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S2: 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,Y15: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X15),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y15),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => 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,S2)),groups766887009212190081x_real(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_8643_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S2: 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,Y15: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X15),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y15),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X15),Y15)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S2))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => 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,S2)),groups73079841787564623at_rat(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_8644_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S2: 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,Y15: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X15),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y15),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X15),Y15)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => 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,S2)),groups1072433553688619179nt_rat(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_8645_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,one_one_rat),one_one_rat))
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X15),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y15),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X15),Y15)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => 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,groups225925009352817453ex_rat(H,S2)),groups225925009352817453ex_rat(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_8646_prod_Orelated,axiom,
    ! [R: fun_nat_fun_nat_bool,S2: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,one_one_nat),one_one_nat))
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X15),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y15),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_int_nat(H,X4)),aa_int_nat(G,X4))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,groups1707563613775114915nt_nat(H,S2)),groups1707563613775114915nt_nat(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_8647_prod_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(X),A2))
         => ( groups713298508707869441omplex(G,insert_real(X,A2)) = groups713298508707869441omplex(G,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A2))
         => ( groups713298508707869441omplex(G,insert_real(X,A2)) = aa_complex_complex(times_times_complex(aa_real_complex(G,X)),groups713298508707869441omplex(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_8648_prod_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( pp(aa_set_nat_bool(member_nat(X),A2))
         => ( groups6464643781859351333omplex(G,insert_nat(X,A2)) = groups6464643781859351333omplex(G,A2) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
         => ( groups6464643781859351333omplex(G,insert_nat(X,A2)) = aa_complex_complex(times_times_complex(aa_nat_complex(G,X)),groups6464643781859351333omplex(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_8649_prod_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( pp(aa_set_int_bool(member_int(X),A2))
         => ( groups7440179247065528705omplex(G,insert_int(X,A2)) = groups7440179247065528705omplex(G,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A2))
         => ( groups7440179247065528705omplex(G,insert_int(X,A2)) = aa_complex_complex(times_times_complex(aa_int_complex(G,X)),groups7440179247065528705omplex(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_8650_prod_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups3708469109370488835omplex(G,insert_complex(X,A2)) = groups3708469109370488835omplex(G,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups3708469109370488835omplex(G,insert_complex(X,A2)) = aa_complex_complex(times_times_complex(aa_complex_complex(G,X)),groups3708469109370488835omplex(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_8651_prod_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(X),A2))
         => ( groups1681761925125756287l_real(G,insert_real(X,A2)) = groups1681761925125756287l_real(G,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A2))
         => ( groups1681761925125756287l_real(G,insert_real(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,X)),groups1681761925125756287l_real(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_8652_prod_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( pp(aa_set_nat_bool(member_nat(X),A2))
         => ( groups129246275422532515t_real(G,insert_nat(X,A2)) = groups129246275422532515t_real(G,A2) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
         => ( groups129246275422532515t_real(G,insert_nat(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,X)),groups129246275422532515t_real(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_8653_prod_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( pp(aa_set_int_bool(member_int(X),A2))
         => ( groups2316167850115554303t_real(G,insert_int(X,A2)) = groups2316167850115554303t_real(G,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A2))
         => ( groups2316167850115554303t_real(G,insert_int(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,X)),groups2316167850115554303t_real(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_8654_prod_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups766887009212190081x_real(G,insert_complex(X,A2)) = groups766887009212190081x_real(G,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
         => ( groups766887009212190081x_real(G,insert_complex(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,X)),groups766887009212190081x_real(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_8655_prod_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(X),A2))
         => ( groups4061424788464935467al_rat(G,insert_real(X,A2)) = groups4061424788464935467al_rat(G,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A2))
         => ( groups4061424788464935467al_rat(G,insert_real(X,A2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(G,X)),groups4061424788464935467al_rat(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_8656_prod_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( pp(aa_set_nat_bool(member_nat(X),A2))
         => ( groups73079841787564623at_rat(G,insert_nat(X,A2)) = groups73079841787564623at_rat(G,A2) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
         => ( groups73079841787564623at_rat(G,insert_nat(X,A2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,X)),groups73079841787564623at_rat(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_8657_fact__split,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_complex(semiri5044797733671781792omplex,N) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K),N)))),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ) ).

% fact_split
tff(fact_8658_fact__split,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K),N)))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ) ).

% fact_split
tff(fact_8659_fact__split,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_int(semiri1406184849735516958ct_int,N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K),N)))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ) ).

% fact_split
tff(fact_8660_fact__split,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_real(semiri2265585572941072030t_real,N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K),N)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ) ).

% fact_split
tff(fact_8661_fact__split,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K),N)))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ) ).

% fact_split
tff(fact_8662_prod__dvd__prod__subset,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups861055069439313189ex_nat(F,A2)),groups861055069439313189ex_nat(F,B2))) ) ) ).

% prod_dvd_prod_subset
tff(fact_8663_prod__dvd__prod__subset,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups705719431365010083at_int(F,A2)),groups705719431365010083at_int(F,B2))) ) ) ).

% prod_dvd_prod_subset
tff(fact_8664_prod__dvd__prod__subset,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups858564598930262913ex_int(F,A2)),groups858564598930262913ex_int(F,B2))) ) ) ).

% prod_dvd_prod_subset
tff(fact_8665_prod__dvd__prod__subset,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_Code_integer] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,groups3455450783089532116nteger(F,A2)),groups3455450783089532116nteger(F,B2))) ) ) ).

% prod_dvd_prod_subset
tff(fact_8666_prod__dvd__prod__subset,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_co8153928105989009288nteger] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,groups8682486955453173170nteger(F,A2)),groups8682486955453173170nteger(F,B2))) ) ) ).

% prod_dvd_prod_subset
tff(fact_8667_prod__dvd__prod__subset,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups1707563613775114915nt_nat(F,A2)),groups1707563613775114915nt_nat(F,B2))) ) ) ).

% prod_dvd_prod_subset
tff(fact_8668_prod__dvd__prod__subset,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups1705073143266064639nt_int(F,A2)),groups1705073143266064639nt_int(F,B2))) ) ) ).

% prod_dvd_prod_subset
tff(fact_8669_prod__dvd__prod__subset,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_Code_integer] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,groups3827104343326376752nteger(F,A2)),groups3827104343326376752nteger(F,B2))) ) ) ).

% prod_dvd_prod_subset
tff(fact_8670_prod__dvd__prod__subset,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)),aa_set_nat_nat(groups708209901874060359at_nat(F),B2))) ) ) ).

% prod_dvd_prod_subset
tff(fact_8671_prod__dvd__prod__subset,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
     => ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups4077766827762148844at_nat(F,A2)),groups4077766827762148844at_nat(F,B2))) ) ) ).

% prod_dvd_prod_subset
tff(fact_8672_prod__dvd__prod__subset2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),A2))
             => 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,A2)),groups4696554848551431203al_nat(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8673_prod__dvd__prod__subset2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),A2))
             => 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,A2)),groups861055069439313189ex_nat(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8674_prod__dvd__prod__subset2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_int,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),A2))
             => 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,A2)),groups4694064378042380927al_int(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8675_prod__dvd__prod__subset2,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
       => ( ! [A3: nat] :
              ( pp(aa_set_nat_bool(member_nat(A3),A2))
             => 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,A2)),groups705719431365010083at_int(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8676_prod__dvd__prod__subset2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),A2))
             => 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,A2)),groups858564598930262913ex_int(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8677_prod__dvd__prod__subset2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_re6978948838062957830nteger,G: fun_re6978948838062957830nteger] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),A2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_real_Code_integer(F,A3)),aa_real_Code_integer(G,A3))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,groups6225526099057966256nteger(F,A2)),groups6225526099057966256nteger(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8678_prod__dvd__prod__subset2,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
       => ( ! [A3: nat] :
              ( pp(aa_set_nat_bool(member_nat(A3),A2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(F,A3)),aa_nat_Code_integer(G,A3))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,groups3455450783089532116nteger(F,A2)),groups3455450783089532116nteger(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8679_prod__dvd__prod__subset2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_co8153928105989009288nteger,G: fun_co8153928105989009288nteger] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),A2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_com4217951221296830281nteger(F,A3)),aa_com4217951221296830281nteger(G,A3))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,groups8682486955453173170nteger(F,A2)),groups8682486955453173170nteger(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8680_prod__dvd__prod__subset2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),A2))
             => 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,A2)),groups1707563613775114915nt_nat(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8681_prod__dvd__prod__subset2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_int,G: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),A2))
             => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(F,A3)),aa_int_int(G,A3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups1705073143266064639nt_int(F,A2)),groups1705073143266064639nt_int(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8682_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T4: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T2: set_real,G: fun_real_int,H: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T4))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_real_real(J,A3)),minus_minus_set_real(T2,T4))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T4)))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T4)))
                   => pp(aa_set_real_bool(member_real(aa_real_real(I,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_int(G,A3) = one_one_int ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T4))
                       => ( aa_real_int(H,B3) = one_one_int ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_real_int(H,aa_real_real(J,A3)) = aa_real_int(G,A3) ) )
                     => ( groups4694064378042380927al_int(G,S2) = groups4694064378042380927al_int(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8683_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T4: set_int,S2: set_real,I: fun_int_real,J: fun_real_int,T2: set_int,G: fun_real_int,H: fun_int_int] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T4))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_int_real(I,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_int_bool(member_int(aa_real_int(J,A3)),minus_minus_set_int(T2,T4))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T4)))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T4)))
                   => pp(aa_set_real_bool(member_real(aa_int_real(I,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_int(G,A3) = one_one_int ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T4))
                       => ( aa_int_int(H,B3) = one_one_int ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_int_int(H,aa_real_int(J,A3)) = aa_real_int(G,A3) ) )
                     => ( groups4694064378042380927al_int(G,S2) = groups1705073143266064639nt_int(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8684_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T4: set_complex,S2: set_real,I: fun_complex_real,J: fun_real_complex,T2: set_complex,G: fun_real_int,H: fun_complex_int] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_complex_real(I,aa_real_complex(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_real_complex(J,A3)),minus_811609699411566653omplex(T2,T4))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T2,T4)))
                 => ( aa_real_complex(J,aa_complex_real(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T2,T4)))
                   => pp(aa_set_real_bool(member_real(aa_complex_real(I,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_int(G,A3) = one_one_int ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T4))
                       => ( aa_complex_int(H,B3) = one_one_int ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_complex_int(H,aa_real_complex(J,A3)) = aa_real_int(G,A3) ) )
                     => ( groups4694064378042380927al_int(G,S2) = groups858564598930262913ex_int(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8685_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T4: set_real,S2: set_int,I: fun_real_int,J: fun_int_real,T2: set_real,G: fun_int_int,H: fun_real_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T4))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
             => ( aa_real_int(I,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_int_real(J,A3)),minus_minus_set_real(T2,T4))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T4)))
                 => ( aa_int_real(J,aa_real_int(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T4)))
                   => pp(aa_set_int_bool(member_int(aa_real_int(I,B3)),minus_minus_set_int(S2,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_int(G,A3) = one_one_int ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T4))
                       => ( aa_real_int(H,B3) = one_one_int ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_real_int(H,aa_int_real(J,A3)) = aa_int_int(G,A3) ) )
                     => ( groups1705073143266064639nt_int(G,S2) = groups4694064378042380927al_int(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8686_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T4: set_int,S2: set_int,I: fun_int_int,J: fun_int_int,T2: set_int,G: fun_int_int,H: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T4))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
             => ( aa_int_int(I,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
               => pp(aa_set_int_bool(member_int(aa_int_int(J,A3)),minus_minus_set_int(T2,T4))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T4)))
                 => ( aa_int_int(J,aa_int_int(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T4)))
                   => pp(aa_set_int_bool(member_int(aa_int_int(I,B3)),minus_minus_set_int(S2,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_int(G,A3) = one_one_int ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T4))
                       => ( aa_int_int(H,B3) = one_one_int ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_int_int(H,aa_int_int(J,A3)) = aa_int_int(G,A3) ) )
                     => ( groups1705073143266064639nt_int(G,S2) = groups1705073143266064639nt_int(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8687_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T4: set_complex,S2: set_int,I: fun_complex_int,J: fun_int_complex,T2: set_complex,G: fun_int_int,H: fun_complex_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
             => ( aa_complex_int(I,aa_int_complex(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_int_complex(J,A3)),minus_811609699411566653omplex(T2,T4))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T2,T4)))
                 => ( aa_int_complex(J,aa_complex_int(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T2,T4)))
                   => pp(aa_set_int_bool(member_int(aa_complex_int(I,B3)),minus_minus_set_int(S2,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_int(G,A3) = one_one_int ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T4))
                       => ( aa_complex_int(H,B3) = one_one_int ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_complex_int(H,aa_int_complex(J,A3)) = aa_int_int(G,A3) ) )
                     => ( groups1705073143266064639nt_int(G,S2) = groups858564598930262913ex_int(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8688_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T4: set_real,S2: set_complex,I: fun_real_complex,J: fun_complex_real,T2: set_real,G: fun_complex_int,H: fun_real_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T4))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
             => ( aa_real_complex(I,aa_complex_real(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_complex_real(J,A3)),minus_minus_set_real(T2,T4))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T4)))
                 => ( aa_complex_real(J,aa_real_complex(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T4)))
                   => pp(aa_set_complex_bool(member_complex(aa_real_complex(I,B3)),minus_811609699411566653omplex(S2,S5))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S5))
                     => ( aa_complex_int(G,A3) = one_one_int ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T4))
                       => ( aa_real_int(H,B3) = one_one_int ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S2))
                         => ( aa_real_int(H,aa_complex_real(J,A3)) = aa_complex_int(G,A3) ) )
                     => ( groups858564598930262913ex_int(G,S2) = groups4694064378042380927al_int(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8689_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T4: set_int,S2: set_complex,I: fun_int_complex,J: fun_complex_int,T2: set_int,G: fun_complex_int,H: fun_int_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T4))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
             => ( aa_int_complex(I,aa_complex_int(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
               => pp(aa_set_int_bool(member_int(aa_complex_int(J,A3)),minus_minus_set_int(T2,T4))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T4)))
                 => ( aa_complex_int(J,aa_int_complex(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T4)))
                   => pp(aa_set_complex_bool(member_complex(aa_int_complex(I,B3)),minus_811609699411566653omplex(S2,S5))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S5))
                     => ( aa_complex_int(G,A3) = one_one_int ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T4))
                       => ( aa_int_int(H,B3) = one_one_int ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S2))
                         => ( aa_int_int(H,aa_complex_int(J,A3)) = aa_complex_int(G,A3) ) )
                     => ( groups858564598930262913ex_int(G,S2) = groups1705073143266064639nt_int(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8690_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T4: set_complex,S2: set_complex,I: fun_complex_complex,J: fun_complex_complex,T2: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
             => ( aa_complex_complex(I,aa_complex_complex(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_complex_complex(J,A3)),minus_811609699411566653omplex(T2,T4))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T2,T4)))
                 => ( aa_complex_complex(J,aa_complex_complex(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T2,T4)))
                   => pp(aa_set_complex_bool(member_complex(aa_complex_complex(I,B3)),minus_811609699411566653omplex(S2,S5))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S5))
                     => ( aa_complex_int(G,A3) = one_one_int ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T4))
                       => ( aa_complex_int(H,B3) = one_one_int ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S2))
                         => ( aa_complex_int(H,aa_complex_complex(J,A3)) = aa_complex_int(G,A3) ) )
                     => ( groups858564598930262913ex_int(G,S2) = groups858564598930262913ex_int(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8691_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T4: set_nat,S2: set_real,I: fun_nat_real,J: fun_real_nat,T2: set_nat,G: fun_real_complex,H: fun_nat_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_nat_bool(finite_finite_nat,T4))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_nat_real(I,aa_real_nat(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_nat_bool(member_nat(aa_real_nat(J,A3)),minus_minus_set_nat(T2,T4))) )
           => ( ! [B3: nat] :
                  ( pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(T2,T4)))
                 => ( aa_real_nat(J,aa_nat_real(I,B3)) = B3 ) )
             => ( ! [B3: nat] :
                    ( pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(T2,T4)))
                   => pp(aa_set_real_bool(member_real(aa_nat_real(I,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: nat] :
                        ( pp(aa_set_nat_bool(member_nat(B3),T4))
                       => ( aa_nat_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_nat_complex(H,aa_real_nat(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups713298508707869441omplex(G,S2) = groups6464643781859351333omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8692_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_8693_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,N4),set_or4665077453230672383an_nat(zero_zero_nat,N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N4)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_8694_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_8695_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)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ly),set_or1269000886237332187st_nat(aa_nat_nat(suc,N),M))) ) ) ).

% fact_eq_fact_times
tff(fact_8696_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)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ly),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),M)) ) ) ).

% fact_div_fact
tff(fact_8697_OR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_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,Y),aa_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(bit_se1409905431419307370or_int(X),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% OR_upper
tff(fact_8698_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),N) = minus_minus_set_nat(set_ord_lessThan_nat(N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_8699_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A: fun_nat_nat,B: 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(A,I2)),aa_nat_nat(A,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(B,J2)),aa_nat_nat(B,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_yr(A),B)),set_or4665077453230672383an_nat(zero_zero_nat,N)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),set_or4665077453230672383an_nat(zero_zero_nat,N))),aa_set_nat_nat(groups3542108847815614940at_nat(B),set_or4665077453230672383an_nat(zero_zero_nat,N))))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_8700_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Node(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_list_VEBT_VEBT(vEBT_size_VEBT,X13)),aa_VEBT_VEBT_nat(vEBT_size_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size_gen(1)
tff(fact_8701_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_ys),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_8702_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,L),U))) ).

% finite_atLeastLessThan_int
tff(fact_8703_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(aa_num_rat(numeral_numeral_rat,K)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),one_one_int) ).

% quotient_of_number(3)
tff(fact_8704_rat__one__code,axiom,
    quotient_of(one_one_rat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int) ).

% rat_one_code
tff(fact_8705_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

% rat_zero_code
tff(fact_8706_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(aa_int2565445551602696444nt_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_8707_quotient__of__number_I4_J,axiom,
    quotient_of(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% quotient_of_number(4)
tff(fact_8708_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : pp(aa_set_int_bool(finite_finite_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,zero_zero_int),U))) ).

% finite_atLeastZeroLessThan_int
tff(fact_8709_quotient__of__div,axiom,
    ! [R2: rat,N: int,D: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,N),D) )
     => ( R2 = aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,N)),aa_int_rat(ring_1_of_int_rat,D)) ) ) ).

% quotient_of_div
tff(fact_8710_quotient__of__denom__pos,axiom,
    ! [R2: rat,P2: int,Q4: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q4)) ) ).

% quotient_of_denom_pos
tff(fact_8711_rat__uminus__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(uminus_uminus_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_yt),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_8712_rat__abs__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(abs_abs_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_yu),quotient_of(P2)) ).

% rat_abs_code
tff(fact_8713_rat__less__code,axiom,
    ! [P2: rat,Q4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,P2),Q4))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_yw(Q4)),quotient_of(P2))) ) ).

% rat_less_code
tff(fact_8714_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: bool,X222: bool] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf(X21,X222)) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_8715_quotient__of__int,axiom,
    ! [A: int] : quotient_of(aa_int_rat(of_int,A)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),one_one_int) ).

% quotient_of_int
tff(fact_8716_rat__plus__code,axiom,
    ! [P2: rat,Q4: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,P2),Q4)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_yy(Q4)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_8717_rat__minus__code,axiom,
    ! [P2: rat,Q4: rat] : quotient_of(aa_rat_rat(minus_minus_rat(P2),Q4)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_za(Q4)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_8718_normalize__negative,axiom,
    ! [Q4: int,P2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Q4),zero_zero_int))
     => ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,P2)),aa_int_int(uminus_uminus_int,Q4))) ) ) ).

% normalize_negative
tff(fact_8719_normalize__denom__zero,axiom,
    ! [P2: int] : aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),zero_zero_int)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

% normalize_denom_zero
tff(fact_8720_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P2: int,Q4: int] :
      ( ( aa_Pro6970504127699319548nt_int(normalize,R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q4)) ) ).

% normalize_denom_pos
tff(fact_8721_normalize__crossproduct,axiom,
    ! [Q4: int,S: int,P2: int,R2: int] :
      ( ( Q4 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R2),S)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P2),S) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),Q4) ) ) ) ) ).

% normalize_crossproduct
tff(fact_8722_rat__divide__code,axiom,
    ! [P2: rat,Q4: rat] : quotient_of(aa_rat_rat(divide_divide_rat(P2),Q4)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_zc(Q4)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_8723_rat__times__code,axiom,
    ! [P2: rat,Q4: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P2),Q4)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_ze(Q4)),quotient_of(P2)) ).

% rat_times_code
tff(fact_8724_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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_8725_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_fd(L),Qr) ).

% divmod_step_integer_def
tff(fact_8726_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),aa_num_int(numeral_numeral_int,L))) = aa_rat_rat(divide_divide_rat(aa_num_rat(numeral_numeral_rat,K)),aa_num_rat(numeral_numeral_rat,L)) ).

% Frct_code_post(6)
tff(fact_8727_card__Collect__less__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bd,N))) = N ).

% card_Collect_less_nat
tff(fact_8728_card__Collect__le__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_be,N))) = aa_nat_nat(suc,N) ).

% card_Collect_le_nat
tff(fact_8729_divmod__integer_H__def,axiom,
    ! [M: num,N: num] : unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N)),modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N))) ).

% divmod_integer'_def
tff(fact_8730_full__exhaustive__integer_H_Ocases,axiom,
    ! [X: produc1616345218596728535nteger] :
      ~ ! [F2: fun_Pr272337284026993017e_term,D4: code_integer,I2: code_integer] : X != aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,F2),produc1086072967326762835nteger(D4,I2)) ).

% full_exhaustive_integer'.cases
tff(fact_8731_exhaustive__integer_H_Ocases,axiom,
    ! [X: produc5207004643709237345nteger] :
      ~ ! [F2: fun_Co6545587727236015215e_term,D4: code_integer,I2: code_integer] : X != aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,F2),produc1086072967326762835nteger(D4,I2)) ).

% exhaustive_integer'.cases
tff(fact_8732_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_8733_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_8734_card__less,axiom,
    ! [M4: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M4))
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_zf(M4),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_8735_card__less__Suc,axiom,
    ! [M4: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M4))
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_zg(M4),I)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_zf(M4),I))) ) ) ).

% card_less_Suc
tff(fact_8736_card__less__Suc2,axiom,
    ! [M4: set_nat,I: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M4))
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_zg(M4),I))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_zf(M4),I))) ) ) ).

% card_less_Suc2
tff(fact_8737_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,N4),set_or4665077453230672383an_nat(zero_zero_nat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N4)),N)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_8738_card__sum__le__nat__sum,axiom,
    ! [S2: set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ly),set_or4665077453230672383an_nat(zero_zero_nat,aa_set_nat_nat(finite_card_nat,S2)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ly),S2))) ).

% card_sum_le_nat_sum
tff(fact_8739_card__nth__roots,axiom,
    ! [C: complex,N: nat] :
      ( ( C != 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_zh(C),N))) = N ) ) ) ).

% card_nth_roots
tff(fact_8740_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_di,N))) = N ) ) ).

% card_roots_unity_eq
tff(fact_8741_Frct__code__post_I2_J,axiom,
    ! [A: int] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),zero_zero_int)) = zero_zero_rat ).

% Frct_code_post(2)
tff(fact_8742_Frct__code__post_I1_J,axiom,
    ! [A: int] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),A)) = zero_zero_rat ).

% Frct_code_post(1)
tff(fact_8743_Frct__code__post_I7_J,axiom,
    ! [A: int,B: int] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,A)),B)) = aa_rat_rat(uminus_uminus_rat,frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ).

% Frct_code_post(7)
tff(fact_8744_Frct__code__post_I8_J,axiom,
    ! [A: int,B: int] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),aa_int_int(uminus_uminus_int,B))) = aa_rat_rat(uminus_uminus_rat,frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ).

% Frct_code_post(8)
tff(fact_8745_Frct__code__post_I3_J,axiom,
    frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)) = one_one_rat ).

% Frct_code_post(3)
tff(fact_8746_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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_8747_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) = if_Code_integer(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_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_8748_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),zero_zero_int))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ).

% signed_take_bit_negative_iff
tff(fact_8749_less__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Xa2),X)) ) ).

% less_integer.abs_eq
tff(fact_8750_less__integer__code_I1_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)) ).

% less_integer_code(1)
tff(fact_8751_abs__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(K) = K ) ) ) ).

% abs_integer_code
tff(fact_8752_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_8753_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_8754_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_8755_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(bit_se2923211474154528505it_int(M),K))) ) ) ).

% bit_imp_take_bit_positive
tff(fact_8756_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M2))
             => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),M2))
              <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N2)) ) )
         => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)))
              <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N2)) ) ) ) ).

% int_bit_bound
tff(fact_8757_int__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( code_int_of_integer(K) = aa_int_int(uminus_uminus_int,code_int_of_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( ( ( K = zero_z3403309356797280102nteger )
           => ( code_int_of_integer(K) = zero_zero_int ) )
          & ( ( K != zero_z3403309356797280102nteger )
           => ( code_int_of_integer(K) = produc1553301316500091796er_int(aTP_Lamm_zi,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ) ) ).

% int_of_integer_code
tff(fact_8758_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = produc5135875973608410785r_bool(divide6298287555418463151nteger(K,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),K))) ).

% bit_cut_integer_def
tff(fact_8759_divmod__integer__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = produc1086072967326762835nteger(divide6298287555418463151nteger(K,L),modulo364778990260209775nteger(K,L)) ).

% divmod_integer_def
tff(fact_8760_less__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Xa2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,code_int_of_integer(X)),code_int_of_integer(Xa2))) ) ).

% less_integer.rep_eq
tff(fact_8761_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),L))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,code_int_of_integer(K)),code_int_of_integer(L))) ) ).

% integer_less_iff
tff(fact_8762_binomial__def,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = finite_card_set_nat(collect_set_nat(aa_nat8857836526276733660t_bool(aTP_Lamm_zj(N),K))) ).

% binomial_def
tff(fact_8763_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( code_bit_cut_integer(K) = produc5135875973608410785r_bool(zero_z3403309356797280102nteger,fFalse) ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( code_bit_cut_integer(K) = produc6117575242473779287r_bool(aTP_Lamm_zk(K),code_divmod_abs(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).

% bit_cut_integer_code
tff(fact_8764_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
       => ( aa_Code_integer_nat(code_nat_of_integer,K) = zero_zero_nat ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
       => ( aa_Code_integer_nat(code_nat_of_integer,K) = produc1555791787009142072er_nat(aTP_Lamm_zl,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).

% nat_of_integer_code
tff(fact_8765_divmod__abs__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_abs(K,L) = produc1086072967326762835nteger(divide6298287555418463151nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L)),modulo364778990260209775nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L))) ).

% divmod_abs_def
tff(fact_8766_pair__leqI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,S),T))
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_8767_pair__leqI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_8768_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
     => ( aa_Code_integer_nat(code_nat_of_integer,K) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_8769_nat__of__integer__code__post_I1_J,axiom,
    aa_Code_integer_nat(code_nat_of_integer,zero_z3403309356797280102nteger) = zero_zero_nat ).

% nat_of_integer_code_post(1)
tff(fact_8770_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] : code_divmod_abs(zero_z3403309356797280102nteger,J) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ).

% divmod_abs_code(6)
tff(fact_8771_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] : code_divmod_abs(J,zero_z3403309356797280102nteger) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,abs_abs_Code_integer(J)) ).

% divmod_abs_code(5)
tff(fact_8772_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L))
           => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K))
               => ( code_divmod_integer(K,L) = code_divmod_abs(K,L) ) )
              & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K))
               => ( code_divmod_integer(K,L) = produc6916734918728496179nteger(aTP_Lamm_zm(L),code_divmod_abs(K,L)) ) ) ) )
          & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L))
           => ( ( ( L = zero_z3403309356797280102nteger )
               => ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,K) ) )
              & ( ( L != zero_z3403309356797280102nteger )
               => ( code_divmod_integer(K,L) = aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_zn(L),code_divmod_abs(K,L)))) ) ) ) ) ) ) ) ).

% divmod_integer_code
tff(fact_8773_wmin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),XS))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
       => ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS)),fun_min_weak))
         => pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y,YS))),fun_min_weak)) ) ) ) ).

% wmin_insertI
tff(fact_8774_wmax__insertI,axiom,
    ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y),YS))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
       => ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS)),fun_max_weak))
         => pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X,XS),YS)),fun_max_weak)) ) ) ) ).

% wmax_insertI
tff(fact_8775_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero_nat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ).

% bezw_0
tff(fact_8776_Suc__0__mod__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_mod_numeral
tff(fact_8777_wmax__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,X6))
     => pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,X6)),fun_max_weak)) ) ).

% wmax_emptyI
tff(fact_8778_wmin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] : pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,bot_bo2099793752762293965at_nat)),fun_min_weak)) ).

% wmin_emptyI
tff(fact_8779_smin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),XS))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_less)
       => ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS)),fun_min_strict))
         => pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y,YS))),fun_min_strict)) ) ) ) ).

% smin_insertI
tff(fact_8780_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y5: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X6: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y),Y5))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_less)
       => ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,Y5)),fun_max_strict))
         => pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X,X6),Y5)),fun_max_strict)) ) ) ) ).

% smax_insertI
tff(fact_8781_prod__decode__aux_Osimps,axiom,
    ! [M: nat,K: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
       => ( nat_prod_decode_aux(K,M) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),M)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
       => ( nat_prod_decode_aux(K,M) = nat_prod_decode_aux(aa_nat_nat(suc,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,K))) ) ) ) ).

% prod_decode_aux.simps
tff(fact_8782_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(X,Xa2) = Y )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
         => ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa2)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
         => ( Y = nat_prod_decode_aux(aa_nat_nat(suc,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X))) ) ) ) ) ).

% prod_decode_aux.elims
tff(fact_8783_Suc__0__div__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_div_numeral
tff(fact_8784_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(quotient_of(R2)))) ).

% quotient_of_denom_pos'
tff(fact_8785_smin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( ( X6 != bot_bo2099793752762293965at_nat )
     => pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,bot_bo2099793752762293965at_nat)),fun_min_strict)) ) ).

% smin_emptyI
tff(fact_8786_smax__emptyI,axiom,
    ! [Y5: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,Y5))
     => ( ( Y5 != bot_bo2099793752762293965at_nat )
       => pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,Y5)),fun_max_strict)) ) ) ).

% smax_emptyI
tff(fact_8787_bezw_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( bezw(X,Y) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ) )
      & ( ( Y != zero_zero_nat )
       => ( bezw(X,Y) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ) ).

% bezw.simps
tff(fact_8788_rat__sgn__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(sgn_sgn_rat,P2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(sgn_sgn_int,product_fst_int_int(quotient_of(P2)))),one_one_int) ).

% rat_sgn_code
tff(fact_8789_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Y))
     => ( bezw(X,Y) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ).

% bezw_non_0
tff(fact_8790_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa2) = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa2))))) ) ) ) ) ).

% bezw.elims
tff(fact_8791_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa2) = Y )
     => ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa2))))) ) ) )
           => ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).

% bezw.pelims
tff(fact_8792_normalize__def,axiom,
    ! [P2: product_prod_int_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(P2)))
       => ( aa_Pro6970504127699319548nt_int(normalize,P2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(divide_divide_int(product_fst_int_int(P2)),aa_int_int(gcd_gcd_int(product_fst_int_int(P2)),product_snd_int_int(P2)))),aa_int_int(divide_divide_int(product_snd_int_int(P2)),aa_int_int(gcd_gcd_int(product_fst_int_int(P2)),product_snd_int_int(P2)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(P2)))
       => ( ( ( product_snd_int_int(P2) = zero_zero_int )
           => ( aa_Pro6970504127699319548nt_int(normalize,P2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ) )
          & ( ( product_snd_int_int(P2) != zero_zero_int )
           => ( aa_Pro6970504127699319548nt_int(normalize,P2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(divide_divide_int(product_fst_int_int(P2)),aa_int_int(uminus_uminus_int,aa_int_int(gcd_gcd_int(product_fst_int_int(P2)),product_snd_int_int(P2))))),aa_int_int(divide_divide_int(product_snd_int_int(P2)),aa_int_int(uminus_uminus_int,aa_int_int(gcd_gcd_int(product_fst_int_int(P2)),product_snd_int_int(P2))))) ) ) ) ) ) ).

% normalize_def
tff(fact_8793_vebt__mint_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(X) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),X))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( ( pp(A3)
                   => ( Y = some_nat(zero_zero_nat) ) )
                  & ( ~ pp(A3)
                   => ( ( pp(B3)
                       => ( Y = some_nat(one_one_nat) ) )
                      & ( ~ pp(B3)
                       => ( Y = none_nat ) ) ) ) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Leaf(A3,B3))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y = none_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2))) ) )
           => ~ ! [Mi3: nat,Ma3: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),Ux2,Uy2,Uz2) )
                 => ( ( Y = some_nat(Mi3) )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),Ux2,Uy2,Uz2))) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_8794_vebt__maxt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(X) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),X))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( ( pp(B3)
                   => ( Y = some_nat(one_one_nat) ) )
                  & ( ~ pp(B3)
                   => ( ( pp(A3)
                       => ( Y = some_nat(zero_zero_nat) ) )
                      & ( ~ pp(A3)
                       => ( Y = none_nat ) ) ) ) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Leaf(A3,B3))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y = none_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2))) ) )
           => ~ ! [Mi3: nat,Ma3: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),Ux2,Uy2,Uz2) )
                 => ( ( Y = some_nat(Ma3) )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),Ux2,Uy2,Uz2))) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_8795_gcd__pos__int,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(gcd_gcd_int(M),N)))
    <=> ( ( M != zero_zero_int )
        | ( N != zero_zero_int ) ) ) ).

% gcd_pos_int
tff(fact_8796_gcd__le1__int,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(gcd_gcd_int(A),B)),A)) ) ).

% gcd_le1_int
tff(fact_8797_gcd__le2__int,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(gcd_gcd_int(A),B)),B)) ) ).

% gcd_le2_int
tff(fact_8798_gcd__non__0__int,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Y))
     => ( aa_int_int(gcd_gcd_int(X),Y) = aa_int_int(gcd_gcd_int(Y),aa_int_int(modulo_modulo_int(X),Y)) ) ) ).

% gcd_non_0_int
tff(fact_8799_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(X,Xa2) = Y )
     => ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
               => ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa2)) ) )
              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
               => ( Y = nat_prod_decode_aux(aa_nat_nat(suc,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X))) ) ) )
           => ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_8800_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_8801_gcd__nat_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),A) = A ).

% gcd_nat.left_neutral
tff(fact_8802_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_8803_gcd__nat_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),zero_zero_nat) = A ).

% gcd_nat.right_neutral
tff(fact_8804_gcd__0__nat,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),zero_zero_nat) = X ).

% gcd_0_nat
tff(fact_8805_gcd__0__left__nat,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),X) = X ).

% gcd_0_left_nat
tff(fact_8806_gcd__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% gcd_Suc_0
tff(fact_8807_gcd__pos__nat,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(gcd_gcd_nat,M),N)))
    <=> ( ( M != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_8808_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_8809_Inf__nat__def1,axiom,
    ! [K4: set_nat] :
      ( ( K4 != bot_bot_set_nat )
     => pp(aa_set_nat_bool(member_nat(aa_set_nat_nat(complete_Inf_Inf_nat,K4)),K4)) ) ).

% Inf_nat_def1
tff(fact_8810_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B)) ) ).

% gcd_le2_nat
tff(fact_8811_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A)) ) ).

% gcd_le1_nat
tff(fact_8812_gcd__diff2__nat,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(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ) ).

% gcd_diff2_nat
tff(fact_8813_gcd__diff1__nat,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(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ) ).

% gcd_diff1_nat
tff(fact_8814_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa2) = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y = X ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)) ) ) ) ) ).

% gcd_nat.elims
tff(fact_8815_gcd__nat_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X ) )
      & ( ( Y != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ) ) ) ).

% gcd_nat.simps
tff(fact_8816_gcd__non__0__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ) ) ).

% gcd_non_0_nat
tff(fact_8817_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X4: nat,Y2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) ) ).

% bezout_nat
tff(fact_8818_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X4: nat,Y2: 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,B),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4)))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) )
      | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4)))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_8819_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa2) = Y )
     => ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y = X ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)) ) ) )
           => ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).

% gcd_nat.pelims
tff(fact_8820_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: bool] :
      ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
      <=> pp(Y) )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X))
       => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
           => ( pp(Y)
             => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fFalse,fFalse))) ) )
         => ( ! [Uv2: bool] :
                ( ( X = vEBT_Leaf(fTrue,Uv2) )
               => ( ~ pp(Y)
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fTrue,Uv2))) ) )
           => ( ! [Uu2: bool] :
                  ( ( X = vEBT_Leaf(Uu2,fTrue) )
                 => ( ~ pp(Y)
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(Uu2,fTrue))) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
                   => ( pp(Y)
                     => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2))) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
                     => ( ~ pp(Y)
                       => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_8821_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X))
       => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
           => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fFalse,fFalse))) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2))) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_8822_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X))
       => ( ! [Uv2: bool] :
              ( ( X = vEBT_Leaf(fTrue,Uv2) )
             => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fTrue,Uv2))) )
         => ( ! [Uu2: bool] :
                ( ( X = vEBT_Leaf(Uu2,fTrue) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(Uu2,fTrue))) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2))) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_8823_finite__enumerate,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ? [R3: fun_nat_nat] :
          ( pp(aa_set_nat_bool(strict1292158309912662752at_nat(R3),set_ord_lessThan_nat(aa_set_nat_nat(finite_card_nat,S2))))
          & ! [N5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N5),aa_set_nat_nat(finite_card_nat,S2)))
             => pp(aa_set_nat_bool(member_nat(aa_nat_nat(R3,N5)),S2)) ) ) ) ).

% finite_enumerate
tff(fact_8824_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( ( ( L = zero_z3403309356797280102nteger )
           => ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,K) ) )
          & ( ( L != zero_z3403309356797280102nteger )
           => ( code_divmod_integer(K,L) = aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L)),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_zo(L),code_divmod_abs(K,L)))) ) ) ) ) ) ).

% divmod_integer_eq_cases
tff(fact_8825_card_Ocomp__fun__commute__on,axiom,
    comp_nat_nat_nat(suc,suc) = comp_nat_nat_nat(suc,suc) ).

% card.comp_fun_commute_on
tff(fact_8826_root__powr__inverse,axiom,
    ! [N: nat,X: 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),X))
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% root_powr_inverse
tff(fact_8827_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se8568078237143864401it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% drop_bit_negative_int_iff
tff(fact_8828_real__root__Suc__0,axiom,
    ! [X: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),X) = X ).

% real_root_Suc_0
tff(fact_8829_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y: 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),X) = aa_real_real(aa_nat_fun_real_real(root,N),Y) )
      <=> ( X = Y ) ) ) ).

% real_root_eq_iff
tff(fact_8830_root__0,axiom,
    ! [X: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),X) = zero_zero_real ).

% root_0
tff(fact_8831_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)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% drop_bit_of_Suc_0
tff(fact_8832_real__root__eq__0__iff,axiom,
    ! [N: nat,X: 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),X) = zero_zero_real )
      <=> ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_8833_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y: 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),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% real_root_less_iff
tff(fact_8834_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y: 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),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% real_root_le_iff
tff(fact_8835_real__root__eq__1__iff,axiom,
    ! [N: nat,X: 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),X) = one_one_real )
      <=> ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_8836_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_8837_real__root__lt__0__iff,axiom,
    ! [N: nat,X: 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),X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ) ).

% real_root_lt_0_iff
tff(fact_8838_real__root__gt__0__iff,axiom,
    ! [N: nat,Y: 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),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y)) ) ) ).

% real_root_gt_0_iff
tff(fact_8839_real__root__le__0__iff,axiom,
    ! [N: nat,X: 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),X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ) ).

% real_root_le_0_iff
tff(fact_8840_real__root__ge__0__iff,axiom,
    ! [N: nat,Y: 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),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y)) ) ) ).

% real_root_ge_0_iff
tff(fact_8841_real__root__lt__1__iff,axiom,
    ! [N: nat,X: 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),X)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% real_root_lt_1_iff
tff(fact_8842_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: 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),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),Y)) ) ) ).

% real_root_gt_1_iff
tff(fact_8843_real__root__le__1__iff,axiom,
    ! [N: nat,X: 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),X)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ).

% real_root_le_1_iff
tff(fact_8844_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: 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),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y)) ) ) ).

% real_root_ge_1_iff
tff(fact_8845_real__root__pow__pos2,axiom,
    ! [N: nat,X: 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),X))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),N) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_8846_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y: 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,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y))) ) ) ).

% real_root_less_mono
tff(fact_8847_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y: 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,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y))) ) ) ).

% real_root_le_mono
tff(fact_8848_real__root__power,axiom,
    ! [N: nat,X: 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,X),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),K) ) ) ).

% real_root_power
tff(fact_8849_real__root__abs,axiom,
    ! [N: nat,X: 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,X)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) ) ) ).

% real_root_abs
tff(fact_8850_sgn__root,axiom,
    ! [N: nat,X: 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),X)) = aa_real_real(sgn_sgn_real,X) ) ) ).

% sgn_root
tff(fact_8851_real__root__gt__zero,axiom,
    ! [N: nat,X: 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),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ).

% real_root_gt_zero
tff(fact_8852_real__root__strict__decreasing,axiom,
    ! [N: nat,N4: nat,X: 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),N4))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N4),X)),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ) ).

% real_root_strict_decreasing
tff(fact_8853_root__abs__power,axiom,
    ! [N: nat,Y: 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,Y),N))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_8854_real__root__pos__pos,axiom,
    ! [N: nat,X: 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),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ).

% real_root_pos_pos
tff(fact_8855_real__root__strict__increasing,axiom,
    ! [N: nat,N4: nat,X: 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),N4))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N4),X))) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_8856_real__root__decreasing,axiom,
    ! [N: nat,N4: nat,X: 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),N4))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N4),X)),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ) ).

% real_root_decreasing
tff(fact_8857_real__root__pow__pos,axiom,
    ! [N: nat,X: 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),X))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),N) = X ) ) ) ).

% real_root_pow_pos
tff(fact_8858_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,X: 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),Y))
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N) = X )
         => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_8859_real__root__power__cancel,axiom,
    ! [N: nat,X: 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),X))
       => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_8860_real__root__increasing,axiom,
    ! [N: nat,N4: nat,X: 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),N4))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N4),X))) ) ) ) ) ).

% real_root_increasing
tff(fact_8861_sgn__power__root,axiom,
    ! [N: nat,X: 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),X))),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),X))),N)) = X ) ) ).

% sgn_power_root
tff(fact_8862_root__sgn__power,axiom,
    ! [N: nat,Y: 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,Y)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y)),N))) = Y ) ) ).

% root_sgn_power
tff(fact_8863_ln__root,axiom,
    ! [N: nat,B: 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),B))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,N),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% ln_root
tff(fact_8864_log__root,axiom,
    ! [N: nat,A: real,B: 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),A))
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_nat_fun_real_real(root,N),A)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),A)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% log_root
tff(fact_8865_log__base__root,axiom,
    ! [N: nat,B: real,X: 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),B))
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_nat_fun_real_real(root,N),B)),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ) ).

% log_base_root
tff(fact_8866_split__root,axiom,
    ! [P: fun_real_bool,N: nat,X: real] :
      ( pp(aa_real_bool(P,aa_real_real(aa_nat_fun_real_real(root,N),X)))
    <=> ( ( ( 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))
         => ! [Y4: real] :
              ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y4)),N)) = X )
             => pp(aa_real_bool(P,Y4)) ) ) ) ) ).

% split_root
tff(fact_8867_Suc__funpow,axiom,
    ! [N: nat] : compow_fun_nat_nat(N,suc) = aa_nat_fun_nat_nat(plus_plus_nat,N) ).

% Suc_funpow
tff(fact_8868_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_be,aTP_Lamm_bd) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_8869_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_8870_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q4: 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),Q4)),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,Q4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_8871_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_zp) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_8872_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_min_strict,fun_min_weak)) ).

% min_rpair_set
tff(fact_8873_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_max_strict,fun_max_weak)) ).

% max_rpair_set
tff(fact_8874_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se545348938243370406it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% push_bit_negative_int_iff
tff(fact_8875_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(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_8876_Gcd__remove0__nat,axiom,
    ! [M4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M4))
     => ( aa_set_nat_nat(gcd_Gcd_nat,M4) = aa_set_nat_nat(gcd_Gcd_nat,minus_minus_set_nat(M4,insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_8877_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_zr),Xa2),X)) ).

% times_int.abs_eq
tff(fact_8878_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X4: nat,Y2: nat] : Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y2)) ).

% eq_Abs_Integ
tff(fact_8879_Gcd__in,axiom,
    ! [A2: set_nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_set_nat_bool(member_nat(A3),A2))
         => ( pp(aa_set_nat_bool(member_nat(B3),A2))
           => pp(aa_set_nat_bool(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A3),B3)),A2)) ) )
     => ( ( A2 != bot_bot_set_nat )
       => pp(aa_set_nat_bool(member_nat(aa_set_nat_nat(gcd_Gcd_nat,A2)),A2)) ) ) ).

% Gcd_in
tff(fact_8880_zero__int__def,axiom,
    zero_zero_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)) ).

% zero_int_def
tff(fact_8881_int__def,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N),zero_zero_nat)) ).

% int_def
tff(fact_8882_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_zs),X)) ).

% uminus_int.abs_eq
tff(fact_8883_one__int__def,axiom,
    one_one_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).

% one_int_def
tff(fact_8884_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_zu),Xa2),X)) ) ).

% less_int.abs_eq
tff(fact_8885_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_zw),Xa2),X)) ) ).

% less_eq_int.abs_eq
tff(fact_8886_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_zy),Xa2),X)) ).

% plus_int.abs_eq
tff(fact_8887_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_aaa),Xa2),X)) ).

% minus_int.abs_eq
tff(fact_8888_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_zw),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa2))) ) ).

% less_eq_int.rep_eq
tff(fact_8889_less__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Xa2))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_zu),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa2))) ) ).

% less_int.rep_eq
tff(fact_8890_rat__floor__lemma,axiom,
    ! [A: int,B: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(A),B))),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),B)),one_one_int)))) ) ).

% rat_floor_lemma
tff(fact_8891_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
       => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aab,C),set_or4665077453230672383an_nat(X,Y)) = set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
           => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aab,C),set_or4665077453230672383an_nat(X,Y)) = insert_nat(zero_zero_nat,bot_bot_set_nat) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
           => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aab,C),set_or4665077453230672383an_nat(X,Y)) = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_8892_less__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C),D)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)))) ) ) ) ).

% less_rat
tff(fact_8893_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] : ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),image_nat_nat(suc,A2))) ).

% zero_notin_Suc_image
tff(fact_8894_Rat__induct__pos,axiom,
    ! [P: fun_rat_bool,Q4: rat] :
      ( ! [A3: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
         => pp(aa_rat_bool(P,aa_int_rat(aa_int_fun_int_rat(fract,A3),B3))) )
     => pp(aa_rat_bool(P,Q4)) ) ).

% Rat_induct_pos
tff(fact_8895_quotient__of__eq,axiom,
    ! [A: int,B: int,P2: int,Q4: int] :
      ( ( quotient_of(aa_int_rat(aa_int_fun_int_rat(fract,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4) )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,P2),Q4) = aa_int_rat(aa_int_fun_int_rat(fract,A),B) ) ) ).

% quotient_of_eq
tff(fact_8896_normalize__eq,axiom,
    ! [A: int,B: int,P2: int,Q4: int] :
      ( ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4) )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,P2),Q4) = aa_int_rat(aa_int_fun_int_rat(fract,A),B) ) ) ).

% normalize_eq
tff(fact_8897_quotient__of__Fract,axiom,
    ! [A: int,B: int] : quotient_of(aa_int_rat(aa_int_fun_int_rat(fract,A),B)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ).

% quotient_of_Fract
tff(fact_8898_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_8899_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_8900_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_ord_lessThan_nat(aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_ord_lessThan_nat(N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_8901_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_ord_atMost_nat(aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_ord_atMost_nat(N))) ).

% atMost_Suc_eq_insert_0
tff(fact_8902_Fract__less__zero__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),zero_zero_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% Fract_less_zero_iff
tff(fact_8903_zero__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ).

% zero_less_Fract_iff
tff(fact_8904_Fract__less__one__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),one_one_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% Fract_less_one_iff
tff(fact_8905_one__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ).

% one_less_Fract_iff
tff(fact_8906_zero__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).

% zero_le_Fract_iff
tff(fact_8907_Fract__le__zero__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),zero_zero_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ) ).

% Fract_le_zero_iff
tff(fact_8908_one__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% one_le_Fract_iff
tff(fact_8909_Fract__le__one__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),one_one_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% Fract_le_one_iff
tff(fact_8910_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_zs)) ).

% uminus_int_def
tff(fact_8911_finite__int__iff__bounded,axiom,
    ! [S2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ? [K3: int] : pp(aa_set_int_bool(ord_less_eq_set_int(image_int_int(abs_abs_int,S2)),set_ord_lessThan_int(K3))) ) ).

% finite_int_iff_bounded
tff(fact_8912_finite__int__iff__bounded__le,axiom,
    ! [S2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ? [K3: int] : pp(aa_set_int_bool(ord_less_eq_set_int(image_int_int(abs_abs_int,S2)),set_ord_atMost_int(K3))) ) ).

% finite_int_iff_bounded_le
tff(fact_8913_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S2: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ~ pp(aa_set_nat_bool(finite_finite_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),S2))) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_8914_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_zr)) ).

% times_int_def
tff(fact_8915_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_aaa)) ).

% minus_int_def
tff(fact_8916_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_zy)) ).

% plus_int_def
tff(fact_8917_infinite__UNIV__nat,axiom,
    ~ pp(aa_set_nat_bool(finite_finite_nat,top_top_set_nat)) ).

% infinite_UNIV_nat
tff(fact_8918_nat__not__finite,axiom,
    ~ pp(aa_set_nat_bool(finite_finite_nat,top_top_set_nat)) ).

% nat_not_finite
tff(fact_8919_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_8920_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_aac,N),top_top_set_nat) = set_or4665077453230672383an_nat(zero_zero_nat,N) ) ) ).

% range_mod
tff(fact_8921_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_8922_range__mult,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( image_real_real(aa_rea7340605660300375871l_real(times_times_real,A),top_top_set_real) = insert_real(zero_zero_real,bot_bot_set_real) ) )
      & ( ( A != zero_zero_real )
       => ( image_real_real(aa_rea7340605660300375871l_real(times_times_real,A),top_top_set_real) = top_top_set_real ) ) ) ).

% range_mult
tff(fact_8923_infinite__UNIV__int,axiom,
    ~ pp(aa_set_int_bool(finite_finite_int,top_top_set_int)) ).

% infinite_UNIV_int
tff(fact_8924_root__def,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero_nat )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = zero_zero_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_aad,N)),X) ) ) ) ).

% root_def
tff(fact_8925_DERIV__even__real__root,axiom,
    ! [N: nat,X: 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,X),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),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_8926_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,X: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S2))
     => ( 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))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)),S2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_8927_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,X: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S2))
     => ( 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))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)),S2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_8928_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S2))
     => ( 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))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)),S2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_8929_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S2))
     => ( 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))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)),S2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_8930_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X4,top_top_set_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,aa_real_real(F,B)),aa_real_real(F,A))) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_8931_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X4,top_top_set_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(F,A)),aa_real_real(F,B))) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_8932_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,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))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_8933_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,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))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_8934_DERIV__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,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))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_8935_DERIV__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,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))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_8936_MVT2,axiom,
    ! [A: real,B: real,F: fun_real_real,F5: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
             => has_fi5821293074295781190e_real(F,aa_real_real(F5,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
       => ? [Z3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z3))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B))
            & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),aa_real_real(F5,Z3)) ) ) ) ) ).

% MVT2
tff(fact_8937_DERIV__local__const,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y2: 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,X),Y2))),D))
             => ( aa_real_real(F,X) = aa_real_real(F,Y2) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_8938_DERIV__ln,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(inverse_inverse_real,X),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_ln
tff(fact_8939_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y2: 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,X),Y2))),D))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,Y2)),aa_real_real(F,X))) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_8940_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y2: 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,X),Y2))),D))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X)),aa_real_real(F,Y2))) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_8941_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_8942_DERIV__pow,axiom,
    ! [N: nat,X: real,S: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_aae,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,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(X,S)) ).

% DERIV_pow
tff(fact_8943_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_aaf,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_8944_DERIV__log,axiom,
    ! [X: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(log,B),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,B)),X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_log
tff(fact_8945_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,M: real,X: real,R2: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(G,X)))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_aag(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,X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),M),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_8946_DERIV__powr,axiom,
    ! [G: fun_real_real,M: real,X: real,F: fun_real_real,R2: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(G,X)))
       => ( has_fi5821293074295781190e_real(F,R2,topolo2177554685111907308n_real(X,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_aah(G),F),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,X)),aa_real_real(F,X))),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,X)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),aa_real_real(F,X))),aa_real_real(G,X)))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_8947_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => 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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X,A2)) ) ).

% arcosh_real_has_field_derivative
tff(fact_8948_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_8949_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D6: real] :
      ( ( X != zero_zero_real )
     => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( D6 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
           => ( D6 = 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,X)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D6,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_8950_artanh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X,A2)) ) ).

% artanh_real_has_field_derivative
tff(fact_8951_DERIV__real__root,axiom,
    ! [N: nat,X: 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),X))
       => 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),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_8952_DERIV__arcsin,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_8953_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: 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)) )
     => ? [T6: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X)))
          & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aai(Diff),X),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),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_8954_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: 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))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aai(Diff),X),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),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_8955_DERIV__odd__real__root,axiom,
    ! [N: nat,X: 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))
     => ( ( X != 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),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_8956_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,T6: 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),T6))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),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)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
           => ? [T6: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),zero_zero_real))
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_td(H),Diff),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),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_8957_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,T6: 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),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),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)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
         => ? [T6: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),H))
              & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_td(H),Diff),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),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_8958_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,T6: 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),T6))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),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)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
           => ? [T6: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),H))
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_td(H),Diff),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),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_8959_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat,X: 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))
       => ( ( X != 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))
           => ? [T6: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T6)))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X)))
                & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aai(Diff),X),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),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_8960_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M5: nat,T6: 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,T6)),aa_real_real(abs_abs_real,X))) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aai(Diff),X),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),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_8961_Taylor__down,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: 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,T6: 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,A),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
             => ? [T6: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),T6))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),C))
                  & ( aa_real_real(F,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_aaj(Diff),A),C),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),T6)),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,A),C)),N))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_8962_Taylor__up,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: 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,T6: 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,A),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
             => ? [T6: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),T6))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),B))
                  & ( aa_real_real(F,B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_aaj(Diff),B),C),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),T6)),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,B),C)),N))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_8963_Taylor,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real,X: 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,T6: 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,A),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),B))
                 => ( ( X != C )
                   => ? [T6: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T6))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),C)) ) )
                        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),T6))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),X)) ) )
                        & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_aak(Diff),C),X),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),T6)),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,X),C)),N))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_8964_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B2: real] :
      ( ! [M5: nat,T6: 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),T6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),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)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
     => ( ( N = aa_nat_nat(suc,K) )
       => ! [M2: nat,T7: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),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(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_aam(N),Diff),B2),M2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T7)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_aan(Diff),M2),T7),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M2))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,T7),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M2)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M2))))))),topolo2177554685111907308n_real(T7,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_8965_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => has_fi5821293074295781190e_real(aTP_Lamm_aap,suminf_real(aa_real_fun_nat_real(aTP_Lamm_aaq,X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_8966_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D6: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( X != 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),X))
             => ( D6 = 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),X)),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,X),zero_zero_real))
               => ( D6 = 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),X)),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))
               => ( D6 = 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),X)),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),D6,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_8967_DERIV__arccos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_8968_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_aar(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_aat(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_aar(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_8969_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,Y)),aa_real_real(arccos,X))) ) ) ) ).

% arccos_less_arccos
tff(fact_8970_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ) ).

% arccos_less_mono
tff(fact_8971_DERIV__isconst3,axiom,
    ! [A: real,B: real,X: real,Y: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_set_real_bool(member_real(X),set_or1633881224788618240n_real(A,B)))
       => ( pp(aa_set_real_bool(member_real(Y),set_or1633881224788618240n_real(A,B)))
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),set_or1633881224788618240n_real(A,B)))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X4,top_top_set_real)) )
           => ( aa_real_real(F,X) = aa_real_real(F,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_8972_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(arccos,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,Y)),pi)) ) ) ) ).

% arccos_lt_bounded
tff(fact_8973_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => ( aa_real_real(sin_real,aa_real_real(arccos,X)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_8974_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,set_or5834768355832116004an_nat(L,U))) ).

% finite_greaterThanLessThan
tff(fact_8975_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or5832277885323065728an_int,L),U))) ).

% finite_greaterThanLessThan_int
tff(fact_8976_LIM__fun__gt__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,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))
            & ! [X2: real] :
                ( ( ( X2 != C )
                  & 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,C),X2))),R3)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,X2))) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_8977_LIM__fun__not__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( ( L != zero_zero_real )
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ! [X2: real] :
                ( ( ( X2 != C )
                  & 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,C),X2))),R3)) )
               => ( aa_real_real(F,X2) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_8978_LIM__fun__less__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,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))
            & ! [X2: real] :
                ( ( ( X2 != C )
                  & 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,C),X2))),R3)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X2)),zero_zero_real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_8979_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I))))
     => ( aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I,J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_8980_summable__Leibniz_I2_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(A,zero_zero_nat)))
         => ! [N5: nat] : pp(aa_set_real_bool(member_real(suminf_real(aTP_Lamm_aau(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_aau(A),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))),N5))),groups6591440286371151544t_real(aTP_Lamm_aau(A),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))),N5)),one_one_nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_8981_summable__Leibniz_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(A,zero_zero_nat)),zero_zero_real))
         => ! [N5: nat] : pp(aa_set_real_bool(member_real(suminf_real(aTP_Lamm_aau(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_aau(A),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))),N5)),one_one_nat))),groups6591440286371151544t_real(aTP_Lamm_aau(A),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))),N5)))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_8982_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
tff(fact_8983_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
     => filterlim_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),at_top_nat,at_top_nat) ) ).

% mult_nat_left_at_top
tff(fact_8984_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
     => filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aav,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_8985_LIMSEQ__inverse__zero,axiom,
    ! [X6: fun_nat_real] :
      ( ! [R3: real] :
        ? [N7: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R3),aa_nat_real(X6,N2))) )
     => filterlim_nat_real(aTP_Lamm_aaw(X6),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_8986_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_aax,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_8987_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),L))
       => ( ! [E: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
             => ? [N5: 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,N5)),E))) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_8988_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_8989_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_aay(X),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_8990_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real))
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,C),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_8991_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real))
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,C)),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_8992_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_aaz,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_8993_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),X4))
         => ? [Y3: real] :
              ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X4,top_top_set_real))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),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,B))) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_8994_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_aba(N),F),at_top_real,F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_8995_eventually__sequentially,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N9: nat] :
        ! [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),N3))
         => pp(aa_nat_bool(P,N3)) ) ) ).

% eventually_sequentially
tff(fact_8996_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),X4))
         => pp(aa_nat_bool(P,X4)) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_8997_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ord_le2510731241096832064er_nat(F3,at_top_nat)
    <=> ! [N9: nat] : eventually_nat(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),F3) ) ).

% le_sequentially
tff(fact_8998_eventually__at__left__real,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => eventually_real(aa_rea7809428808904475482l_bool(aTP_Lamm_abb(B),A),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ).

% eventually_at_left_real
tff(fact_8999_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
         => ? [Y3: real] :
              ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X4,top_top_set_real))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3)) ) )
     => ( 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,B))) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_9000_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_aba(N),F),at_bot_real,F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_9001_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,set_or6659071591806873216st_nat(L,U))) ).

% finite_greaterThanAtMost
tff(fact_9002_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_bool,B: nat] :
      ( ? [X_12: nat] : pp(aa_nat_bool(P,X_12))
     => ( ! [Y2: nat] :
            ( pp(aa_nat_bool(P,Y2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),B)) )
       => pp(aa_nat_bool(P,order_Greatest_nat(P))) ) ) ).

% GreatestI_ex_nat
tff(fact_9003_Greatest__le__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y2: nat] :
            ( pp(aa_nat_bool(P,Y2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),B)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),order_Greatest_nat(P))) ) ) ).

% Greatest_le_nat
tff(fact_9004_GreatestI__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y2: nat] :
            ( pp(aa_nat_bool(P,Y2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),B)) )
       => pp(aa_nat_bool(P,order_Greatest_nat(P))) ) ) ).

% GreatestI_nat
tff(fact_9005_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)))
     => ( aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I,J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9006_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or6656581121297822940st_int,L),U))) ).

% finite_greaterThanAtMost_int
tff(fact_9007_isCont__inverse__function2,axiom,
    ! [A: real,X: real,B: real,G: fun_real_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
       => ( ! [Z3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z3))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z3),B))
               => ( aa_real_real(G,aa_real_real(F,Z3)) = Z3 ) ) )
         => ( ! [Z3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z3),B))
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_9008_isCont__arcosh,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arcosh_real) ) ).

% isCont_arcosh
tff(fact_9009_DERIV__inverse__function,axiom,
    ! [F: fun_real_real,D6: real,G: fun_real_real,X: real,A: real,B: real] :
      ( has_fi5821293074295781190e_real(F,D6,topolo2177554685111907308n_real(aa_real_real(G,X),top_top_set_real))
     => ( ( D6 != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
           => ( ! [Y2: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Y2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),B))
                   => ( aa_real_real(F,aa_real_real(G,Y2)) = Y2 ) ) )
             => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),G)
               => has_fi5821293074295781190e_real(G,aa_real_real(inverse_inverse_real,D6),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_9010_isCont__arccos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arccos) ) ) ).

% isCont_arccos
tff(fact_9011_isCont__arcsin,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arcsin) ) ) ).

% isCont_arcsin
tff(fact_9012_LIM__less__bound,axiom,
    ! [B: real,X: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),X))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),set_or1633881224788618240n_real(B,X)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),F)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X))) ) ) ) ).

% LIM_less_bound
tff(fact_9013_isCont__artanh,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),artanh_real) ) ) ).

% isCont_artanh
tff(fact_9014_isCont__inverse__function,axiom,
    ! [D: real,X: real,G: fun_real_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
     => ( ! [Z3: 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,Z3),X))),D))
           => ( aa_real_real(G,aa_real_real(F,Z3)) = Z3 ) )
       => ( ! [Z3: 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,Z3),X))),D))
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_9015_GMVT_H,axiom,
    ! [A: real,B: 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,A),B))
     => ( ! [Z3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z3))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z3),B))
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) ) )
       => ( ! [Z3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z3))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z3),B))
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),G) ) )
         => ( ! [Z3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B))
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z3),topolo2177554685111907308n_real(Z3,top_top_set_real)) ) )
           => ( ! [Z3: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z3))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B))
                   => has_fi5821293074295781190e_real(F,aa_real_real(F5,Z3),topolo2177554685111907308n_real(Z3,top_top_set_real)) ) )
             => ? [C3: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C3),B))
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),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,B)),aa_real_real(G,A))),aa_real_real(F5,C3)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_9016_INT__greaterThan__UNIV,axiom,
    comple7806235888213564991et_nat(image_nat_set_nat(set_or1210151606488870762an_nat,top_top_set_nat)) = bot_bot_set_nat ).

% INT_greaterThan_UNIV
tff(fact_9017_greaterThan__0,axiom,
    aa_nat_set_nat(set_or1210151606488870762an_nat,zero_zero_nat) = image_nat_nat(suc,top_top_set_nat) ).

% greaterThan_0
tff(fact_9018_eventually__at__right__real,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => eventually_real(aa_rea7809428808904475482l_bool(aTP_Lamm_abb(A),B),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ).

% eventually_at_right_real
tff(fact_9019_greaterThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_or1210151606488870762an_nat,K),insert_nat(aa_nat_nat(suc,K),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_9020_GMVT,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X4: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B)) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X4,top_top_set_real),F) )
       => ( ! [X4: real] :
              ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B)) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) )
         => ( ! [X4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B)) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X4,top_top_set_real),G) )
           => ( ! [X4: real] :
                  ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B)) )
                 => 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,A),C3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C3),B))
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),G_c) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B)),aa_real_real(G,A))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_9021_atLeast__0,axiom,
    set_ord_atLeast_nat(zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_9022_atLeast__Suc,axiom,
    ! [K: nat] : set_ord_atLeast_nat(aa_nat_nat(suc,K)) = minus_minus_set_nat(set_ord_atLeast_nat(K),insert_nat(K,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_9023_Gcd__eq__Max,axiom,
    ! [M4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M4))
     => ( ( M4 != bot_bot_set_nat )
       => ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M4))
         => ( aa_set_nat_nat(gcd_Gcd_nat,M4) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_abc,M4))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_9024_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ey,N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_9025_bdd__above__nat,axiom,
    ! [X6: set_nat] :
      ( pp(aa_set_nat_bool(condit2214826472909112428ve_nat,X6))
    <=> pp(aa_set_nat_bool(finite_finite_nat,X6)) ) ).

% bdd_above_nat
tff(fact_9026_card__le__Suc__Max,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,S2)),aa_nat_nat(suc,aa_set_nat_nat(lattic8265883725875713057ax_nat,S2)))) ) ).

% card_le_Suc_Max
tff(fact_9027_Sup__nat__def,axiom,
    ! [X6: set_nat] :
      ( ( ( X6 = bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X6) = zero_zero_nat ) )
      & ( ( X6 != bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X6) = aa_set_nat_nat(lattic8265883725875713057ax_nat,X6) ) ) ) ).

% Sup_nat_def
tff(fact_9028_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) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_abd(N),M))) ) ) ) ).

% divide_nat_def
tff(fact_9029_gcd__is__Max__divisors__nat,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(gcd_gcd_nat,M),N) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_abe(N),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_9030_MVT,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ? [L3: real,Z3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B))
              & has_fi5821293074295781190e_real(F,L3,topolo2177554685111907308n_real(Z3,top_top_set_real))
              & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),L3) ) ) ) ) ) ).

% MVT
tff(fact_9031_uniformity__complex__def,axiom,
    topolo896644834953643431omplex = comple8358262395181532106omplex(image_5971271580939081552omplex(aTP_Lamm_abg,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_complex_def
tff(fact_9032_uniformity__real__def,axiom,
    topolo1511823702728130853y_real = comple2936214249959783750l_real(image_2178119161166701260l_real(aTP_Lamm_abi,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_real_def
tff(fact_9033_Rolle__deriv,axiom,
    ! [A: real,B: real,F: fun_real_real,F5: fun_re7681423839302404990l_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_real(F,A) = aa_real_real(F,B) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => ( ! [X4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
                 => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F5,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
           => ? [Z3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z3))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B))
                & ! [X2: real] : aa_real_real(aa_rea7340605660300375871l_real(F5,Z3),X2) = zero_zero_real ) ) ) ) ) ).

% Rolle_deriv
tff(fact_9034_mvt,axiom,
    ! [A: real,B: real,F: fun_real_real,F5: fun_re7681423839302404990l_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
               => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F5,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ~ ! [Xi: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Xi))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Xi),B))
                 => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) != aa_real_real(aa_rea7340605660300375871l_real(F5,Xi),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ) ) ) ) ) ) ).

% mvt
tff(fact_9035_DERIV__pos__imp__increasing__open,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3)) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B))) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_9036_DERIV__neg__imp__decreasing__open,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),zero_zero_real)) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A))) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_9037_DERIV__isconst__end,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ( aa_real_real(F,B) = aa_real_real(F,A) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_9038_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: fun_real_real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
               => 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,A),X))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),B))
             => ( aa_real_real(F,X) = aa_real_real(F,A) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_9039_Rolle,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_real(F,A) = aa_real_real(F,B) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => ( ! [X4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
                 => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
           => ? [Z3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z3))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B))
                & has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(Z3,top_top_set_real)) ) ) ) ) ) ).

% Rolle
tff(fact_9040_eventually__prod__sequentially,axiom,
    ! [P: fun_Pr7757163693017731481t_bool] :
      ( eventu1038000079068216329at_nat(P,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N9: nat] :
        ! [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),M3))
         => ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),N3))
             => pp(aa_Pro2232559814647311250t_bool(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N3),M3))) ) ) ) ).

% eventually_prod_sequentially
tff(fact_9041_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_9042_mono__times__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N)) ) ).

% mono_times_nat
tff(fact_9043_mono__ge2__power__minus__self,axiom,
    ! [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))),K))
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abj,K)) ) ).

% mono_ge2_power_minus_self
tff(fact_9044_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_aad,N)),top_top_set_real)) ) ).

% inj_sgn_power
tff(fact_9045_log__inj,axiom,
    ! [B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => pp(aa_set_real_bool(inj_on_real_real(aa_rea7340605660300375871l_real(log,B)),set_or5849166863359141190n_real(zero_zero_real))) ) ).

% log_inj
tff(fact_9046_inj__Suc,axiom,
    ! [N4: set_nat] : pp(aa_set_nat_bool(inj_on_nat_nat(suc),N4)) ).

% inj_Suc
tff(fact_9047_inj__on__diff__nat,axiom,
    ! [N4: set_nat,K: nat] :
      ( ! [N2: nat] :
          ( pp(aa_set_nat_bool(member_nat(N2),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2)) )
     => pp(aa_set_nat_bool(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aab,K)),N4)) ) ).

% inj_on_diff_nat
tff(fact_9048_inj__on__set__encode,axiom,
    pp(aa_set_set_nat_bool(inj_on_set_nat_nat(nat_set_encode),collect_set_nat(finite_finite_nat))) ).

% inj_on_set_encode
tff(fact_9049_powr__real__of__int_H,axiom,
    ! [X: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( ( X != zero_zero_real )
          | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N)) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,N)) = aa_int_real(power_int_real(X),N) ) ) ) ).

% powr_real_of_int'
tff(fact_9050_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
             => ( ( Deg2 = Xa2 )
                & ! [X4: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                   => pp(aa_nat_bool(vEBT_VEBT_valid(X4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                & pp(aa_nat_bool(vEBT_VEBT_valid(Summary3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                & ( 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))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_abk(Deg2),TreeList2),Summary3))),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_9051_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( set_or4665077453230672383an_nat(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = sup_sup_set_nat(set_or4665077453230672383an_nat(I,J),set_or4665077453230672383an_nat(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_9052_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(vEBT_Node(Mima2,Deg,TreeList,Summary)),Deg4))
    <=> ( ( Deg = Deg4 )
        & ! [X3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
           => pp(aa_nat_bool(vEBT_VEBT_valid(X3),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(vEBT_VEBT_valid(Summary),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),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_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,Deg),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_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_abk(Deg),TreeList),Summary))),Mima2)) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_9053_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( pp(Y)
          <=> ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
             => ( pp(Y)
              <=> ~ ( ( Deg2 = Xa2 )
                    & ! [X3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                       => pp(aa_nat_bool(vEBT_VEBT_valid(X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & pp(aa_nat_bool(vEBT_VEBT_valid(Summary3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                    & ( 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))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_abk(Deg2),TreeList2),Summary3))),Mima)) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_9054_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X2: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                     => pp(aa_nat_bool(vEBT_VEBT_valid(X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                  & pp(aa_nat_bool(vEBT_VEBT_valid(Summary3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                  & ( 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))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                  & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_abk(Deg2),TreeList2),Summary3))),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_9055_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2))
               => ( Xa2 = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary3),Xa2))
                 => ( ( Deg2 = Xa2 )
                    & ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                       => pp(aa_nat_bool(vEBT_VEBT_valid(X4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & pp(aa_nat_bool(vEBT_VEBT_valid(Summary3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                    & ( 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))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_abk(Deg2),TreeList2),Summary3))),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_9056_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2))
               => ( Xa2 != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary3),Xa2))
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X2: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                         => pp(aa_nat_bool(vEBT_VEBT_valid(X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & pp(aa_nat_bool(vEBT_VEBT_valid(Summary3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                      & ( 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))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_abk(Deg2),TreeList2),Summary3))),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_9057_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ( pp(Y)
                <=> ( Xa2 = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
               => ( ( pp(Y)
                  <=> ( ( Deg2 = Xa2 )
                      & ! [X3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                         => pp(aa_nat_bool(vEBT_VEBT_valid(X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & pp(aa_nat_bool(vEBT_VEBT_valid(Summary3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                      & ( 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))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_abk(Deg2),TreeList2),Summary3))),Mima)) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary3),Xa2)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_9058_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_9059_min__weak__def,axiom,
    fun_min_weak = sup_su5525570899277871387at_nat(min_ex6901939911449802026at_nat(fun_pair_leq),insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat),bot_bo228742789529271731at_nat)) ).

% min_weak_def
tff(fact_9060_max__weak__def,axiom,
    fun_max_weak = sup_su5525570899277871387at_nat(max_ex8135407076693332796at_nat(fun_pair_leq),insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat),bot_bo228742789529271731at_nat)) ).

% max_weak_def
tff(fact_9061_take__bit__num__simps_I1_J,axiom,
    ! [M: num] : bit_take_bit_num(zero_zero_nat,M) = none_num ).

% take_bit_num_simps(1)
tff(fact_9062_take__bit__num__simps_I2_J,axiom,
    ! [N: nat] : bit_take_bit_num(aa_nat_nat(suc,N),one) = some_num(one) ).

% take_bit_num_simps(2)
tff(fact_9063_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),one) = some_num(one) ).

% take_bit_num_simps(5)
tff(fact_9064_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,N),aa_num_num(bit0,M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_abl,bit_take_bit_num(N,M)) ).

% take_bit_num_simps(3)
tff(fact_9065_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,N),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(N,M))) ).

% take_bit_num_simps(4)
tff(fact_9066_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,aa_num_num(bit1,M)) = case_nat_option_num(none_num,aTP_Lamm_abm(M),N) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_9067_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit0,M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_abl,bit_take_bit_num(aa_num_nat(pred_numeral,R2),M)) ).

% take_bit_num_simps(6)
tff(fact_9068_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(aa_num_nat(pred_numeral,R2),M))) ).

% take_bit_num_simps(7)
tff(fact_9069_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_9070_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_9071_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_9072_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_9073_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_9074_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_9075_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N: nat] : bit_take_bit_num(N,one) = case_nat_option_num(none_num,aTP_Lamm_abn,N) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_9076_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,aa_num_num(bit0,M)) = case_nat_option_num(none_num,aTP_Lamm_abo(M),N) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_9077_Bit__Operations_Otake__bit__num__code,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,M) = produc478579273971653890on_num(aTP_Lamm_abs,product_Pair_nat_num(N,M)) ).

% Bit_Operations.take_bit_num_code
tff(fact_9078_take__bit__num__def,axiom,
    ! [N: nat,M: num] :
      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)) = zero_zero_nat )
       => ( bit_take_bit_num(N,M) = none_num ) )
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)) != zero_zero_nat )
       => ( bit_take_bit_num(N,M) = some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)))) ) ) ) ).

% take_bit_num_def
tff(fact_9079_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fTrue,aTP_Lamm_cf),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_9080_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aTP_Lamm_abt),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_9081_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9082_less__eq__nat_Osimps_I2_J,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(case_nat_bool(fFalse,aa_nat_fun_nat_bool(ord_less_eq_nat,M)),N)) ) ).

% less_eq_nat.simps(2)
tff(fact_9083_max__Suc1,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_abu,N)),M) ).

% max_Suc1
tff(fact_9084_max__Suc2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_abv,N)),M) ).

% max_Suc2
tff(fact_9085_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_9086_num__of__nat__plus__distrib,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,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_nat_num(num_of_nat,M)),aa_nat_num(num_of_nat,N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9087_num__of__nat__One,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),one_one_nat))
     => ( aa_nat_num(num_of_nat,N) = one ) ) ).

% num_of_nat_One
tff(fact_9088_diff__Suc,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(case_nat_nat(zero_zero_nat,aTP_Lamm_ly),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ).

% diff_Suc
tff(fact_9089_num__of__nat__double,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)) = aa_num_num(bit0,aa_nat_num(num_of_nat,N)) ) ) ).

% num_of_nat_double
tff(fact_9090_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_ly),Nat) ).

% pred_def
tff(fact_9091_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_9092_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_9093_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_9094_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_9095_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q4: 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)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4)) ).

% nat_mult_min_left
tff(fact_9096_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q4: 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),Q4)) = 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),Q4)) ).

% nat_mult_min_right
tff(fact_9097_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_9098_min__Suc2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_abw,N)),M) ).

% min_Suc2
tff(fact_9099_min__Suc1,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_abx,N)),M) ).

% min_Suc1
tff(fact_9100_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa_num_num(bit1,M),aa_num_num(bit0,N)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_aby,bit_and_not_num(M,N)) ).

% and_not_num.simps(8)
tff(fact_9101_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one,one) = none_num ).

% and_not_num.simps(1)
tff(fact_9102_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] : bit_and_not_num(aa_num_num(bit0,M),one) = some_num(aa_num_num(bit0,M)) ).

% and_not_num.simps(4)
tff(fact_9103_and__not__num_Osimps_I2_J,axiom,
    ! [N: num] : bit_and_not_num(one,aa_num_num(bit0,N)) = some_num(one) ).

% and_not_num.simps(2)
tff(fact_9104_and__not__num_Osimps_I3_J,axiom,
    ! [N: num] : bit_and_not_num(one,aa_num_num(bit1,N)) = none_num ).

% and_not_num.simps(3)
tff(fact_9105_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] : bit_and_not_num(aa_num_num(bit1,M),one) = some_num(aa_num_num(bit0,M)) ).

% and_not_num.simps(7)
tff(fact_9106_and__not__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( bit_and_not_num(X,Xa2) = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != none_num ) ) )
       => ( ( ( X = one )
           => ( ? [N2: num] : Xa2 = aa_num_num(bit0,N2)
             => ( Y != some_num(one) ) ) )
         => ( ( ( X = one )
             => ( ? [N2: num] : Xa2 = aa_num_num(bit1,N2)
               => ( Y != none_num ) ) )
           => ( ! [M5: num] :
                  ( ( X = aa_num_num(bit0,M5) )
                 => ( ( Xa2 = one )
                   => ( Y != some_num(aa_num_num(bit0,M5)) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ! [N2: num] :
                        ( ( Xa2 = aa_num_num(bit0,N2) )
                       => ( Y != map_option_num_num(bit0,bit_and_not_num(M5,N2)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N2: num] :
                          ( ( Xa2 = aa_num_num(bit1,N2) )
                         => ( Y != map_option_num_num(bit0,bit_and_not_num(M5,N2)) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa_num_num(bit1,M5) )
                       => ( ( Xa2 = one )
                         => ( Y != some_num(aa_num_num(bit0,M5)) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ! [N2: num] :
                              ( ( Xa2 = aa_num_num(bit0,N2) )
                             => ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_aby,bit_and_not_num(M5,N2)) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N2: num] :
                                ( ( Xa2 = aa_num_num(bit1,N2) )
                               => ( Y != map_option_num_num(bit0,bit_and_not_num(M5,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_9107_xor__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : bit_un2480387367778600638or_num(aa_num_num(bit1,M),aa_num_num(bit0,N)) = some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M,N))) ).

% xor_num.simps(8)
tff(fact_9108_xor__num_Osimps_I1_J,axiom,
    bit_un2480387367778600638or_num(one,one) = none_num ).

% xor_num.simps(1)
tff(fact_9109_xor__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( bit_un2480387367778600638or_num(X,Xa2) = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != none_num ) ) )
       => ( ( ( X = one )
           => ! [N2: num] :
                ( ( Xa2 = aa_num_num(bit0,N2) )
               => ( Y != some_num(aa_num_num(bit1,N2)) ) ) )
         => ( ( ( X = one )
             => ! [N2: num] :
                  ( ( Xa2 = aa_num_num(bit1,N2) )
                 => ( Y != some_num(aa_num_num(bit0,N2)) ) ) )
           => ( ! [M5: num] :
                  ( ( X = aa_num_num(bit0,M5) )
                 => ( ( Xa2 = one )
                   => ( Y != some_num(aa_num_num(bit1,M5)) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ! [N2: num] :
                        ( ( Xa2 = aa_num_num(bit0,N2) )
                       => ( Y != map_option_num_num(bit0,bit_un2480387367778600638or_num(M5,N2)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N2: num] :
                          ( ( Xa2 = aa_num_num(bit1,N2) )
                         => ( Y != some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M5,N2))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa_num_num(bit1,M5) )
                       => ( ( Xa2 = one )
                         => ( Y != some_num(aa_num_num(bit0,M5)) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ! [N2: num] :
                              ( ( Xa2 = aa_num_num(bit0,N2) )
                             => ( Y != some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M5,N2))) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N2: num] :
                                ( ( Xa2 = aa_num_num(bit1,N2) )
                               => ( Y != map_option_num_num(bit0,bit_un2480387367778600638or_num(M5,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
tff(fact_9110_xor__num_Osimps_I2_J,axiom,
    ! [N: num] : bit_un2480387367778600638or_num(one,aa_num_num(bit0,N)) = some_num(aa_num_num(bit1,N)) ).

% xor_num.simps(2)
tff(fact_9111_xor__num_Osimps_I3_J,axiom,
    ! [N: num] : bit_un2480387367778600638or_num(one,aa_num_num(bit1,N)) = some_num(aa_num_num(bit0,N)) ).

% xor_num.simps(3)
tff(fact_9112_xor__num_Osimps_I4_J,axiom,
    ! [M: num] : bit_un2480387367778600638or_num(aa_num_num(bit0,M),one) = some_num(aa_num_num(bit1,M)) ).

% xor_num.simps(4)
tff(fact_9113_xor__num_Osimps_I7_J,axiom,
    ! [M: num] : bit_un2480387367778600638or_num(aa_num_num(bit1,M),one) = some_num(aa_num_num(bit0,M)) ).

% xor_num.simps(7)
tff(fact_9114_xor__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] : bit_un2480387367778600638or_num(aa_num_num(bit0,M),aa_num_num(bit1,N)) = some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M,N))) ).

% xor_num.simps(6)
tff(fact_9115_and__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( bit_un7362597486090784418nd_num(X,Xa2) = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != some_num(one) ) ) )
       => ( ( ( X = one )
           => ( ? [N2: num] : Xa2 = aa_num_num(bit0,N2)
             => ( Y != none_num ) ) )
         => ( ( ( X = one )
             => ( ? [N2: num] : Xa2 = aa_num_num(bit1,N2)
               => ( Y != some_num(one) ) ) )
           => ( ( ? [M5: num] : X = aa_num_num(bit0,M5)
               => ( ( Xa2 = one )
                 => ( Y != none_num ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ! [N2: num] :
                        ( ( Xa2 = aa_num_num(bit0,N2) )
                       => ( Y != map_option_num_num(bit0,bit_un7362597486090784418nd_num(M5,N2)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N2: num] :
                          ( ( Xa2 = aa_num_num(bit1,N2) )
                         => ( Y != map_option_num_num(bit0,bit_un7362597486090784418nd_num(M5,N2)) ) ) )
                 => ( ( ? [M5: num] : X = aa_num_num(bit1,M5)
                     => ( ( Xa2 = one )
                       => ( Y != some_num(one) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ! [N2: num] :
                              ( ( Xa2 = aa_num_num(bit0,N2) )
                             => ( Y != map_option_num_num(bit0,bit_un7362597486090784418nd_num(M5,N2)) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N2: num] :
                                ( ( Xa2 = aa_num_num(bit1,N2) )
                               => ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_aby,bit_un7362597486090784418nd_num(M5,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_9116_and__num_Osimps_I1_J,axiom,
    bit_un7362597486090784418nd_num(one,one) = some_num(one) ).

% and_num.simps(1)
tff(fact_9117_and__num_Osimps_I7_J,axiom,
    ! [M: num] : bit_un7362597486090784418nd_num(aa_num_num(bit1,M),one) = some_num(one) ).

% and_num.simps(7)
tff(fact_9118_and__num_Osimps_I3_J,axiom,
    ! [N: num] : bit_un7362597486090784418nd_num(one,aa_num_num(bit1,N)) = some_num(one) ).

% and_num.simps(3)
tff(fact_9119_and__num_Osimps_I4_J,axiom,
    ! [M: num] : bit_un7362597486090784418nd_num(aa_num_num(bit0,M),one) = none_num ).

% and_num.simps(4)
tff(fact_9120_and__num_Osimps_I2_J,axiom,
    ! [N: num] : bit_un7362597486090784418nd_num(one,aa_num_num(bit0,N)) = none_num ).

% and_num.simps(2)
tff(fact_9121_and__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : bit_un7362597486090784418nd_num(aa_num_num(bit1,M),aa_num_num(bit1,N)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_aby,bit_un7362597486090784418nd_num(M,N)) ).

% and_num.simps(9)
tff(fact_9122_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = aa_num_num(inc,aa_nat_num(num_of_nat,N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = one ) ) ) ).

% num_of_nat.simps(2)
tff(fact_9123_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_abz) ).

% Rats_eq_int_div_nat
tff(fact_9124_Rats__no__bot__less,axiom,
    ! [X: real] :
    ? [X4: real] :
      ( pp(aa_set_real_bool(member_real(X4),field_5140801741446780682s_real))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),X)) ) ).

% Rats_no_bot_less
tff(fact_9125_Rats__dense__in__real,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ? [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),field_5140801741446780682s_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X4))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y)) ) ) ).

% Rats_dense_in_real
tff(fact_9126_pos__deriv__imp__strict__mono,axiom,
    ! [F: fun_real_real,F5: fun_real_real] :
      ( ! [X4: real] : has_fi5821293074295781190e_real(F,aa_real_real(F5,X4),topolo2177554685111907308n_real(X4,top_top_set_real))
     => ( ! [X4: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F5,X4)))
       => order_7092887310737990675l_real(F) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_9127_positive__rat,axiom,
    ! [A: int,B: int] :
      ( pp(aa_rat_bool(positive,aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
    <=> 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,A),B))) ) ).

% positive_rat
tff(fact_9128_strict__mono__imp__increasing,axiom,
    ! [F: fun_nat_nat,N: nat] :
      ( order_5726023648592871131at_nat(F)
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(F,N))) ) ).

% strict_mono_imp_increasing
tff(fact_9129_infinite__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ? [R3: fun_nat_nat] :
          ( order_5726023648592871131at_nat(R3)
          & ! [N5: nat] : pp(aa_set_nat_bool(member_nat(aa_nat_nat(R3,N5)),S2)) ) ) ).

% infinite_enumerate
tff(fact_9130_less__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> pp(aa_rat_bool(positive,aa_rat_rat(minus_minus_rat(Y),X))) ) ).

% less_rat_def
tff(fact_9131_Rat_Opositive_Orep__eq,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(positive,X))
    <=> 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,product_fst_int_int(aa_rat7754553326175492643nt_int(rep_Rat,X))),product_snd_int_int(aa_rat7754553326175492643nt_int(rep_Rat,X))))) ) ).

% Rat.positive.rep_eq
tff(fact_9132_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,I)),J))
     => ( linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I,J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9133_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),J))
     => ( linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I,J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9134_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(set_ord_lessThan_nat(aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(set_ord_lessThan_nat(K)),cons_nat(K,nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_9135_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(set_ord_atMost_nat(aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(set_ord_atMost_nat(K)),cons_nat(aa_nat_nat(suc,K),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_9136_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_9137_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( upto(X,Xa2) = Y )
     => ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa2)))
       => ~ ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
               => ( Y = cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa2)) ) )
              & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
               => ( Y = nil_int ) ) )
           => ~ pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa2))) ) ) ) ).

% upto.pelims
tff(fact_9138_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I),J)))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
         => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
         => ( upto(I,J) = nil_int ) ) ) ) ).

% upto.psimps
tff(fact_9139_upto__empty,axiom,
    ! [J: int,I: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
     => ( upto(I,J) = nil_int ) ) ).

% upto_empty
tff(fact_9140_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int = upto(I,J) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I)) ) ).

% upto_Nil2
tff(fact_9141_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I)) ) ).

% upto_Nil
tff(fact_9142_upto__single,axiom,
    ! [I: int] : upto(I,I) = cons_int(I,nil_int) ).

% upto_single
tff(fact_9143_nth__upto,axiom,
    ! [I: int,K: nat,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),aa_nat_int(semiri1314217659103216013at_int,K))),J))
     => ( aa_nat_int(nth_int(upto(I,J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_9144_length__upto,axiom,
    ! [I: int,J: int] : size_size_list_int(upto(I,J)) = 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,J),I)),one_one_int)) ).

% length_upto
tff(fact_9145_upto__rec__numeral_I1_J,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)))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N)) = cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),aa_num_int(numeral_numeral_int,N))) ) )
      & ( ~ 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)))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N)) = nil_int ) ) ) ).

% upto_rec_numeral(1)
tff(fact_9146_upto__rec__numeral_I2_J,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_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = nil_int ) ) ) ).

% upto_rec_numeral(2)
tff(fact_9147_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int),aa_num_int(numeral_numeral_int,N))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = nil_int ) ) ) ).

% upto_rec_numeral(3)
tff(fact_9148_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_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))))
       => ( upto(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))) = cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_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))))
       => ( upto(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))) = nil_int ) ) ) ).

% upto_rec_numeral(4)
tff(fact_9149_upto__code,axiom,
    ! [I: int,J: int] : upto(I,J) = upto_aux(I,J,nil_int) ).

% upto_code
tff(fact_9150_upto__aux__def,axiom,
    ! [I: int,J: int,Js: list_int] : upto_aux(I,J,Js) = append_int(upto(I,J),Js) ).

% upto_aux_def
tff(fact_9151_distinct__upto,axiom,
    ! [I: int,J: int] : pp(aa_list_int_bool(distinct_int,upto(I,J))) ).

% distinct_upto
tff(fact_9152_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,I),J) = set_int2(upto(I,J)) ).

% atLeastAtMost_upto
tff(fact_9153_upto__split2,axiom,
    ! [I: int,J: int,K: 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,J),K))
       => ( upto(I,K) = append_int(upto(I,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_9154_upto__split1,axiom,
    ! [I: int,J: int,K: 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,J),K))
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_9155_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,I),J) = set_int2(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_9156_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or6656581121297822940st_int,I),J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_9157_upto_Oelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( upto(X,Xa2) = Y )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
         => ( Y = cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa2)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
tff(fact_9158_upto_Osimps,axiom,
    ! [I: int,J: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
       => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
       => ( upto(I,J) = nil_int ) ) ) ).

% upto.simps
tff(fact_9159_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_9160_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( upto(I,J) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_9161_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or5832277885323065728an_int,I),J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_9162_upto__split3,axiom,
    ! [I: int,J: int,K: 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,J),K))
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K))) ) ) ) ).

% upto_split3
tff(fact_9163_remdups__upt,axiom,
    ! [M: nat,N: nat] : remdups_nat(upt(M,N)) = upt(M,N) ).

% remdups_upt
tff(fact_9164_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( hd_nat(upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_9165_drop__upt,axiom,
    ! [M: nat,I: nat,J: nat] : drop_nat(M,upt(I,J)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M),J) ).

% drop_upt
tff(fact_9166_length__upt,axiom,
    ! [I: nat,J: nat] : size_size_list_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) ).

% length_upt
tff(fact_9167_take__upt,axiom,
    ! [I: 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,I),M)),N))
     => ( take_nat(M,upt(I,N)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)) ) ) ).

% take_upt
tff(fact_9168_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( upt(I,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_9169_sorted__list__of__set__range,axiom,
    ! [M: nat,N: nat] : linord2614967742042102400et_nat(set_or4665077453230672383an_nat(M,N)) = upt(M,N) ).

% sorted_list_of_set_range
tff(fact_9170_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I)) ) ) ).

% upt_eq_Nil_conv
tff(fact_9171_nth__upt,axiom,
    ! [I: nat,K: 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),K)),J))
     => ( aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,upt(I,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).

% nth_upt
tff(fact_9172_upt__rec__numeral,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)))
       => ( upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,N)) = cons_nat(aa_num_nat(numeral_numeral_nat,M),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,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)))
       => ( upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,N)) = nil_nat ) ) ) ).

% upt_rec_numeral
tff(fact_9173_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_9174_greaterThanAtMost__upt,axiom,
    ! [N: nat,M: nat] : set_or6659071591806873216st_nat(N,M) = set_nat2(upt(aa_nat_nat(suc,N),aa_nat_nat(suc,M))) ).

% greaterThanAtMost_upt
tff(fact_9175_atLeast__upt,axiom,
    ! [N: nat] : set_ord_lessThan_nat(N) = set_nat2(upt(zero_zero_nat,N)) ).

% atLeast_upt
tff(fact_9176_atLeastLessThan__upt,axiom,
    ! [I: nat,J: nat] : set_or4665077453230672383an_nat(I,J) = set_nat2(upt(I,J)) ).

% atLeastLessThan_upt
tff(fact_9177_atLeastAtMost__upt,axiom,
    ! [N: nat,M: nat] : set_or1269000886237332187st_nat(N,M) = set_nat2(upt(N,aa_nat_nat(suc,M))) ).

% atLeastAtMost_upt
tff(fact_9178_map__add__upt,axiom,
    ! [N: nat,M: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aca,N),upt(zero_zero_nat,M)) = upt(N,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% map_add_upt
tff(fact_9179_distinct__upt,axiom,
    ! [I: nat,J: nat] : pp(aa_list_nat_bool(distinct_nat,upt(I,J))) ).

% distinct_upt
tff(fact_9180_map__Suc__upt,axiom,
    ! [M: nat,N: nat] : map_nat_nat(suc,upt(M,N)) = upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N)) ).

% map_Suc_upt
tff(fact_9181_greaterThanLessThan__upt,axiom,
    ! [N: nat,M: nat] : set_or5834768355832116004an_nat(N,M) = set_nat2(upt(aa_nat_nat(suc,N),M)) ).

% greaterThanLessThan_upt
tff(fact_9182_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list_nat,Q4: nat] :
      ( ( cons_nat(M,cons_nat(N,Ns)) = upt(M,Q4) )
    <=> ( cons_nat(N,Ns) = upt(aa_nat_nat(suc,M),Q4) ) ) ).

% upt_conv_Cons_Cons
tff(fact_9183_atMost__upto,axiom,
    ! [N: nat] : set_ord_atMost_nat(N) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,N))) ).

% atMost_upto
tff(fact_9184_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_9185_map__decr__upt,axiom,
    ! [M: nat,N: nat] : map_nat_nat(aTP_Lamm_acb,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = upt(M,N) ).

% map_decr_upt
tff(fact_9186_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_9187_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(X,Xs) )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & ( I = X )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_9188_upt__rec,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( upt(I,J) = nil_nat ) ) ) ).

% upt_rec
tff(fact_9189_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( upt(I,aa_nat_nat(suc,J)) = nil_nat ) ) ) ).

% upt_Suc
tff(fact_9190_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_9191_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups4561878855575611511st_nat(upt(M,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ly),set_or4665077453230672383an_nat(M,N)) ) ) ).

% sum_list_upt
tff(fact_9192_card__length__sum__list__rec,axiom,
    ! [M: nat,N4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
     => ( finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_acc(M),N4))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_acd(M),N4)))),finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_ace(M),N4)))) ) ) ).

% card_length_sum_list_rec
tff(fact_9193_card__length__sum__list,axiom,
    ! [M: nat,N4: nat] : finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_acc(M),N4))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),M)),one_one_nat)),N4) ).

% card_length_sum_list
tff(fact_9194_tl__upt,axiom,
    ! [M: nat,N: nat] : tl_nat(upt(M,N)) = upt(aa_nat_nat(suc,M),N) ).

% tl_upt
tff(fact_9195_sorted__wrt__upt,axiom,
    ! [M: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),upt(M,N))) ).

% sorted_wrt_upt
tff(fact_9196_sorted__upt,axiom,
    ! [M: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),upt(M,N))) ).

% sorted_upt
tff(fact_9197_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),Ns))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Ns)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_lis1299702591413342011at_nat(nth_nat,Ns),I))) ) ) ).

% sorted_wrt_less_idx
tff(fact_9198_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),upto(I,J))) ).

% sorted_wrt_upto
tff(fact_9199_sorted__upto,axiom,
    ! [M: int,N: int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),upto(M,N))) ).

% sorted_upto
tff(fact_9200_pairs__le__eq__Sigma,axiom,
    ! [M: nat] : collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ny(M))) = produc457027306803732586at_nat(set_ord_atMost_nat(M),aTP_Lamm_acf(M)) ).

% pairs_le_eq_Sigma
tff(fact_9201_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat)) ).

% natLess_def
tff(fact_9202_finite__vimage__Suc__iff,axiom,
    ! [F3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,vimage_nat_nat(suc,F3)))
    <=> pp(aa_set_nat_bool(finite_finite_nat,F3)) ) ).

% finite_vimage_Suc_iff
tff(fact_9203_vimage__Suc__insert__0,axiom,
    ! [A2: set_nat] : vimage_nat_nat(suc,insert_nat(zero_zero_nat,A2)) = vimage_nat_nat(suc,A2) ).

% vimage_Suc_insert_0
tff(fact_9204_Restr__natLeq,axiom,
    ! [N: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bd,N)),aTP_Lamm_acg(N))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ach(N))) ).

% Restr_natLeq
tff(fact_9205_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_9206_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_eq_nat)) ).

% natLeq_def
tff(fact_9207_Restr__natLeq2,axiom,
    ! [N: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq,N),aTP_Lamm_aci(N))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ach(N))) ).

% Restr_natLeq2
tff(fact_9208_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_9209_natLeq__underS__less,axiom,
    ! [N: nat] : order_underS_nat(bNF_Ca8665028551170535155natLeq,N) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bd,N)) ).

% natLeq_underS_less
tff(fact_9210_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,X) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
         => ( aa_complex_real(arg,Z) = X ) ) ) ) ).

% cis_Arg_unique
tff(fact_9211_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_acj(N),set_ord_lessThan_nat(N)),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_di,N)))) ) ).

% bij_betw_roots_unity
tff(fact_9212_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N: nat] :
      ( ( C != 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(times_times_complex(aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_nat_fun_real_real(root,N),aa_complex_real(real_V1022390504157884413omplex,C)))),aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,N))))),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_di,N))),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_zh(C),N)))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9213_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_ack(Z)) ) ) ) ).

% Arg_def
tff(fact_9214_bij__betw__Suc,axiom,
    ! [M4: set_nat,N4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(bij_betw_nat_nat(suc),M4),N4))
    <=> ( image_nat_nat(suc,M4) = N4 ) ) ).

% bij_betw_Suc
tff(fact_9215_quotient__of__def,axiom,
    ! [X: rat] : quotient_of(X) = the_Pr4378521158711661632nt_int(aTP_Lamm_acl(X)) ).

% quotient_of_def
tff(fact_9216_normalize__stable,axiom,
    ! [Q4: int,P2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,P2),Q4))
       => ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4) ) ) ) ).

% normalize_stable
tff(fact_9217_normalize__coprime,axiom,
    ! [R2: product_prod_int_int,P2: int,Q4: int] :
      ( ( aa_Pro6970504127699319548nt_int(normalize,R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4) )
     => pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,P2),Q4)) ) ).

% normalize_coprime
tff(fact_9218_quotient__of__coprime,axiom,
    ! [R2: rat,P2: int,Q4: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4) )
     => pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,P2),Q4)) ) ).

% quotient_of_coprime
tff(fact_9219_Rat__induct,axiom,
    ! [P: fun_rat_bool,Q4: rat] :
      ( ! [A3: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A3),B3))
           => pp(aa_rat_bool(P,aa_int_rat(aa_int_fun_int_rat(fract,A3),B3))) ) )
     => pp(aa_rat_bool(P,Q4)) ) ).

% Rat_induct
tff(fact_9220_Rat__cases,axiom,
    ! [Q4: rat] :
      ~ ! [A3: int,B3: int] :
          ( ( Q4 = aa_int_rat(aa_int_fun_int_rat(fract,A3),B3) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A3),B3)) ) ) ).

% Rat_cases
tff(fact_9221_Rat__cases__nonzero,axiom,
    ! [Q4: rat] :
      ( ! [A3: int,B3: int] :
          ( ( Q4 = aa_int_rat(aa_int_fun_int_rat(fract,A3),B3) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
           => ( ( A3 != zero_zero_int )
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A3),B3)) ) ) )
     => ( Q4 = zero_zero_rat ) ) ).

% Rat_cases_nonzero
tff(fact_9222_quotient__of__unique,axiom,
    ! [R2: rat] :
    ? [X4: product_prod_int_int] :
      ( ( R2 = aa_int_rat(aa_int_fun_int_rat(fract,product_fst_int_int(X4)),product_snd_int_int(X4)) )
      & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(X4)))
      & pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,product_fst_int_int(X4)),product_snd_int_int(X4)))
      & ! [Y3: product_prod_int_int] :
          ( ( ( R2 = aa_int_rat(aa_int_fun_int_rat(fract,product_fst_int_int(Y3)),product_snd_int_int(Y3)) )
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(Y3)))
            & pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,product_fst_int_int(Y3)),product_snd_int_int(Y3))) )
         => ( Y3 = X4 ) ) ) ).

% quotient_of_unique
tff(fact_9223_coprime__Suc__0__right,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_nat_nat(suc,zero_zero_nat))) ).

% coprime_Suc_0_right
tff(fact_9224_coprime__Suc__0__left,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(suc,zero_zero_nat)),N)) ).

% coprime_Suc_0_left
tff(fact_9225_coprime__diff__one__right__nat,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(algebr934650988132801477me_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).

% coprime_diff_one_right_nat
tff(fact_9226_coprime__diff__one__left__nat,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(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),N)) ) ).

% coprime_diff_one_left_nat
tff(fact_9227_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] : aa_set_nat_nat(gcd_Gcd_nat,set_nat2(Xs)) = aa_nat_nat(aa_lis1299702591413342011at_nat(fold_nat_nat(gcd_gcd_nat),Xs),zero_zero_nat) ).

% Gcd_nat_set_eq_fold
tff(fact_9228_Rats__abs__nat__div__natE,axiom,
    ! [X: real] :
      ( pp(aa_set_real_bool(member_real(X),field_5140801741446780682s_real))
     => ~ ! [M5: nat,N2: nat] :
            ( ( N2 != zero_zero_nat )
           => ( ( aa_real_real(abs_abs_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,M5)),aa_nat_real(semiri5074537144036343181t_real,N2)) )
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,M5),N2)) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_9229_sort__upt,axiom,
    ! [M: nat,N: nat] : linord738340561235409698at_nat(aTP_Lamm_ly,upt(M,N)) = upt(M,N) ).

% sort_upt
tff(fact_9230_sort__upto,axiom,
    ! [I: int,J: int] : linord1735203802627413978nt_int(aTP_Lamm_acm,upto(I,J)) = upto(I,J) ).

% sort_upto
tff(fact_9231_vanishes__mult__bounded,axiom,
    ! [X6: fun_nat_rat,Y5: fun_nat_rat] :
      ( ? [A8: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A8))
          & ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N2))),A8)) )
     => ( vanishes(Y5)
       => vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_acn(X6),Y5)) ) ) ).

% vanishes_mult_bounded
tff(fact_9232_vanishes__def,axiom,
    ! [X6: fun_nat_rat] :
      ( vanishes(X6)
    <=> ! [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
         => ? [K3: nat] :
            ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N3))),R5)) ) ) ) ).

% vanishes_def
tff(fact_9233_vanishesI,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [R3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R3))
         => ? [K6: nat] :
            ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),N2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N2))),R3)) ) )
     => vanishes(X6) ) ).

% vanishesI
tff(fact_9234_vanishesD,axiom,
    ! [X6: fun_nat_rat,R2: rat] :
      ( vanishes(X6)
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R2))
       => ? [K2: nat] :
          ! [N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N5))),R2)) ) ) ) ).

% vanishesD
tff(fact_9235_and__not__num_Opelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( bit_and_not_num(X,Xa2) = Y )
     => ( accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(X,Xa2))
       => ( ( ( X = one )
           => ( ( Xa2 = one )
             => ( ( Y = none_num )
               => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [N2: num] :
                  ( ( Xa2 = aa_num_num(bit0,N2) )
                 => ( ( Y = some_num(one) )
                   => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,aa_num_num(bit0,N2))) ) ) )
           => ( ( ( X = one )
               => ! [N2: num] :
                    ( ( Xa2 = aa_num_num(bit1,N2) )
                   => ( ( Y = none_num )
                     => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,aa_num_num(bit1,N2))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ( ( Xa2 = one )
                     => ( ( Y = some_num(aa_num_num(bit0,M5)) )
                       => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M5),one)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N2: num] :
                          ( ( Xa2 = aa_num_num(bit0,N2) )
                         => ( ( Y = map_option_num_num(bit0,bit_and_not_num(M5,N2)) )
                           => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit0,N2))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa_num_num(bit0,M5) )
                       => ! [N2: num] :
                            ( ( Xa2 = aa_num_num(bit1,N2) )
                           => ( ( Y = map_option_num_num(bit0,bit_and_not_num(M5,N2)) )
                             => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit1,N2))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ( ( Xa2 = one )
                           => ( ( Y = some_num(aa_num_num(bit0,M5)) )
                             => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M5),one)) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N2: num] :
                                ( ( Xa2 = aa_num_num(bit0,N2) )
                               => ( ( Y = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_aby,bit_and_not_num(M5,N2)) )
                                 => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit0,N2))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa_num_num(bit1,M5) )
                             => ! [N2: num] :
                                  ( ( Xa2 = aa_num_num(bit1,N2) )
                                 => ( ( Y = map_option_num_num(bit0,bit_and_not_num(M5,N2)) )
                                   => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
tff(fact_9236_and__num_Opelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( bit_un7362597486090784418nd_num(X,Xa2) = Y )
     => ( accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(X,Xa2))
       => ( ( ( X = one )
           => ( ( Xa2 = one )
             => ( ( Y = some_num(one) )
               => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [N2: num] :
                  ( ( Xa2 = aa_num_num(bit0,N2) )
                 => ( ( Y = none_num )
                   => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,aa_num_num(bit0,N2))) ) ) )
           => ( ( ( X = one )
               => ! [N2: num] :
                    ( ( Xa2 = aa_num_num(bit1,N2) )
                   => ( ( Y = some_num(one) )
                     => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,aa_num_num(bit1,N2))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ( ( Xa2 = one )
                     => ( ( Y = none_num )
                       => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M5),one)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N2: num] :
                          ( ( Xa2 = aa_num_num(bit0,N2) )
                         => ( ( Y = map_option_num_num(bit0,bit_un7362597486090784418nd_num(M5,N2)) )
                           => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit0,N2))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa_num_num(bit0,M5) )
                       => ! [N2: num] :
                            ( ( Xa2 = aa_num_num(bit1,N2) )
                           => ( ( Y = map_option_num_num(bit0,bit_un7362597486090784418nd_num(M5,N2)) )
                             => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit1,N2))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ( ( Xa2 = one )
                           => ( ( Y = some_num(one) )
                             => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M5),one)) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N2: num] :
                                ( ( Xa2 = aa_num_num(bit0,N2) )
                               => ( ( Y = map_option_num_num(bit0,bit_un7362597486090784418nd_num(M5,N2)) )
                                 => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit0,N2))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa_num_num(bit1,M5) )
                             => ! [N2: num] :
                                  ( ( Xa2 = aa_num_num(bit1,N2) )
                                 => ( ( Y = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_aby,bit_un7362597486090784418nd_num(M5,N2)) )
                                   => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
tff(fact_9237_xor__num_Opelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( bit_un2480387367778600638or_num(X,Xa2) = Y )
     => ( accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(X,Xa2))
       => ( ( ( X = one )
           => ( ( Xa2 = one )
             => ( ( Y = none_num )
               => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [N2: num] :
                  ( ( Xa2 = aa_num_num(bit0,N2) )
                 => ( ( Y = some_num(aa_num_num(bit1,N2)) )
                   => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(one,aa_num_num(bit0,N2))) ) ) )
           => ( ( ( X = one )
               => ! [N2: num] :
                    ( ( Xa2 = aa_num_num(bit1,N2) )
                   => ( ( Y = some_num(aa_num_num(bit0,N2)) )
                     => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(one,aa_num_num(bit1,N2))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ( ( Xa2 = one )
                     => ( ( Y = some_num(aa_num_num(bit1,M5)) )
                       => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit0,M5),one)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N2: num] :
                          ( ( Xa2 = aa_num_num(bit0,N2) )
                         => ( ( Y = map_option_num_num(bit0,bit_un2480387367778600638or_num(M5,N2)) )
                           => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit0,N2))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa_num_num(bit0,M5) )
                       => ! [N2: num] :
                            ( ( Xa2 = aa_num_num(bit1,N2) )
                           => ( ( Y = some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M5,N2))) )
                             => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit1,N2))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ( ( Xa2 = one )
                           => ( ( Y = some_num(aa_num_num(bit0,M5)) )
                             => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit1,M5),one)) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N2: num] :
                                ( ( Xa2 = aa_num_num(bit0,N2) )
                               => ( ( Y = some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M5,N2))) )
                                 => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit0,N2))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa_num_num(bit1,M5) )
                             => ! [N2: num] :
                                  ( ( Xa2 = aa_num_num(bit1,N2) )
                                 => ( ( Y = map_option_num_num(bit0,bit_un2480387367778600638or_num(M5,N2)) )
                                   => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
tff(fact_9238_Field__natLeq__on,axiom,
    ! [N: nat] : field_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ach(N)))) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bd,N)) ).

% Field_natLeq_on
tff(fact_9239_wf__less,axiom,
    pp(aa_set3135850010578026034t_bool(wf_nat,collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat)))) ).

% wf_less
tff(fact_9240_cauchyD,axiom,
    ! [X6: fun_nat_rat,R2: rat] :
      ( cauchy(X6)
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R2))
       => ? [K2: nat] :
          ! [M2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),M2))
           => ! [N5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X6,M2)),aa_nat_rat(X6,N5)))),R2)) ) ) ) ) ).

% cauchyD
tff(fact_9241_cauchy__imp__bounded,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ? [B3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
          & ! [N5: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N5))),B3)) ) ) ).

% cauchy_imp_bounded
tff(fact_9242_cauchy__not__vanishes__cases,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ( ~ vanishes(X6)
       => ? [B3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
            & ? [K2: nat] :
                ( ! [N5: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(X6,N5)))) )
                | ! [N5: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_nat_rat(X6,N5))) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
tff(fact_9243_cauchy__not__vanishes,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ( ~ vanishes(X6)
       => ? [B3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
            & ? [K2: nat] :
              ! [N5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N5)))) ) ) ) ) ).

% cauchy_not_vanishes
tff(fact_9244_cauchy__def,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
    <=> ! [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
         => ? [K3: nat] :
            ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),M3))
             => ! [N3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X6,M3)),aa_nat_rat(X6,N3)))),R5)) ) ) ) ) ).

% cauchy_def
tff(fact_9245_cauchyI,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [R3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R3))
         => ? [K6: nat] :
            ! [M5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),M5))
             => ! [N2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),N2))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X6,M5)),aa_nat_rat(X6,N2)))),R3)) ) ) )
     => cauchy(X6) ) ).

% cauchyI
tff(fact_9246_le__Real,axiom,
    ! [X6: fun_nat_rat,Y5: fun_nat_rat] :
      ( cauchy(X6)
     => ( cauchy(Y5)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,real2(X6)),real2(Y5)))
        <=> ! [R5: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
             => ? [K3: nat] :
                ! [N3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,N3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Y5,N3)),R5))) ) ) ) ) ) ).

% le_Real
tff(fact_9247_not__positive__Real,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ( ~ pp(aa_real_bool(positive2,real2(X6)))
      <=> ! [R5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
           => ? [K3: nat] :
              ! [N3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,N3)),R5)) ) ) ) ) ).

% not_positive_Real
tff(fact_9248_less__real__def,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> pp(aa_real_bool(positive2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),X))) ) ).

% less_real_def
tff(fact_9249_positive__Real,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ( pp(aa_real_bool(positive2,real2(X6)))
      <=> ? [R5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
            & ? [K3: nat] :
              ! [N3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(X6,N3))) ) ) ) ) ).

% positive_Real
tff(fact_9250_Real_Opositive_Orep__eq,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(positive2,X))
    <=> ? [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
          & ? [K3: nat] :
            ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(rep_real(X),N3))) ) ) ) ).

% Real.positive.rep_eq
tff(fact_9251_strict__mono__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => order_5726023648592871131at_nat(aa_set6537093552120046177at_nat(infini8530281810654367211te_nat,S2)) ) ).

% strict_mono_enumerate
tff(fact_9252_enumerate__Ex,axiom,
    ! [S2: set_nat,S: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_nat_bool(member_nat(S),S2))
       => ? [N2: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(infini8530281810654367211te_nat,S2),N2) = S ) ) ).

% enumerate_Ex
tff(fact_9253_le__enumerate,axiom,
    ! [S2: set_nat,N: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_set6537093552120046177at_nat(infini8530281810654367211te_nat,S2),N))) ) ).

% le_enumerate
tff(fact_9254_range__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( image_nat_nat(aa_set6537093552120046177at_nat(infini8530281810654367211te_nat,S2),top_top_set_nat) = S2 ) ) ).

% range_enumerate
tff(fact_9255_finite__le__enumerate,axiom,
    ! [S2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_set_nat_nat(finite_card_nat,S2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_set6537093552120046177at_nat(infini8530281810654367211te_nat,S2),N))) ) ) ).

% finite_le_enumerate
tff(fact_9256_bij__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(bij_betw_nat_nat(aa_set6537093552120046177at_nat(infini8530281810654367211te_nat,S2)),top_top_set_nat),S2)) ) ).

% bij_enumerate
tff(fact_9257_Least__eq__0,axiom,
    ! [P: fun_nat_bool] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ord_Least_nat(P) = zero_zero_nat ) ) ).

% Least_eq_0
tff(fact_9258_Least__Suc,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_aco(P))) ) ) ) ).

% Least_Suc
tff(fact_9259_Least__Suc2,axiom,
    ! [P: fun_nat_bool,N: nat,Q: fun_nat_bool,M: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( pp(aa_nat_bool(Q,M))
       => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
         => ( ! [K2: nat] :
                ( pp(aa_nat_bool(P,aa_nat_nat(suc,K2)))
              <=> pp(aa_nat_bool(Q,K2)) )
           => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_9260_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_9261_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_9262_not__mask__negative__int,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,N)),zero_zero_int)) ).

% not_mask_negative_int
tff(fact_9263_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_9264_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_9265_divmod__nat__code,axiom,
    ! [M: nat,N: nat] : divmod_nat(M,N) = produc8678311845419106900er_nat(code_nat_of_integer,code_nat_of_integer,if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,aa_nat_Code_integer(code_integer_of_nat,M)),zero_z3403309356797280102nteger),produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,aa_nat_Code_integer(code_integer_of_nat,N)),zero_z3403309356797280102nteger),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_nat_Code_integer(code_integer_of_nat,M)),code_divmod_abs(aa_nat_Code_integer(code_integer_of_nat,M),aa_nat_Code_integer(code_integer_of_nat,N))))) ).

% divmod_nat_code
tff(fact_9266_integer__of__nat__0,axiom,
    aa_nat_Code_integer(code_integer_of_nat,zero_zero_nat) = zero_z3403309356797280102nteger ).

% integer_of_nat_0
tff(fact_9267_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( last_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),one_one_nat) ) ) ).

% last_upt
tff(fact_9268_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_9269_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_9270_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N: num,Q4: num] :
      ( ( bit_and_not_num(M,N) = some_num(Q4) )
    <=> ( aa_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))) = aa_num_int(numeral_numeral_int,Q4) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_9271_and__not__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num(M,N) = none_num )
    <=> ( aa_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))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_9272_Real_Opositive_Oabs__eq,axiom,
    ! [X: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
     => ( pp(aa_real_bool(positive2,real2(X)))
      <=> ? [R5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
            & ? [K3: nat] :
              ! [N3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(X,N3))) ) ) ) ) ).

% Real.positive.abs_eq
tff(fact_9273_Real_Opositive_Orsp,axiom,
    bNF_re2717234250851183977l_bool(realrel,fequal_bool,aTP_Lamm_acp,aTP_Lamm_acp) ).

% Real.positive.rsp
tff(fact_9274_less__eq__natural_Orsp,axiom,
    bNF_re1480548845665215743t_bool(fequal_nat,bNF_re6223507645442431167l_bool(fequal_nat,fequal_bool),ord_less_eq_nat,ord_less_eq_nat) ).

% less_eq_natural.rsp
tff(fact_9275_less__natural_Orsp,axiom,
    bNF_re1480548845665215743t_bool(fequal_nat,bNF_re6223507645442431167l_bool(fequal_nat,fequal_bool),ord_less_nat,ord_less_nat) ).

% less_natural.rsp
tff(fact_9276_less__integer_Orsp,axiom,
    bNF_re5780769757191362231t_bool(fequal_int,bNF_re9007465950226028919l_bool(fequal_int,fequal_bool),ord_less_int,ord_less_int) ).

% less_integer.rsp
tff(fact_9277_Real_Opositive_Otransfer,axiom,
    bNF_re1458715823057739248l_bool(pcr_real,fequal_bool,aTP_Lamm_acp,positive2) ).

% Real.positive.transfer
tff(fact_9278_plus__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_acq,plus_plus_rat) ).

% plus_rat.transfer
tff(fact_9279_one__rat_Otransfer,axiom,
    pp(aa_rat_bool(aa_Pro6332413336323567195t_bool(pcr_rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)),one_one_rat)) ).

% one_rat.transfer
tff(fact_9280_zero__rat_Otransfer,axiom,
    pp(aa_rat_bool(aa_Pro6332413336323567195t_bool(pcr_rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)),zero_zero_rat)) ).

% zero_rat.transfer
tff(fact_9281_Fract_Otransfer,axiom,
    bNF_re2486238095283335024nt_rat(fequal_int,bNF_re2214769303045360666nt_rat(fequal_int,pcr_rat),aTP_Lamm_acr,fract) ).

% Fract.transfer
tff(fact_9282_uminus__rat_Otransfer,axiom,
    pp(aa_fun_rat_rat_bool(aa_fun2103967415933125506t_bool(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_acs),uminus_uminus_rat)) ).

% uminus_rat.transfer
tff(fact_9283_times__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_act,times_times_rat) ).

% times_rat.transfer
tff(fact_9284_Rat_Opositive_Otransfer,axiom,
    bNF_re3535474257078034868l_bool(pcr_rat,fequal_bool,aTP_Lamm_acu,positive) ).

% Rat.positive.transfer
tff(fact_9285_inverse__rat_Otransfer,axiom,
    pp(aa_fun_rat_rat_bool(aa_fun2103967415933125506t_bool(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_acv),inverse_inverse_rat)) ).

% inverse_rat.transfer
tff(fact_9286_times__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_zr),times_times_int) ).

% times_int.transfer
tff(fact_9287_minus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_aaa),minus_minus_int) ).

% minus_int.transfer
tff(fact_9288_zero__int_Otransfer,axiom,
    pp(aa_int_bool(aa_Pro2928840338864870335t_bool(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),zero_zero_int)) ).

% zero_int.transfer
tff(fact_9289_int__transfer,axiom,
    bNF_re6830278522597306478at_int(fequal_nat,pcr_int,aTP_Lamm_acw,semiri1314217659103216013at_int) ).

% int_transfer
tff(fact_9290_uminus__int_Otransfer,axiom,
    pp(aa_fun_int_int_bool(aa_fun7527181275634263258t_bool(bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc2626176000494625587at_nat(aTP_Lamm_zs)),uminus_uminus_int)) ).

% uminus_int.transfer
tff(fact_9291_one__int_Otransfer,axiom,
    pp(aa_int_bool(aa_Pro2928840338864870335t_bool(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),one_one_int)) ).

% one_int.transfer
tff(fact_9292_less__int_Otransfer,axiom,
    bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_zu),ord_less_int) ).

% less_int.transfer
tff(fact_9293_less__eq__int_Otransfer,axiom,
    bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_zw),ord_less_eq_int) ).

% less_eq_int.transfer
tff(fact_9294_plus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_zy),plus_plus_int) ).

% plus_int.transfer
tff(fact_9295_times__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_zr),produc4528391901023682385at_nat(aTP_Lamm_zr)) ).

% times_int.rsp
tff(fact_9296_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),V)))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),V) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,U),Y) ) ) ).

% intrel_iff
tff(fact_9297_uminus__int_Orsp,axiom,
    pp(aa_fun4519828053337421206t_bool(aa_fun2399787666232550060t_bool(bNF_re2241393799969408733at_nat(intrel,intrel),produc2626176000494625587at_nat(aTP_Lamm_zs)),produc2626176000494625587at_nat(aTP_Lamm_zs))) ).

% uminus_int.rsp
tff(fact_9298_zero__int_Orsp,axiom,
    pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat))) ).

% zero_int.rsp
tff(fact_9299_one__int_Orsp,axiom,
    pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat))) ).

% one_int.rsp
tff(fact_9300_less__int_Orsp,axiom,
    bNF_re2219331435504962769t_bool(intrel,bNF_re3189209160384574813l_bool(intrel,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_zu),produc5023667602222594023t_bool(aTP_Lamm_zu)) ).

% less_int.rsp
tff(fact_9301_less__eq__int_Orsp,axiom,
    bNF_re2219331435504962769t_bool(intrel,bNF_re3189209160384574813l_bool(intrel,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_zw),produc5023667602222594023t_bool(aTP_Lamm_zw)) ).

% less_eq_int.rsp
tff(fact_9302_plus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_zy),produc4528391901023682385at_nat(aTP_Lamm_zy)) ).

% plus_int.rsp
tff(fact_9303_minus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_aaa),produc4528391901023682385at_nat(aTP_Lamm_aaa)) ).

% minus_int.rsp
tff(fact_9304_plus__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_acq,aTP_Lamm_acq) ).

% plus_rat.rsp
tff(fact_9305_inverse__rat_Orsp,axiom,
    pp(aa_fun1403452029515102102t_bool(aa_fun8298786006102058668t_bool(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_acv),aTP_Lamm_acv)) ).

% inverse_rat.rsp
tff(fact_9306_zero__rat_Orsp,axiom,
    pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int))) ).

% zero_rat.rsp
tff(fact_9307_one__rat_Orsp,axiom,
    pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int))) ).

% one_rat.rsp
tff(fact_9308_Fract_Orsp,axiom,
    bNF_re4808202430149235895nt_int(fequal_int,bNF_re6250860962936578807nt_int(fequal_int,ratrel),aTP_Lamm_acr,aTP_Lamm_acr) ).

% Fract.rsp
tff(fact_9309_uminus__rat_Orsp,axiom,
    pp(aa_fun1403452029515102102t_bool(aa_fun8298786006102058668t_bool(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_acs),aTP_Lamm_acs)) ).

% uminus_rat.rsp
tff(fact_9310_times__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_act,aTP_Lamm_act) ).

% times_rat.rsp
tff(fact_9311_Rat_Opositive_Orsp,axiom,
    bNF_re1394333895916556125l_bool(ratrel,fequal_bool,aTP_Lamm_acu,aTP_Lamm_acu) ).

% Rat.positive.rsp
tff(fact_9312_plus__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod_int_int,X: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,Xa2),Xa2))
     => ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xa2)),aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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,product_fst_int_int(Xa2)),product_snd_int_int(X))),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X)),product_snd_int_int(Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Xa2)),product_snd_int_int(X)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_9313_inverse__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
     => ( aa_rat_rat(inverse_inverse_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_Pro6970504127699319548nt_int(if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,product_fst_int_int(X)),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_snd_int_int(X)),product_fst_int_int(X)))) ) ) ).

% inverse_rat.abs_eq
tff(fact_9314_one__rat__def,axiom,
    one_one_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)) ).

% one_rat_def
tff(fact_9315_Fract_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa_int_rat(aa_int_fun_int_rat(fract,Xa2),X) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_Pro6970504127699319548nt_int(if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa2),X))) ).

% Fract.abs_eq
tff(fact_9316_zero__rat__def,axiom,
    zero_zero_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)) ).

% zero_rat_def
tff(fact_9317_uminus__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
     => ( aa_rat_rat(uminus_uminus_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,product_fst_int_int(X))),product_snd_int_int(X))) ) ) ).

% uminus_rat.abs_eq
tff(fact_9318_times__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod_int_int,X: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,Xa2),Xa2))
     => ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xa2)),aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xa2)),product_fst_int_int(X))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Xa2)),product_snd_int_int(X)))) ) ) ) ).

% times_rat.abs_eq
tff(fact_9319_Rat_Opositive_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
     => ( pp(aa_rat_bool(positive,aa_Pro6166618844887768661nt_rat(abs_Rat,X)))
      <=> 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,product_fst_int_int(X)),product_snd_int_int(X)))) ) ) ).

% Rat.positive.abs_eq
tff(fact_9320_inverse__rat__def,axiom,
    inverse_inverse_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_acv) ).

% inverse_rat_def
tff(fact_9321_uminus__rat__def,axiom,
    uminus_uminus_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_acs) ).

% uminus_rat_def
tff(fact_9322_plus__rat__def,axiom,
    plus_plus_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_acq) ).

% plus_rat_def
tff(fact_9323_times__rat__def,axiom,
    times_times_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_act) ).

% times_rat_def
tff(fact_9324_natLeq__on__wo__rel,axiom,
    ! [N: nat] : pp(aa_set3135850010578026034t_bool(bNF_We3818239936649020644el_nat,collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ach(N))))) ).

% natLeq_on_wo_rel
tff(fact_9325_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N)),transi2905341329935302413cl_nat(pred_nat)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% pred_nat_trancl_eq_le
tff(fact_9326_less__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N)),transi6264000038957366511cl_nat(pred_nat)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% less_eq
tff(fact_9327_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(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_9328_concat__bit__0,axiom,
    ! [K: int,L: int] : aa_int_int(bit_concat_bit(zero_zero_nat,K),L) = L ).

% concat_bit_0
tff(fact_9329_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_concat_bit(N,K),L)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ).

% concat_bit_negative_iff
tff(fact_9330_gcd__nat_Oordering__top__axioms,axiom,
    pp(aa_nat_bool(ordering_top_nat(dvd_dvd_nat,aTP_Lamm_zp),zero_zero_nat)) ).

% gcd_nat.ordering_top_axioms
tff(fact_9331_bot__nat__0_Oordering__top__axioms,axiom,
    pp(aa_nat_bool(ordering_top_nat(aTP_Lamm_be,aTP_Lamm_bd),zero_zero_nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_9332_ATP_Olambda__1,axiom,
    ! [Uu: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aTP_Lamm_acv,Uu) = aa_Pro6970504127699319548nt_int(if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,product_fst_int_int(Uu)),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_snd_int_int(Uu)),product_fst_int_int(Uu))) ).

% ATP.lambda_1
tff(fact_9333_ATP_Olambda__2,axiom,
    ! [Uu: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aTP_Lamm_acs,Uu) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,product_fst_int_int(Uu))),product_snd_int_int(Uu)) ).

% ATP.lambda_2
tff(fact_9334_ATP_Olambda__3,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_acb,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_3
tff(fact_9335_ATP_Olambda__4,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_mt,Uu) = aa_complex_complex(plus_plus_complex(Uu),one_one_complex) ).

% ATP.lambda_4
tff(fact_9336_ATP_Olambda__5,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_mx,Uu) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),one_one_real) ).

% ATP.lambda_5
tff(fact_9337_ATP_Olambda__6,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_mu,Uu) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),one_one_rat) ).

% ATP.lambda_6
tff(fact_9338_ATP_Olambda__7,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_mv,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_7
tff(fact_9339_ATP_Olambda__8,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_mw,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).

% ATP.lambda_8
tff(fact_9340_ATP_Olambda__9,axiom,
    ! [Uu: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_acw,Uu) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),zero_zero_nat) ).

% ATP.lambda_9
tff(fact_9341_ATP_Olambda__10,axiom,
    ! [Uu: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_acu,Uu))
    <=> 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,product_fst_int_int(Uu)),product_snd_int_int(Uu)))) ) ).

% ATP.lambda_10
tff(fact_9342_ATP_Olambda__11,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_aap,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_aao,Uu)) ).

% ATP.lambda_11
tff(fact_9343_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_abc,Uu) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ey,Uu)) ).

% ATP.lambda_12
tff(fact_9344_ATP_Olambda__13,axiom,
    ! [Uu: real] : aa_rea5428163979925208277l_real(aTP_Lamm_abi,Uu) = princi6114159922880469582l_real(collec3799799289383736868l_real(produc5257810830037637649l_bool(aTP_Lamm_abh(Uu)))) ).

% ATP.lambda_13
tff(fact_9345_ATP_Olambda__14,axiom,
    ! [Uu: real] : aa_rea5117059003799598681omplex(aTP_Lamm_abg,Uu) = princi3496590319149328850omplex(collec8663557070575231912omplex(produc9214369023355538061x_bool(aTP_Lamm_abf(Uu)))) ).

% ATP.lambda_14
tff(fact_9346_ATP_Olambda__15,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_aby,Uu) = some_num(aa_num_num(bit1,Uu)) ).

% ATP.lambda_15
tff(fact_9347_ATP_Olambda__16,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_abl,Uu) = some_num(aa_num_num(bit0,Uu)) ).

% ATP.lambda_16
tff(fact_9348_ATP_Olambda__17,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_yt,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,Uu)) ).

% ATP.lambda_17
tff(fact_9349_ATP_Olambda__18,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_yu,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_18
tff(fact_9350_ATP_Olambda__19,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_fm,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_19
tff(fact_9351_ATP_Olambda__20,axiom,
    ! [Uu: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aTP_Lamm_acp,Uu))
    <=> ? [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
          & ? [K3: nat] :
            ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(Uu,N3))) ) ) ) ).

% ATP.lambda_20
tff(fact_9352_ATP_Olambda__21,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_abz,Uu))
    <=> ? [I4: int,N3: nat] :
          ( ( Uu = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I4)),aa_nat_real(semiri5074537144036343181t_real,N3)) )
          & ( N3 != zero_zero_nat ) ) ) ).

% ATP.lambda_21
tff(fact_9353_ATP_Olambda__22,axiom,
    ! [Uu: nat] : aa_nat_option_num(aTP_Lamm_abn,Uu) = some_num(one) ).

% ATP.lambda_22
tff(fact_9354_ATP_Olambda__23,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_abr(Uu),Uua) = case_num_option_num(some_num(one),aTP_Lamm_abp(Uua),aTP_Lamm_abq(Uua),Uu) ).

% ATP.lambda_23
tff(fact_9355_ATP_Olambda__24,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_tw(Uu),Uua) = aa_complex_complex(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_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_complex(power_power_complex(Uu),Uua))),zero_zero_complex) ).

% ATP.lambda_24
tff(fact_9356_ATP_Olambda__25,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tv,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_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua))),zero_zero_real) ).

% ATP.lambda_25
tff(fact_9357_ATP_Olambda__26,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ps(Uu),Uua) = aa_complex_complex(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_26
tff(fact_9358_ATP_Olambda__27,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_pv,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_27
tff(fact_9359_ATP_Olambda__28,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_pt(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_28
tff(fact_9360_ATP_Olambda__29,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aTP_Lamm_pu(Uu),Uua) = aa_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_29
tff(fact_9361_ATP_Olambda__30,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_uh(Uu),Uua) = aa_complex_complex(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),zero_zero_complex),aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_complex(power_power_complex(Uu),Uua))) ).

% ATP.lambda_30
tff(fact_9362_ATP_Olambda__31,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ug,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_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua))) ).

% ATP.lambda_31
tff(fact_9363_ATP_Olambda__32,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_zi,Uu),Uua) = aa_int_int(if_int(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Uu))),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))),code_int_of_integer(Uu))),one_one_int)) ).

% ATP.lambda_32
tff(fact_9364_ATP_Olambda__33,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Extended_enat(aa_nat2025370463774033377d_enat(aTP_Lamm_mi,Uu),Uua) = if_Extended_enat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uu)),plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uu)),one_on7984719198319812577d_enat)) ).

% ATP.lambda_33
tff(fact_9365_ATP_Olambda__34,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_mh,Uu),Uua) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Uu))),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Uu))),one_one_complex)) ).

% ATP.lambda_34
tff(fact_9366_ATP_Olambda__35,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_mm,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat)),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,Uu))),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,Uu))),one_one_real)) ).

% ATP.lambda_35
tff(fact_9367_ATP_Olambda__36,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_mj,Uu),Uua) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uu))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uu))),one_one_rat)) ).

% ATP.lambda_36
tff(fact_9368_ATP_Olambda__37,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mk,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_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(semiri1316708129612266289at_nat,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))),aa_nat_nat(semiri1316708129612266289at_nat,Uu))),one_one_nat)) ).

% ATP.lambda_37
tff(fact_9369_ATP_Olambda__38,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ml,Uu),Uua) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),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(semiri1314217659103216013at_int,Uu))),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_nat_int(semiri1314217659103216013at_int,Uu))),one_one_int)) ).

% ATP.lambda_38
tff(fact_9370_ATP_Olambda__39,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_zl,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_integer_nat(code_nat_of_integer,Uu)),aa_Code_integer_nat(code_nat_of_integer,Uu))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_integer_nat(code_nat_of_integer,Uu)),aa_Code_integer_nat(code_nat_of_integer,Uu))),one_one_nat)) ).

% ATP.lambda_39
tff(fact_9371_ATP_Olambda__40,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ys,Uu),Uua) = aa_Pro6970504127699319548nt_int(if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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_40
tff(fact_9372_ATP_Olambda__41,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_acr,Uu),Uua) = aa_Pro6970504127699319548nt_int(if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uua),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),Uua)) ).

% ATP.lambda_41
tff(fact_9373_ATP_Olambda__42,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_po(Uu),Uua) = aa_complex_complex(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_42
tff(fact_9374_ATP_Olambda__43,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_pr,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_43
tff(fact_9375_ATP_Olambda__44,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_pp(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_44
tff(fact_9376_ATP_Olambda__45,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aTP_Lamm_pq(Uu),Uua) = aa_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_45
tff(fact_9377_ATP_Olambda__46,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_abs,Uu),Uua) = case_nat_option_num(none_num,aTP_Lamm_abr(Uua),Uu) ).

% ATP.lambda_46
tff(fact_9378_ATP_Olambda__47,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_abp(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_abl,bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_47
tff(fact_9379_ATP_Olambda__48,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_abo(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_abl,bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_48
tff(fact_9380_ATP_Olambda__49,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_oc(Uu),Uua) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ob(Uu),Uua),set_ord_atMost_nat(Uua)) ).

% ATP.lambda_49
tff(fact_9381_ATP_Olambda__50,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_oa(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nz(Uu),Uua)),set_ord_atMost_nat(Uua)) ).

% ATP.lambda_50
tff(fact_9382_ATP_Olambda__51,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_acq,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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,product_fst_int_int(Uu)),product_snd_int_int(Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uua)),product_snd_int_int(Uu)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Uu)),product_snd_int_int(Uua))) ).

% ATP.lambda_51
tff(fact_9383_ATP_Olambda__52,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tl,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_52
tff(fact_9384_ATP_Olambda__53,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_qd(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_53
tff(fact_9385_ATP_Olambda__54,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_qb(Uu),Uua) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)),aa_nat_complex(semiri8010041392384452111omplex,Uua))),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_54
tff(fact_9386_ATP_Olambda__55,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_qf,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_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_55
tff(fact_9387_ATP_Olambda__56,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_qc(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,Uua))),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_56
tff(fact_9388_ATP_Olambda__57,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aTP_Lamm_qe(Uu),Uua) = 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(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Uua)),aa_nat_int(semiri1314217659103216013at_int,Uua))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_57
tff(fact_9389_ATP_Olambda__58,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_uj,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_58
tff(fact_9390_ATP_Olambda__59,axiom,
    ! [Uu: complex,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ack(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_59
tff(fact_9391_ATP_Olambda__60,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_act,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu)),product_fst_int_int(Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Uu)),product_snd_int_int(Uua))) ).

% ATP.lambda_60
tff(fact_9392_ATP_Olambda__61,axiom,
    ! [Uu: real,Uua: int] :
      ( pp(aa_int_bool(aa_real_fun_int_bool(aTP_Lamm_vs,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_61
tff(fact_9393_ATP_Olambda__62,axiom,
    ! [Uu: rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_vy(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_62
tff(fact_9394_ATP_Olambda__63,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aao,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_63
tff(fact_9395_ATP_Olambda__64,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aaq,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_64
tff(fact_9396_ATP_Olambda__65,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aau(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_65
tff(fact_9397_ATP_Olambda__66,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_py(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_66
tff(fact_9398_ATP_Olambda__67,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_pw(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_67
tff(fact_9399_ATP_Olambda__68,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_qa,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(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_68
tff(fact_9400_ATP_Olambda__69,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_px(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_69
tff(fact_9401_ATP_Olambda__70,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aTP_Lamm_pz(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Uua)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_70
tff(fact_9402_ATP_Olambda__71,axiom,
    ! [Uu: rat,Uua: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_acl(Uu),Uua))
    <=> ( ( Uu = aa_int_rat(aa_int_fun_int_rat(fract,product_fst_int_int(Uua)),product_snd_int_int(Uua)) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(Uua)))
        & pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,product_fst_int_int(Uua)),product_snd_int_int(Uua))) ) ) ).

% ATP.lambda_71
tff(fact_9403_ATP_Olambda__72,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_rg(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(Uu),Uua)),aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Uu),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_72
tff(fact_9404_ATP_Olambda__73,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ri,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_73
tff(fact_9405_ATP_Olambda__74,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_rh(Uu),Uua) = aa_rat_rat(aa_rat_fun_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_74
tff(fact_9406_ATP_Olambda__75,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_zp,Uu),Uua))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uu),Uua))
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_75
tff(fact_9407_ATP_Olambda__76,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tb,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_76
tff(fact_9408_ATP_Olambda__77,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abj,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uua)),Uua) ).

% ATP.lambda_77
tff(fact_9409_ATP_Olambda__78,axiom,
    ! [Uu: nat,Uua: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_di,Uu),Uua))
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_78
tff(fact_9410_ATP_Olambda__79,axiom,
    ! [Uu: nat,Uua: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aTP_Lamm_dh,Uu),Uua))
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_79
tff(fact_9411_ATP_Olambda__80,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lq(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_80
tff(fact_9412_ATP_Olambda__81,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lo(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,aa_nat_nat(suc,Uua))),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_81
tff(fact_9413_ATP_Olambda__82,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lp(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_82
tff(fact_9414_ATP_Olambda__83,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_yb(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu,Uua)),set_or4665077453230672383an_nat(zero_zero_nat,Uua)) ).

% ATP.lambda_83
tff(fact_9415_ATP_Olambda__84,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_gv(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uua)),aa_nat_complex(power_power_complex(zero_zero_complex),Uua)) ).

% ATP.lambda_84
tff(fact_9416_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_gx(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_85
tff(fact_9417_ATP_Olambda__86,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_gw(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ).

% ATP.lambda_86
tff(fact_9418_ATP_Olambda__87,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_uq(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)),aa_nat_int(power_power_int(zero_zero_int),Uua)) ).

% ATP.lambda_87
tff(fact_9419_ATP_Olambda__88,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lx(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_88
tff(fact_9420_ATP_Olambda__89,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lv(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_89
tff(fact_9421_ATP_Olambda__90,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lw(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_90
tff(fact_9422_ATP_Olambda__91,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_lr(Uu),Uua) = aa_complex_complex(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_91
tff(fact_9423_ATP_Olambda__92,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lu(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_92
tff(fact_9424_ATP_Olambda__93,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ls(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_93
tff(fact_9425_ATP_Olambda__94,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lt(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_94
tff(fact_9426_ATP_Olambda__95,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nf(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_95
tff(fact_9427_ATP_Olambda__96,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_nd(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_96
tff(fact_9428_ATP_Olambda__97,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ne(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_97
tff(fact_9429_ATP_Olambda__98,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_le(Uu),Uua))
    <=> ( aa_complex_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_98
tff(fact_9430_ATP_Olambda__99,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_lh(Uu),Uua))
    <=> ( aa_complex_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_99
tff(fact_9431_ATP_Olambda__100,axiom,
    ! [Uu: fun_real_complex,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_la(Uu),Uua))
    <=> ( aa_real_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_100
tff(fact_9432_ATP_Olambda__101,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_lc(Uu),Uua))
    <=> ( aa_real_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_101
tff(fact_9433_ATP_Olambda__102,axiom,
    ! [Uu: fun_real_rat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_lf(Uu),Uua))
    <=> ( aa_real_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_102
tff(fact_9434_ATP_Olambda__103,axiom,
    ! [Uu: fun_real_nat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_li(Uu),Uua))
    <=> ( aa_real_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_103
tff(fact_9435_ATP_Olambda__104,axiom,
    ! [Uu: fun_int_complex,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_lb(Uu),Uua))
    <=> ( aa_int_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_104
tff(fact_9436_ATP_Olambda__105,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ld(Uu),Uua))
    <=> ( aa_int_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_105
tff(fact_9437_ATP_Olambda__106,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_lg(Uu),Uua))
    <=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_106
tff(fact_9438_ATP_Olambda__107,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_lj(Uu),Uua))
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_107
tff(fact_9439_ATP_Olambda__108,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_nh(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Uua))),aa_nat_complex(power_power_complex(Uu),Uua)) ).

% ATP.lambda_108
tff(fact_9440_ATP_Olambda__109,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ng,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_109
tff(fact_9441_ATP_Olambda__110,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_aad,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_110
tff(fact_9442_ATP_Olambda__111,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uf,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_111
tff(fact_9443_ATP_Olambda__112,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ui,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_112
tff(fact_9444_ATP_Olambda__113,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_rj(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),Uu) ).

% ATP.lambda_113
tff(fact_9445_ATP_Olambda__114,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_rl,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),Uu) ).

% ATP.lambda_114
tff(fact_9446_ATP_Olambda__115,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_rk(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu) ).

% ATP.lambda_115
tff(fact_9447_ATP_Olambda__116,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_fp,Uu),Uua) = produc1086072967326762835nteger(Uu,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),one_one_Code_integer)) ).

% ATP.lambda_116
tff(fact_9448_ATP_Olambda__117,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_fo,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(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_117
tff(fact_9449_ATP_Olambda__118,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_fn,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),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)) ).

% ATP.lambda_118
tff(fact_9450_ATP_Olambda__119,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_fl,Uu),Uua) = produc1086072967326762835nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_119
tff(fact_9451_ATP_Olambda__120,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_fk,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_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)) ).

% ATP.lambda_120
tff(fact_9452_ATP_Olambda__121,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_fj,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_121
tff(fact_9453_ATP_Olambda__122,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_vv(Uu),Uua) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) ).

% ATP.lambda_122
tff(fact_9454_ATP_Olambda__123,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vx,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_123
tff(fact_9455_ATP_Olambda__124,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_vw(Uu),Uua) = aa_rat_rat(aa_rat_fun_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_124
tff(fact_9456_ATP_Olambda__125,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: produc3843707927480180839at_nat] :
      ( pp(aa_Pro1809015098595223084t_bool(aTP_Lamm_ut(Uu),Uua))
    <=> ( Uu = insert9069300056098147895at_nat(Uua,bot_bo228742789529271731at_nat) ) ) ).

% ATP.lambda_125
tff(fact_9457_ATP_Olambda__126,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aTP_Lamm_uu(Uu),Uua))
    <=> ( Uu = insert8211810215607154385at_nat(Uua,bot_bo2099793752762293965at_nat) ) ) ).

% ATP.lambda_126
tff(fact_9458_ATP_Olambda__127,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_us(Uu),Uua))
    <=> ( Uu = insert5033312907999012233nt_int(Uua,bot_bo1796632182523588997nt_int) ) ) ).

% ATP.lambda_127
tff(fact_9459_ATP_Olambda__128,axiom,
    ! [Uu: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_uv(Uu),Uua))
    <=> ( Uu = insert_real(Uua,bot_bot_set_real) ) ) ).

% ATP.lambda_128
tff(fact_9460_ATP_Olambda__129,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_uw,Uu),Uua))
    <=> ( Uu = insert_nat(Uua,bot_bot_set_nat) ) ) ).

% ATP.lambda_129
tff(fact_9461_ATP_Olambda__130,axiom,
    ! [Uu: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ux(Uu),Uua))
    <=> ( Uu = insert_int(Uua,bot_bot_set_int) ) ) ).

% ATP.lambda_130
tff(fact_9462_ATP_Olambda__131,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yf,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_131
tff(fact_9463_ATP_Olambda__132,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_yd(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_132
tff(fact_9464_ATP_Olambda__133,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yg,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_133
tff(fact_9465_ATP_Olambda__134,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_ye(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_134
tff(fact_9466_ATP_Olambda__135,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_nu,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),Uu)) ) ).

% ATP.lambda_135
tff(fact_9467_ATP_Olambda__136,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aTP_Lamm_az(Uu),Uua))
    <=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(Uua),Uu)) ) ).

% ATP.lambda_136
tff(fact_9468_ATP_Olambda__137,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( pp(aa_set_complex_bool(aTP_Lamm_ay(Uu),Uua))
    <=> pp(aa_set_complex_bool(ord_le211207098394363844omplex(Uua),Uu)) ) ).

% ATP.lambda_137
tff(fact_9469_ATP_Olambda__138,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_ax,Uu),Uua))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Uua),Uu)) ) ).

% ATP.lambda_138
tff(fact_9470_ATP_Olambda__139,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( pp(aa_set_int_bool(aTP_Lamm_ba(Uu),Uua))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(Uua),Uu)) ) ).

% ATP.lambda_139
tff(fact_9471_ATP_Olambda__140,axiom,
    ! [Uu: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_nv,Uu),Uua))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Uua),Uu)) ) ).

% ATP.lambda_140
tff(fact_9472_ATP_Olambda__141,axiom,
    ! [Uu: num,Uua: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(aTP_Lamm_nw,Uu),Uua))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Uua),Uu)) ) ).

% ATP.lambda_141
tff(fact_9473_ATP_Olambda__142,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_be,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uu)) ) ).

% ATP.lambda_142
tff(fact_9474_ATP_Olambda__143,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_nx,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uua),Uu)) ) ).

% ATP.lambda_143
tff(fact_9475_ATP_Olambda__144,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aac,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_144
tff(fact_9476_ATP_Olambda__145,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_nc,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uua),Uu)) ) ).

% ATP.lambda_145
tff(fact_9477_ATP_Olambda__146,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_my,Uu),Uua))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,Uua),Uu)) ) ).

% ATP.lambda_146
tff(fact_9478_ATP_Olambda__147,axiom,
    ! [Uu: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_mz,Uu),Uua))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Uua),Uu)) ) ).

% ATP.lambda_147
tff(fact_9479_ATP_Olambda__148,axiom,
    ! [Uu: num,Uua: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(aTP_Lamm_na,Uu),Uua))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Uua),Uu)) ) ).

% ATP.lambda_148
tff(fact_9480_ATP_Olambda__149,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bd,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu)) ) ).

% ATP.lambda_149
tff(fact_9481_ATP_Olambda__150,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_nb,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uu)) ) ).

% ATP.lambda_150
tff(fact_9482_ATP_Olambda__151,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aav,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_151
tff(fact_9483_ATP_Olambda__152,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aab,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_152
tff(fact_9484_ATP_Olambda__153,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_aae,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_153
tff(fact_9485_ATP_Olambda__154,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aca,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_154
tff(fact_9486_ATP_Olambda__155,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_aaf,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu) ).

% ATP.lambda_155
tff(fact_9487_ATP_Olambda__156,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(aTP_Lamm_ez,Uu),Uua))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,Uua),Uu)) ) ).

% ATP.lambda_156
tff(fact_9488_ATP_Olambda__157,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_ex,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,Uua),Uu)) ) ).

% ATP.lambda_157
tff(fact_9489_ATP_Olambda__158,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ey,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uua),Uu)) ) ).

% ATP.lambda_158
tff(fact_9490_ATP_Olambda__159,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_fa,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uua),Uu)) ) ).

% ATP.lambda_159
tff(fact_9491_ATP_Olambda__160,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_zs,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu) ).

% ATP.lambda_160
tff(fact_9492_ATP_Olambda__161,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aax,Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_161
tff(fact_9493_ATP_Olambda__162,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aTP_Lamm_a(Uu),Uua))
    <=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Uua),Uu)) ) ).

% ATP.lambda_162
tff(fact_9494_ATP_Olambda__163,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_163
tff(fact_9495_ATP_Olambda__164,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_ab(Uu),Uua))
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_164
tff(fact_9496_ATP_Olambda__165,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_ac(Uu),Uua))
    <=> pp(aa_set_set_nat_bool(member_set_nat(Uua),Uu)) ) ).

% ATP.lambda_165
tff(fact_9497_ATP_Olambda__166,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_ad,Uu),Uua))
    <=> pp(aa_set_nat_bool(member_nat(Uua),Uu)) ) ).

% ATP.lambda_166
tff(fact_9498_ATP_Olambda__167,axiom,
    ! [Uu: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ae(Uu),Uua))
    <=> pp(aa_set_int_bool(member_int(Uua),Uu)) ) ).

% ATP.lambda_167
tff(fact_9499_ATP_Olambda__168,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: produc3843707927480180839at_nat] :
      ( pp(aa_Pro1809015098595223084t_bool(aTP_Lamm_fu(Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_168
tff(fact_9500_ATP_Olambda__169,axiom,
    ! [Uu: product_prod_nat_nat,Uua: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(aTP_Lamm_fx,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_169
tff(fact_9501_ATP_Olambda__170,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_fy,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_170
tff(fact_9502_ATP_Olambda__171,axiom,
    ! [Uu: list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_fv(Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_171
tff(fact_9503_ATP_Olambda__172,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_fw,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_172
tff(fact_9504_ATP_Olambda__173,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_fz,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_173
tff(fact_9505_ATP_Olambda__174,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ga,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_174
tff(fact_9506_ATP_Olambda__175,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_aci(Uu),Uua) = order_underS_nat(bNF_Ca8665028551170535155natLeq,Uu) ).

% ATP.lambda_175
tff(fact_9507_ATP_Olambda__176,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_qp(Uu),Uua) = aa_nat_complex(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_176
tff(fact_9508_ATP_Olambda__177,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ln(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_177
tff(fact_9509_ATP_Olambda__178,axiom,
    ! [Uu: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_aco(Uu),Uua))
    <=> pp(aa_nat_bool(Uu,aa_nat_nat(suc,Uua))) ) ).

% ATP.lambda_178
tff(fact_9510_ATP_Olambda__179,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lk(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_179
tff(fact_9511_ATP_Olambda__180,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_lm(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_180
tff(fact_9512_ATP_Olambda__181,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ll(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_181
tff(fact_9513_ATP_Olambda__182,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_abq(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_182
tff(fact_9514_ATP_Olambda__183,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_abm(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu))) ).

% ATP.lambda_183
tff(fact_9515_ATP_Olambda__184,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_aaa,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_zz(Uu),Uua)) ).

% ATP.lambda_184
tff(fact_9516_ATP_Olambda__185,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_zy,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_zx(Uu),Uua)) ).

% ATP.lambda_185
tff(fact_9517_ATP_Olambda__186,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_zw,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_zv(Uu),Uua)) ).

% ATP.lambda_186
tff(fact_9518_ATP_Olambda__187,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_zu,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_zt(Uu),Uua)) ).

% ATP.lambda_187
tff(fact_9519_ATP_Olambda__188,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_zr,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_zq(Uu),Uua)) ).

% ATP.lambda_188
tff(fact_9520_ATP_Olambda__189,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_aat(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_aas(Uu),Uua)) ).

% ATP.lambda_189
tff(fact_9521_ATP_Olambda__190,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_acj(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_190
tff(fact_9522_ATP_Olambda__191,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_ti(Uu),Uua) = times_times_complex(aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_191
tff(fact_9523_ATP_Olambda__192,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_tk(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_192
tff(fact_9524_ATP_Olambda__193,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_tj(Uu),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_193
tff(fact_9525_ATP_Olambda__194,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_vh(Uu),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_194
tff(fact_9526_ATP_Olambda__195,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_vk(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_195
tff(fact_9527_ATP_Olambda__196,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_vi(Uu),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_196
tff(fact_9528_ATP_Olambda__197,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_vl(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_197
tff(fact_9529_ATP_Olambda__198,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_vj(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_198
tff(fact_9530_ATP_Olambda__199,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_acf(Uu),Uua) = set_ord_atMost_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uua)) ).

% ATP.lambda_199
tff(fact_9531_ATP_Olambda__200,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aaz,Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_200
tff(fact_9532_ATP_Olambda__201,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abx,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_201
tff(fact_9533_ATP_Olambda__202,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abw,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_202
tff(fact_9534_ATP_Olambda__203,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abu,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_203
tff(fact_9535_ATP_Olambda__204,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abv,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_204
tff(fact_9536_ATP_Olambda__205,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aTP_Lamm_hg(Uu),Uua))
    <=> ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Uua),Uu)) ) ).

% ATP.lambda_205
tff(fact_9537_ATP_Olambda__206,axiom,
    ! [Uu: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_hh(Uu),Uua))
    <=> ~ pp(aa_set_real_bool(member_real(Uua),Uu)) ) ).

% ATP.lambda_206
tff(fact_9538_ATP_Olambda__207,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_hi(Uu),Uua))
    <=> ~ member_list_nat(Uua,Uu) ) ).

% ATP.lambda_207
tff(fact_9539_ATP_Olambda__208,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_hj(Uu),Uua))
    <=> ~ pp(aa_set_set_nat_bool(member_set_nat(Uua),Uu)) ) ).

% ATP.lambda_208
tff(fact_9540_ATP_Olambda__209,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_hk,Uu),Uua))
    <=> ~ pp(aa_set_nat_bool(member_nat(Uua),Uu)) ) ).

% ATP.lambda_209
tff(fact_9541_ATP_Olambda__210,axiom,
    ! [Uu: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_hl(Uu),Uua))
    <=> ~ pp(aa_set_int_bool(member_int(Uua),Uu)) ) ).

% ATP.lambda_210
tff(fact_9542_ATP_Olambda__211,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_ur(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_211
tff(fact_9543_ATP_Olambda__212,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aaw(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_212
tff(fact_9544_ATP_Olambda__213,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_tq(Uu),Uua) = plus_plus_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_213
tff(fact_9545_ATP_Olambda__214,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_tu(Uu),Uua) = aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_214
tff(fact_9546_ATP_Olambda__215,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_tr(Uu),Uua) = aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_215
tff(fact_9547_ATP_Olambda__216,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_tt(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_216
tff(fact_9548_ATP_Olambda__217,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_ts(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_217
tff(fact_9549_ATP_Olambda__218,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_rm(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_218
tff(fact_9550_ATP_Olambda__219,axiom,
    ! [Uu: fun_real_bool,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_hb(Uu),Uua))
    <=> ~ pp(aa_real_bool(Uu,Uua)) ) ).

% ATP.lambda_219
tff(fact_9551_ATP_Olambda__220,axiom,
    ! [Uu: fun_list_nat_bool,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_hc(Uu),Uua))
    <=> ~ pp(aa_list_nat_bool(Uu,Uua)) ) ).

% ATP.lambda_220
tff(fact_9552_ATP_Olambda__221,axiom,
    ! [Uu: fun_set_nat_bool,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_hd(Uu),Uua))
    <=> ~ pp(aa_set_nat_bool(Uu,Uua)) ) ).

% ATP.lambda_221
tff(fact_9553_ATP_Olambda__222,axiom,
    ! [Uu: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_he(Uu),Uua))
    <=> ~ pp(aa_nat_bool(Uu,Uua)) ) ).

% ATP.lambda_222
tff(fact_9554_ATP_Olambda__223,axiom,
    ! [Uu: fun_int_bool,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_hf(Uu),Uua))
    <=> ~ pp(aa_int_bool(Uu,Uua)) ) ).

% ATP.lambda_223
tff(fact_9555_ATP_Olambda__224,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_acg(Uu),Uua) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bd,Uu)) ).

% ATP.lambda_224
tff(fact_9556_ATP_Olambda__225,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_fd(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu)),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,numera6620942414471956472nteger(Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_225
tff(fact_9557_ATP_Olambda__226,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_fb(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_226
tff(fact_9558_ATP_Olambda__227,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_fc(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(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(aa_int2565445551602696444nt_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(aa_int2565445551602696444nt_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_227
tff(fact_9559_ATP_Olambda__228,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qw(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_set_nat_bool(member_nat(Uub),Uu),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_228
tff(fact_9560_ATP_Olambda__229,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_qz(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_229
tff(fact_9561_ATP_Olambda__230,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qy(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_230
tff(fact_9562_ATP_Olambda__231,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_qx(Uu),Uua),Uub) = aa_int_int(if_int(aa_set_nat_bool(member_nat(Uub),Uu),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_231
tff(fact_9563_ATP_Olambda__232,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_gq(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_232
tff(fact_9564_ATP_Olambda__233,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_gu(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub),aa_complex_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_233
tff(fact_9565_ATP_Olambda__234,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_gl(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_234
tff(fact_9566_ATP_Olambda__235,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_go(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_235
tff(fact_9567_ATP_Olambda__236,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_gr(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_236
tff(fact_9568_ATP_Olambda__237,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_gm(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_237
tff(fact_9569_ATP_Olambda__238,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gs(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub),aa_nat_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_238
tff(fact_9570_ATP_Olambda__239,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_gn(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub),aa_int_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_239
tff(fact_9571_ATP_Olambda__240,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_gp(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_240
tff(fact_9572_ATP_Olambda__241,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_gt(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub),aa_int_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_241
tff(fact_9573_ATP_Olambda__242,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_wc(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub),aa_complex_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_242
tff(fact_9574_ATP_Olambda__243,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_wg(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)),one_one_real) ).

% ATP.lambda_243
tff(fact_9575_ATP_Olambda__244,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_vz(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_244
tff(fact_9576_ATP_Olambda__245,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_wd(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_245
tff(fact_9577_ATP_Olambda__246,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_wh(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_rat(Uua,Uub)),one_one_rat) ).

% ATP.lambda_246
tff(fact_9578_ATP_Olambda__247,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_wa(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub),aa_nat_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_247
tff(fact_9579_ATP_Olambda__248,axiom,
    ! [Uu: nat,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(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub)),aa_nat_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_248
tff(fact_9580_ATP_Olambda__249,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_wi(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub),aa_nat_rat(Uua,Uub)),one_one_rat) ).

% ATP.lambda_249
tff(fact_9581_ATP_Olambda__250,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_wb(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub),aa_int_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_250
tff(fact_9582_ATP_Olambda__251,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_wf(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_251
tff(fact_9583_ATP_Olambda__252,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_gg(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_252
tff(fact_9584_ATP_Olambda__253,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_gk(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu),aa_complex_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_253
tff(fact_9585_ATP_Olambda__254,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_gb(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_254
tff(fact_9586_ATP_Olambda__255,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ge(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_255
tff(fact_9587_ATP_Olambda__256,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_gh(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_256
tff(fact_9588_ATP_Olambda__257,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_gc(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_257
tff(fact_9589_ATP_Olambda__258,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_qi(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_258
tff(fact_9590_ATP_Olambda__259,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gi(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),aa_nat_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_259
tff(fact_9591_ATP_Olambda__260,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qj(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_260
tff(fact_9592_ATP_Olambda__261,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_qk(Uu),Uua),Uub) = aa_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_261
tff(fact_9593_ATP_Olambda__262,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_gd(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu),aa_int_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_262
tff(fact_9594_ATP_Olambda__263,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_gf(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_263
tff(fact_9595_ATP_Olambda__264,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_gj(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu),aa_int_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_264
tff(fact_9596_ATP_Olambda__265,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_wm(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu),aa_complex_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_265
tff(fact_9597_ATP_Olambda__266,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_wq(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)),one_one_real) ).

% ATP.lambda_266
tff(fact_9598_ATP_Olambda__267,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_wj(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_267
tff(fact_9599_ATP_Olambda__268,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_wn(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_268
tff(fact_9600_ATP_Olambda__269,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_wr(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_rat(Uua,Uub)),one_one_rat) ).

% ATP.lambda_269
tff(fact_9601_ATP_Olambda__270,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_wk(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),aa_nat_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_270
tff(fact_9602_ATP_Olambda__271,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_wo(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_271
tff(fact_9603_ATP_Olambda__272,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ws(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),aa_nat_rat(Uua,Uub)),one_one_rat) ).

% ATP.lambda_272
tff(fact_9604_ATP_Olambda__273,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_wl(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu),aa_int_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_273
tff(fact_9605_ATP_Olambda__274,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_wp(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_274
tff(fact_9606_ATP_Olambda__275,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_zn(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu),Uub))) ).

% ATP.lambda_275
tff(fact_9607_ATP_Olambda__276,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_zo(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(abs_abs_Code_integer(Uu),Uub))) ).

% ATP.lambda_276
tff(fact_9608_ATP_Olambda__277,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_zm(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uu,Uub))) ).

% ATP.lambda_277
tff(fact_9609_ATP_Olambda__278,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qs(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(Uu,Uub),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_278
tff(fact_9610_ATP_Olambda__279,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_qv(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_279
tff(fact_9611_ATP_Olambda__280,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qu(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_280
tff(fact_9612_ATP_Olambda__281,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_qt(Uu),Uua),Uub) = aa_int_int(if_int(aa_nat_bool(Uu,Uub),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_281
tff(fact_9613_ATP_Olambda__282,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_bool,Uub: complex] : aa_complex_real(aa_fun2902006202505214429x_real(aTP_Lamm_kq(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_282
tff(fact_9614_ATP_Olambda__283,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_bool,Uub: complex] : aa_complex_rat(aa_fun5529999419959482889ex_rat(aTP_Lamm_ku(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_complex_bool(Uua,Uub),aa_complex_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_283
tff(fact_9615_ATP_Olambda__284,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_bool,Uub: real] : aa_real_complex(aa_fun6813637976755515871omplex(aTP_Lamm_kk(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(Uua,Uub),aa_real_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_284
tff(fact_9616_ATP_Olambda__285,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_bool,Uub: real] : aa_real_real(aa_fun6348708079608492509l_real(aTP_Lamm_kn(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_285
tff(fact_9617_ATP_Olambda__286,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_bool,Uub: real] : aa_real_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_kr(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_real_bool(Uua,Uub),aa_real_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_286
tff(fact_9618_ATP_Olambda__287,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_bool,Uub: nat] : aa_nat_complex(aa_fun8588439219086927967omplex(aTP_Lamm_kl(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(Uua,Uub),aa_nat_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_287
tff(fact_9619_ATP_Olambda__288,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_bool,Uub: nat] : aa_nat_rat(aa_fun5656162108953708681at_rat(aTP_Lamm_ks(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_nat_bool(Uua,Uub),aa_nat_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_288
tff(fact_9620_ATP_Olambda__289,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_bool,Uub: int] : aa_int_complex(aa_fun3606824691168040799omplex(aTP_Lamm_km(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(Uua,Uub),aa_int_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_289
tff(fact_9621_ATP_Olambda__290,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_bool,Uub: int] : aa_int_real(aa_fun6974987029372589917t_real(aTP_Lamm_ko(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_290
tff(fact_9622_ATP_Olambda__291,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_bool,Uub: int] : aa_int_rat(aa_fun4400602662665189257nt_rat(aTP_Lamm_kt(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_int_bool(Uua,Uub),aa_int_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_291
tff(fact_9623_ATP_Olambda__292,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_bool,Uub: complex] : aa_complex_complex(aa_fun1034237174562290143omplex(aTP_Lamm_yk(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_complex_bool(Uua,Uub),aa_complex_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_292
tff(fact_9624_ATP_Olambda__293,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_bool,Uub: complex] : aa_complex_real(aa_fun2902006202505214429x_real(aTP_Lamm_yo(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(Uua,Uub)),aa_complex_real(Uu,Uub)),one_one_real) ).

% ATP.lambda_293
tff(fact_9625_ATP_Olambda__294,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_bool,Uub: real] : aa_real_complex(aa_fun6813637976755515871omplex(aTP_Lamm_yh(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(Uua,Uub),aa_real_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_294
tff(fact_9626_ATP_Olambda__295,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_bool,Uub: real] : aa_real_real(aa_fun6348708079608492509l_real(aTP_Lamm_yl(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_295
tff(fact_9627_ATP_Olambda__296,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_bool,Uub: real] : aa_real_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_yp(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_real_bool(Uua,Uub),aa_real_rat(Uu,Uub)),one_one_rat) ).

% ATP.lambda_296
tff(fact_9628_ATP_Olambda__297,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_bool,Uub: nat] : aa_nat_complex(aa_fun8588439219086927967omplex(aTP_Lamm_yi(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(Uua,Uub),aa_nat_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_297
tff(fact_9629_ATP_Olambda__298,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_bool,Uub: nat] : aa_nat_real(aa_fun1750223370215434333t_real(aTP_Lamm_ym(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_298
tff(fact_9630_ATP_Olambda__299,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_bool,Uub: nat] : aa_nat_rat(aa_fun5656162108953708681at_rat(aTP_Lamm_yq(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_nat_bool(Uua,Uub),aa_nat_rat(Uu,Uub)),one_one_rat) ).

% ATP.lambda_299
tff(fact_9631_ATP_Olambda__300,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_bool,Uub: int] : aa_int_complex(aa_fun3606824691168040799omplex(aTP_Lamm_yj(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(Uua,Uub),aa_int_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_300
tff(fact_9632_ATP_Olambda__301,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_bool,Uub: int] : aa_int_real(aa_fun6974987029372589917t_real(aTP_Lamm_yn(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_301
tff(fact_9633_ATP_Olambda__302,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ob(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)) ).

% ATP.lambda_302
tff(fact_9634_ATP_Olambda__303,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nz(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)) ).

% ATP.lambda_303
tff(fact_9635_ATP_Olambda__304,axiom,
    ! [Uu: fun_co8378349014452122532t_real,Uua: nat,Uub: complex] : aa_complex_real(aa_nat480766339200631653x_real(aTP_Lamm_jf(Uu),Uua),Uub) = aa_nat_real(aa_com20334259897932645t_real(Uu,Uub),Uua) ).

% ATP.lambda_304
tff(fact_9636_ATP_Olambda__305,axiom,
    ! [Uu: fun_co636942507509095880at_nat,Uua: nat,Uub: complex] : aa_complex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_ik(Uu),Uua),Uub) = aa_nat_nat(aa_com6383720667567721865at_nat(Uu,Uub),Uua) ).

% ATP.lambda_305
tff(fact_9637_ATP_Olambda__306,axiom,
    ! [Uu: fun_re7208826350674946562omplex,Uua: complex,Uub: real] : aa_real_complex(aa_com4766338864777664963omplex(aTP_Lamm_io(Uu),Uua),Uub) = aa_complex_complex(aa_rea2618301233319580611omplex(Uu,Uub),Uua) ).

% ATP.lambda_306
tff(fact_9638_ATP_Olambda__307,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_iz(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_307
tff(fact_9639_ATP_Olambda__308,axiom,
    ! [Uu: fun_real_fun_nat_nat,Uua: nat,Uub: real] : aa_real_nat(aa_nat_fun_real_nat(aTP_Lamm_ic(Uu),Uua),Uub) = aa_nat_nat(aa_real_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_308
tff(fact_9640_ATP_Olambda__309,axiom,
    ! [Uu: fun_li2771310203728655226at_nat,Uua: nat,Uub: list_nat] : aa_list_nat_nat(aa_nat3122681260899759163at_nat(aTP_Lamm_xo(Uu),Uua),Uub) = aa_nat_nat(aa_lis1299702591413342011at_nat(Uu,Uub),Uua) ).

% ATP.lambda_309
tff(fact_9641_ATP_Olambda__310,axiom,
    ! [Uu: fun_se1179353343636554400at_nat,Uua: nat,Uub: set_nat] : aa_set_nat_nat(aa_nat6910189468943142497at_nat(aTP_Lamm_xt(Uu),Uua),Uub) = aa_nat_nat(aa_set6537093552120046177at_nat(Uu,Uub),Uua) ).

% ATP.lambda_310
tff(fact_9642_ATP_Olambda__311,axiom,
    ! [Uu: fun_na1591161823130732966omplex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_is(Uu),Uua),Uub) = aa_complex_complex(aa_nat3829984526574172007omplex(Uu,Uub),Uua) ).

% ATP.lambda_311
tff(fact_9643_ATP_Olambda__312,axiom,
    ! [Uu: fun_na8457291176974909128ex_nat,Uua: complex,Uub: nat] : aa_nat_nat(aa_com6383720667567721865at_nat(aTP_Lamm_xg(Uu),Uua),Uub) = aa_complex_nat(aa_nat3391284541783913865ex_nat(Uu,Uub),Uua) ).

% ATP.lambda_312
tff(fact_9644_ATP_Olambda__313,axiom,
    ! [Uu: fun_nat_fun_real_nat,Uua: real,Uub: nat] : aa_nat_nat(aa_real_fun_nat_nat(aTP_Lamm_jj(Uu),Uua),Uub) = aa_real_nat(aa_nat_fun_real_nat(Uu,Uub),Uua) ).

% ATP.lambda_313
tff(fact_9645_ATP_Olambda__314,axiom,
    ! [Uu: fun_na7671319193681171066at_nat,Uua: list_nat,Uub: nat] : aa_nat_nat(aa_lis1299702591413342011at_nat(aTP_Lamm_xy(Uu),Uua),Uub) = aa_list_nat_nat(aa_nat3122681260899759163at_nat(Uu,Uub),Uua) ).

% ATP.lambda_314
tff(fact_9646_ATP_Olambda__315,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xk(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_315
tff(fact_9647_ATP_Olambda__316,axiom,
    ! [Uu: fun_nat_fun_int_nat,Uua: int,Uub: nat] : aa_nat_nat(aa_int_fun_nat_nat(aTP_Lamm_xd(Uu),Uua),Uub) = aa_int_nat(aa_nat_fun_int_nat(Uu,Uub),Uua) ).

% ATP.lambda_316
tff(fact_9648_ATP_Olambda__317,axiom,
    ! [Uu: fun_in1307110748760623106omplex,Uua: complex,Uub: int] : aa_int_complex(aa_com9156536935073639875omplex(aTP_Lamm_iw(Uu),Uua),Uub) = aa_complex_complex(aa_int3545933452204062147omplex(Uu,Uub),Uua) ).

% ATP.lambda_317
tff(fact_9649_ATP_Olambda__318,axiom,
    ! [Uu: fun_int_fun_nat_real,Uua: nat,Uub: int] : aa_int_real(aa_nat_fun_int_real(aTP_Lamm_jc(Uu),Uua),Uub) = aa_nat_real(aa_int_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_318
tff(fact_9650_ATP_Olambda__319,axiom,
    ! [Uu: fun_int_fun_nat_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_ig(Uu),Uua),Uub) = aa_nat_nat(aa_int_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_319
tff(fact_9651_ATP_Olambda__320,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ue(Uu),Uua),Uub) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ud(Uu),Uua),Uub),set_ord_atMost_nat(Uub)) ).

% ATP.lambda_320
tff(fact_9652_ATP_Olambda__321,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_uc(Uu),Uua),Uub) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ub(Uu),Uua),Uub),set_ord_atMost_nat(Uub)) ).

% ATP.lambda_321
tff(fact_9653_ATP_Olambda__322,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ua(Uu),Uua),Uub) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_tz(Uu),Uua),Uub),set_ord_atMost_nat(Uub)) ).

% ATP.lambda_322
tff(fact_9654_ATP_Olambda__323,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ty(Uu),Uua),Uub) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_tx(Uu),Uua),Uub),set_ord_atMost_nat(Uub)) ).

% ATP.lambda_323
tff(fact_9655_ATP_Olambda__324,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tp(Uu),Uua),Uub) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_to(Uu),Uua),Uub),set_ord_atMost_nat(Uub)) ).

% ATP.lambda_324
tff(fact_9656_ATP_Olambda__325,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_tn(Uu),Uua),Uub) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_tm(Uu),Uua),Uub),set_ord_atMost_nat(Uub)) ).

% ATP.lambda_325
tff(fact_9657_ATP_Olambda__326,axiom,
    ! [Uu: nat,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ra(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),one_one_complex),zero_zero_complex)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_326
tff(fact_9658_ATP_Olambda__327,axiom,
    ! [Uu: nat,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rb(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_327
tff(fact_9659_ATP_Olambda__328,axiom,
    ! [Uu: nat,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_rc(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_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(power_power_int(Uua),Uub)) ).

% ATP.lambda_328
tff(fact_9660_ATP_Olambda__329,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod2825494507867081354r_bool(aa_Cod4981436349852597092r_bool(aTP_Lamm_zk(Uu),Uua),Uub) = produc5135875973608410785r_bool(if_Code_integer(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub)),aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),one_one_Code_integer)) ).

% ATP.lambda_329
tff(fact_9661_ATP_Olambda__330,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yc(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_xk(Uu),Uub)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_330
tff(fact_9662_ATP_Olambda__331,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ze(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_zd(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_331
tff(fact_9663_ATP_Olambda__332,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_zc(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_zb(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_332
tff(fact_9664_ATP_Olambda__333,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_za(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_yz(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_333
tff(fact_9665_ATP_Olambda__334,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_yy(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_yx(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_334
tff(fact_9666_ATP_Olambda__335,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_yw(Uu),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_yv(Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_335
tff(fact_9667_ATP_Olambda__336,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aai(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_336
tff(fact_9668_ATP_Olambda__337,axiom,
    ! [Uu: real,Uua: fun_na5723442673449079972x_real,Uub: nat] : aa_nat_real(aa_fun5746040925223427547t_real(aTP_Lamm_tc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(aa_nat480766339200631653x_real(Uua,Uub),zero_zero_complex)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_337
tff(fact_9669_ATP_Olambda__338,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_td(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_338
tff(fact_9670_ATP_Olambda__339,axiom,
    ! [Uu: real,Uua: fun_nat_fun_rat_real,Uub: nat] : aa_nat_real(aa_fun507281639146497029t_real(aTP_Lamm_te(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_rat_real(aa_nat_fun_rat_real(Uua,Uub),zero_zero_rat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_339
tff(fact_9671_ATP_Olambda__340,axiom,
    ! [Uu: real,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_tf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_nat_fun_nat_real(Uua,Uub),zero_zero_nat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_340
tff(fact_9672_ATP_Olambda__341,axiom,
    ! [Uu: real,Uua: fun_nat_fun_int_real,Uub: nat] : aa_nat_real(aa_fun241848180832411225t_real(aTP_Lamm_tg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(aa_nat_fun_int_real(Uua,Uub),zero_zero_int)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_341
tff(fact_9673_ATP_Olambda__342,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aTP_Lamm_oe(Uu),Uua),Uub))
    <=> ( groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nr(Uu),Uub),set_ord_atMost_nat(Uua)) = zero_zero_real ) ) ).

% ATP.lambda_342
tff(fact_9674_ATP_Olambda__343,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_od(Uu),Uua),Uub))
    <=> ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ni(Uu),Uub),set_ord_atMost_nat(Uua)) = zero_zero_complex ) ) ).

% ATP.lambda_343
tff(fact_9675_ATP_Olambda__344,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_gy(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(zero_zero_complex),Uub))),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_344
tff(fact_9676_ATP_Olambda__345,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ha(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_345
tff(fact_9677_ATP_Olambda__346,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gz(Uu),Uua),Uub) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_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_346
tff(fact_9678_ATP_Olambda__347,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aar(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_347
tff(fact_9679_ATP_Olambda__348,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_th(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_348
tff(fact_9680_ATP_Olambda__349,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_acd(Uu),Uua),Uub))
    <=> ( ( size_size_list_nat(Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),one_one_nat) )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_349
tff(fact_9681_ATP_Olambda__350,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ve(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),aa_nat_complex(Uu,Uub))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_350
tff(fact_9682_ATP_Olambda__351,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),aa_nat_real(Uu,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_351
tff(fact_9683_ATP_Olambda__352,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_ace(Uu),Uua),Uub))
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_352
tff(fact_9684_ATP_Olambda__353,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_dc(Uu),Uua),Uub))
    <=> ( 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_353
tff(fact_9685_ATP_Olambda__354,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: list_P6011104703257516679at_nat] :
      ( pp(aa_lis6649954313659033868t_bool(aa_nat1772366067986465313t_bool(aTP_Lamm_db(Uu),Uua),Uub))
    <=> ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s5460976970255530739at_nat(Uub)),Uua)) ) ) ).

% ATP.lambda_354
tff(fact_9686_ATP_Olambda__355,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_da(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_355
tff(fact_9687_ATP_Olambda__356,axiom,
    ! [Uu: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_dd(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_356
tff(fact_9688_ATP_Olambda__357,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_de(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_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_357
tff(fact_9689_ATP_Olambda__358,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_df(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_358
tff(fact_9690_ATP_Olambda__359,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_cw(Uu),Uua),Uub))
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_359
tff(fact_9691_ATP_Olambda__360,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: list_P6011104703257516679at_nat] :
      ( pp(aa_lis6649954313659033868t_bool(aa_nat1772366067986465313t_bool(aTP_Lamm_cv(Uu),Uua),Uub))
    <=> ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Uub)),Uu))
        & ( size_s5460976970255530739at_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_360
tff(fact_9692_ATP_Olambda__361,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_cu(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uu))
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_361
tff(fact_9693_ATP_Olambda__362,axiom,
    ! [Uu: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_cx(Uu),Uua),Uub))
    <=> ( ord_less_eq_set_bool(set_bool2(Uub),Uu)
        & ( size_size_list_bool(Uub) = Uua ) ) ) ).

% ATP.lambda_362
tff(fact_9694_ATP_Olambda__363,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_cy(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Uub)),Uu))
        & ( size_size_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_363
tff(fact_9695_ATP_Olambda__364,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_cz(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_364
tff(fact_9696_ATP_Olambda__365,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_acc(Uu),Uua),Uub))
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_365
tff(fact_9697_ATP_Olambda__366,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_zg(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_366
tff(fact_9698_ATP_Olambda__367,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_nat8857836526276733660t_bool(aTP_Lamm_zj(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(Uub),pow_nat(set_or4665077453230672383an_nat(zero_zero_nat,Uu))))
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_367
tff(fact_9699_ATP_Olambda__368,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ach(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uub)) ) ) ).

% ATP.lambda_368
tff(fact_9700_ATP_Olambda__369,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_zf(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_369
tff(fact_9701_ATP_Olambda__370,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_vd(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uu),Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_370
tff(fact_9702_ATP_Olambda__371,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(diffs_real(Uu),Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_371
tff(fact_9703_ATP_Olambda__372,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vu(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uua),Uub)),aa_nat_complex(power_power_complex(Uu),Uub)) ).

% ATP.lambda_372
tff(fact_9704_ATP_Olambda__373,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_vt(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(diffs_real(Uua),Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_373
tff(fact_9705_ATP_Olambda__374,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rf(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_374
tff(fact_9706_ATP_Olambda__375,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: set_Pr4329608150637261639at_nat,Uub: produc3843707927480180839at_nat] :
      ( pp(aa_Pro1809015098595223084t_bool(aa_set377272637606474814t_bool(aTP_Lamm_ht(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(Uub),Uua)) ) ) ).

% ATP.lambda_375
tff(fact_9707_ATP_Olambda__376,axiom,
    ! [Uu: product_prod_nat_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_set5223363028666124222t_bool(aTP_Lamm_hu(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Uub),Uua)) ) ) ).

% ATP.lambda_376
tff(fact_9708_ATP_Olambda__377,axiom,
    ! [Uu: real,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_hv(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_real_bool(member_real(Uub),Uua)) ) ) ).

% ATP.lambda_377
tff(fact_9709_ATP_Olambda__378,axiom,
    ! [Uu: list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_set257547205302447420t_bool(aTP_Lamm_hw(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_378
tff(fact_9710_ATP_Olambda__379,axiom,
    ! [Uu: set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set9121396911425987952t_bool(aTP_Lamm_hx(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_set_nat_bool(member_set_nat(Uub),Uua)) ) ) ).

% ATP.lambda_379
tff(fact_9711_ATP_Olambda__380,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_hy(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_nat_bool(member_nat(Uub),Uua)) ) ) ).

% ATP.lambda_380
tff(fact_9712_ATP_Olambda__381,axiom,
    ! [Uu: int,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_hz(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_int_bool(member_int(Uub),Uua)) ) ) ).

% ATP.lambda_381
tff(fact_9713_ATP_Olambda__382,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_fr(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_382
tff(fact_9714_ATP_Olambda__383,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_at(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_383
tff(fact_9715_ATP_Olambda__384,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_fq(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_384
tff(fact_9716_ATP_Olambda__385,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_bb(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_385
tff(fact_9717_ATP_Olambda__386,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_bc(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_386
tff(fact_9718_ATP_Olambda__387,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_au(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_387
tff(fact_9719_ATP_Olambda__388,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_aw(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(vEBT_vebt_member(Uu),Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uub)) ) ) ).

% ATP.lambda_388
tff(fact_9720_ATP_Olambda__389,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_av(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(vEBT_vebt_member(Uu),Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_389
tff(fact_9721_ATP_Olambda__390,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_abe(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uub),Uua))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uub),Uu)) ) ) ).

% ATP.lambda_390
tff(fact_9722_ATP_Olambda__391,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: fun_Pr7757163693017731481t_bool,Uub: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_fun830836278467416944t_bool(aTP_Lamm_bl(Uu),Uua),Uub))
    <=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Uub),Uu))
        & pp(aa_Pro2232559814647311250t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_391
tff(fact_9723_ATP_Olambda__392,axiom,
    ! [Uu: set_complex,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_kp(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_392
tff(fact_9724_ATP_Olambda__393,axiom,
    ! [Uu: set_real,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_bm(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_393
tff(fact_9725_ATP_Olambda__394,axiom,
    ! [Uu: set_list_nat,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_bn(Uu),Uua),Uub))
    <=> ( member_list_nat(Uub,Uu)
        & pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_394
tff(fact_9726_ATP_Olambda__395,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_bo(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(Uub),Uu))
        & pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_395
tff(fact_9727_ATP_Olambda__396,axiom,
    ! [Uu: set_nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_bp(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_396
tff(fact_9728_ATP_Olambda__397,axiom,
    ! [Uu: set_int,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_bq(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_397
tff(fact_9729_ATP_Olambda__398,axiom,
    ! [Uu: fun_Pr1607443439640623155t_bool,Uua: produc3843707927480180839at_nat,Uub: produc3843707927480180839at_nat] :
      ( pp(aa_Pro1809015098595223084t_bool(aa_Pro2859564254541897054t_bool(aTP_Lamm_jt(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_Pro1809015098595223084t_bool(Uu,Uub)) ) ) ).

% ATP.lambda_398
tff(fact_9730_ATP_Olambda__399,axiom,
    ! [Uu: fun_Pr7757163693017731481t_bool,Uua: product_prod_nat_nat,Uub: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(aTP_Lamm_jw(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_Pro2232559814647311250t_bool(Uu,Uub)) ) ) ).

% ATP.lambda_399
tff(fact_9731_ATP_Olambda__400,axiom,
    ! [Uu: fun_real_bool,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_jx(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_real_bool(Uu,Uub)) ) ) ).

% ATP.lambda_400
tff(fact_9732_ATP_Olambda__401,axiom,
    ! [Uu: fun_list_nat_bool,Uua: list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(aTP_Lamm_ju(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_list_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_401
tff(fact_9733_ATP_Olambda__402,axiom,
    ! [Uu: fun_set_nat_bool,Uua: set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_jv(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_set_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_402
tff(fact_9734_ATP_Olambda__403,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_jy(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_403
tff(fact_9735_ATP_Olambda__404,axiom,
    ! [Uu: fun_int_bool,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_jz(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_int_bool(Uu,Uub)) ) ) ).

% ATP.lambda_404
tff(fact_9736_ATP_Olambda__405,axiom,
    ! [Uu: fun_Pr1607443439640623155t_bool,Uua: produc3843707927480180839at_nat,Uub: produc3843707927480180839at_nat] :
      ( pp(aa_Pro1809015098595223084t_bool(aa_Pro2859564254541897054t_bool(aTP_Lamm_jm(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_Pro1809015098595223084t_bool(Uu,Uub)) ) ) ).

% ATP.lambda_405
tff(fact_9737_ATP_Olambda__406,axiom,
    ! [Uu: fun_Pr7757163693017731481t_bool,Uua: product_prod_nat_nat,Uub: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(aTP_Lamm_jp(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_Pro2232559814647311250t_bool(Uu,Uub)) ) ) ).

% ATP.lambda_406
tff(fact_9738_ATP_Olambda__407,axiom,
    ! [Uu: fun_real_bool,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_jq(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_real_bool(Uu,Uub)) ) ) ).

% ATP.lambda_407
tff(fact_9739_ATP_Olambda__408,axiom,
    ! [Uu: fun_list_nat_bool,Uua: list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(aTP_Lamm_jn(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_list_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_408
tff(fact_9740_ATP_Olambda__409,axiom,
    ! [Uu: fun_set_nat_bool,Uua: set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_jo(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_set_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_409
tff(fact_9741_ATP_Olambda__410,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_jr(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_410
tff(fact_9742_ATP_Olambda__411,axiom,
    ! [Uu: fun_int_bool,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_js(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_int_bool(Uu,Uub)) ) ) ).

% ATP.lambda_411
tff(fact_9743_ATP_Olambda__412,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aTP_Lamm_ej(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_412
tff(fact_9744_ATP_Olambda__413,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aTP_Lamm_er(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_413
tff(fact_9745_ATP_Olambda__414,axiom,
    ! [Uu: set_complex,Uua: fun_complex_rat,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2759627087638585025x_bool(aTP_Lamm_rx(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_414
tff(fact_9746_ATP_Olambda__415,axiom,
    ! [Uu: set_complex,Uua: fun_complex_int,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun4637897056364002669x_bool(aTP_Lamm_ry(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_int(Uua,Uub) != zero_zero_int ) ) ) ).

% ATP.lambda_415
tff(fact_9747_ATP_Olambda__416,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aTP_Lamm_ed(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_416
tff(fact_9748_ATP_Olambda__417,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_el(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_417
tff(fact_9749_ATP_Olambda__418,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aTP_Lamm_et(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_418
tff(fact_9750_ATP_Olambda__419,axiom,
    ! [Uu: set_real,Uua: fun_real_int,Uub: real] :
      ( pp(aa_real_bool(aa_fun2745684120339038577l_bool(aTP_Lamm_rz(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_int(Uua,Uub) != zero_zero_int ) ) ) ).

% ATP.lambda_419
tff(fact_9751_ATP_Olambda__420,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aTP_Lamm_ef(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_420
tff(fact_9752_ATP_Olambda__421,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_en(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_421
tff(fact_9753_ATP_Olambda__422,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aTP_Lamm_ev(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_422
tff(fact_9754_ATP_Olambda__423,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aTP_Lamm_eh(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_423
tff(fact_9755_ATP_Olambda__424,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_ep(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_424
tff(fact_9756_ATP_Olambda__425,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: int] :
      ( pp(aa_int_bool(aa_fun258058769140594373t_bool(aTP_Lamm_ta(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_425
tff(fact_9757_ATP_Olambda__426,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aTP_Lamm_dp(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_426
tff(fact_9758_ATP_Olambda__427,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aTP_Lamm_dx(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_427
tff(fact_9759_ATP_Olambda__428,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aTP_Lamm_dj(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_428
tff(fact_9760_ATP_Olambda__429,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_dr(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_429
tff(fact_9761_ATP_Olambda__430,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aTP_Lamm_dz(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_430
tff(fact_9762_ATP_Olambda__431,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aTP_Lamm_dl(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_431
tff(fact_9763_ATP_Olambda__432,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_dt(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_432
tff(fact_9764_ATP_Olambda__433,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aTP_Lamm_eb(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_433
tff(fact_9765_ATP_Olambda__434,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aTP_Lamm_dn(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_434
tff(fact_9766_ATP_Olambda__435,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_dv(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_435
tff(fact_9767_ATP_Olambda__436,axiom,
    ! [Uu: fun_real_complex,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_sa(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uua))
        & ( aa_real_complex(Uu,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_436
tff(fact_9768_ATP_Olambda__437,axiom,
    ! [Uu: fun_real_real,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_sd(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uua))
        & ( aa_real_real(Uu,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_437
tff(fact_9769_ATP_Olambda__438,axiom,
    ! [Uu: fun_real_rat,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_sg(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uua))
        & ( aa_real_rat(Uu,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_438
tff(fact_9770_ATP_Olambda__439,axiom,
    ! [Uu: fun_real_nat,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_sj(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uua))
        & ( aa_real_nat(Uu,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_439
tff(fact_9771_ATP_Olambda__440,axiom,
    ! [Uu: fun_nat_complex,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_sb(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uua))
        & ( aa_nat_complex(Uu,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_440
tff(fact_9772_ATP_Olambda__441,axiom,
    ! [Uu: fun_nat_real,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_se(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uua))
        & ( aa_nat_real(Uu,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_441
tff(fact_9773_ATP_Olambda__442,axiom,
    ! [Uu: fun_nat_rat,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_sh(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uua))
        & ( aa_nat_rat(Uu,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_442
tff(fact_9774_ATP_Olambda__443,axiom,
    ! [Uu: fun_int_complex,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_sc(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uua))
        & ( aa_int_complex(Uu,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_443
tff(fact_9775_ATP_Olambda__444,axiom,
    ! [Uu: fun_int_real,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_sf(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uua))
        & ( aa_int_real(Uu,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_444
tff(fact_9776_ATP_Olambda__445,axiom,
    ! [Uu: fun_int_rat,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_si(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uua))
        & ( aa_int_rat(Uu,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_445
tff(fact_9777_ATP_Olambda__446,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_set5223363028666124222t_bool(aTP_Lamm_bf(Uu),Uua),Uub))
    <=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Uub),Uu))
        & ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Uub),Uua)) ) ) ).

% ATP.lambda_446
tff(fact_9778_ATP_Olambda__447,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_bg(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ~ pp(aa_set_real_bool(member_real(Uub),Uua)) ) ) ).

% ATP.lambda_447
tff(fact_9779_ATP_Olambda__448,axiom,
    ! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_set257547205302447420t_bool(aTP_Lamm_bh(Uu),Uua),Uub))
    <=> ( member_list_nat(Uub,Uu)
        & ~ member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_448
tff(fact_9780_ATP_Olambda__449,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set9121396911425987952t_bool(aTP_Lamm_bi(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(Uub),Uu))
        & ~ pp(aa_set_set_nat_bool(member_set_nat(Uub),Uua)) ) ) ).

% ATP.lambda_449
tff(fact_9781_ATP_Olambda__450,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_bk(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ~ pp(aa_set_nat_bool(member_nat(Uub),Uua)) ) ) ).

% ATP.lambda_450
tff(fact_9782_ATP_Olambda__451,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_bj(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ~ pp(aa_set_int_bool(member_int(Uub),Uua)) ) ) ).

% ATP.lambda_451
tff(fact_9783_ATP_Olambda__452,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_abd(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_452
tff(fact_9784_ATP_Olambda__453,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ny(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu)) ) ).

% ATP.lambda_453
tff(fact_9785_ATP_Olambda__454,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_abf(Uu),Uua),Uub))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V3694042436643373181omplex(Uua),Uub)),Uu)) ) ).

% ATP.lambda_454
tff(fact_9786_ATP_Olambda__455,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_abh(Uu),Uua),Uub))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,Uua),Uub)),Uu)) ) ).

% ATP.lambda_455
tff(fact_9787_ATP_Olambda__456,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_of(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu)) ) ).

% ATP.lambda_456
tff(fact_9788_ATP_Olambda__457,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cm(Uu),Uua),Uub))
    <=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)),Uu)) ) ).

% ATP.lambda_457
tff(fact_9789_ATP_Olambda__458,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_cn(Uu),Uua),Uub))
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub),Uu) ) ).

% ATP.lambda_458
tff(fact_9790_ATP_Olambda__459,axiom,
    ! [Uu: set_Pr7592006421409863191nteger,Uua: fun_Co6545587727236015215e_term,Uub: produc8923325533196201883nteger] :
      ( pp(aa_Pro5843519176478241656r_bool(aa_fun5646951941296221740r_bool(aTP_Lamm_co(Uu),Uua),Uub))
    <=> member4581088367259704696nteger(aa_Pro6252757739892038976nteger(aa_fun3790775640809306146nteger(produc4282985567423050387nteger,Uua),Uub),Uu) ) ).

% ATP.lambda_459
tff(fact_9791_ATP_Olambda__460,axiom,
    ! [Uu: set_Pr5663546166630115725nteger,Uua: fun_Pr272337284026993017e_term,Uub: produc8923325533196201883nteger] :
      ( pp(aa_Pro5843519176478241656r_bool(aa_fun5449300841655810358r_bool(aTP_Lamm_cp(Uu),Uua),Uub))
    <=> member6423679177046879214nteger(aa_Pro490869892464487350nteger(aa_fun4101202993852558094nteger(produc7384524951051686153nteger,Uua),Uub),Uu) ) ).

% ATP.lambda_460
tff(fact_9792_ATP_Olambda__461,axiom,
    ! [Uu: set_Pr233528076387193594nt_int,Uua: fun_Pr9164807099015279978e_term,Uub: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_fun5300525129335604105t_bool(aTP_Lamm_cq(Uu),Uua),Uub))
    <=> member690604328116145115nt_int(aa_Pro3513184699142849729nt_int(aa_fun1388719692539776744nt_int(produc1560874099174258678nt_int,Uua),Uub),Uu) ) ).

% ATP.lambda_461
tff(fact_9793_ATP_Olambda__462,axiom,
    ! [Uu: set_Pr5033440245338114598nt_int,Uua: fun_in6429348457993719614e_term,Uub: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_fun4446166836006121821t_bool(aTP_Lamm_cr(Uu),Uua),Uub))
    <=> member500491123954006279nt_int(aa_Pro835648289009615341nt_int(aa_fun7940596247606734144nt_int(produc4852315488734385442nt_int,Uua),Uub),Uu) ) ).

% ATP.lambda_462
tff(fact_9794_ATP_Olambda__463,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_dg(Uu),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_463
tff(fact_9795_ATP_Olambda__464,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_zh(Uu),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_464
tff(fact_9796_ATP_Olambda__465,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_uy(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(suc,Uub))),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_465
tff(fact_9797_ATP_Olambda__466,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(suc,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_466
tff(fact_9798_ATP_Olambda__467,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aas(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_467
tff(fact_9799_ATP_Olambda__468,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ni(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_468
tff(fact_9800_ATP_Olambda__469,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_nr(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_469
tff(fact_9801_ATP_Olambda__470,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_nl(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(Uua),Uub)) ).

% ATP.lambda_470
tff(fact_9802_ATP_Olambda__471,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_oo(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_471
tff(fact_9803_ATP_Olambda__472,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_no(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uub)),aa_nat_int(power_power_int(Uua),Uub)) ).

% ATP.lambda_472
tff(fact_9804_ATP_Olambda__473,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cs(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_473
tff(fact_9805_ATP_Olambda__474,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_acn(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_474
tff(fact_9806_ATP_Olambda__475,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_yr(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_475
tff(fact_9807_ATP_Olambda__476,axiom,
    ! [Uu: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_rn(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_real_Code_integer(Uua,Uub)),aa_real_Code_integer(Uu,Uub)) ).

% ATP.lambda_476
tff(fact_9808_ATP_Olambda__477,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_rq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_477
tff(fact_9809_ATP_Olambda__478,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_rs(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(Uua,Uub)),aa_real_rat(Uu,Uub)) ).

% ATP.lambda_478
tff(fact_9810_ATP_Olambda__479,axiom,
    ! [Uu: fun_real_int,Uua: fun_real_int,Uub: real] : aa_real_int(aa_fun8270430325945277160al_int(aTP_Lamm_rv(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(Uua,Uub)),aa_real_int(Uu,Uub)) ).

% ATP.lambda_479
tff(fact_9811_ATP_Olambda__480,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_ro(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu,Uub)) ).

% ATP.lambda_480
tff(fact_9812_ATP_Olambda__481,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_rt(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu,Uub)) ).

% ATP.lambda_481
tff(fact_9813_ATP_Olambda__482,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_rw(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uua,Uub)),aa_nat_int(Uu,Uub)) ).

% ATP.lambda_482
tff(fact_9814_ATP_Olambda__483,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_rp(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu,Uub)) ).

% ATP.lambda_483
tff(fact_9815_ATP_Olambda__484,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_rr(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uub)),aa_int_real(Uu,Uub)) ).

% ATP.lambda_484
tff(fact_9816_ATP_Olambda__485,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ru(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(Uua,Uub)),aa_int_rat(Uu,Uub)) ).

% ATP.lambda_485
tff(fact_9817_ATP_Olambda__486,axiom,
    ! [Uu: fun_Pr2602117396471556196at_nat,Uua: fun_Pr2602117396471556196at_nat,Uub: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(aa_fun2351638177499875132at_nat(aTP_Lamm_kv(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Pro4243173770044397973at_nat(Uua,Uub)),aa_Pro4243173770044397973at_nat(Uu,Uub)) ).

% ATP.lambda_486
tff(fact_9818_ATP_Olambda__487,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_kw(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_487
tff(fact_9819_ATP_Olambda__488,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_kx(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(Uua,Uub)),aa_set_nat_nat(Uu,Uub)) ).

% ATP.lambda_488
tff(fact_9820_ATP_Olambda__489,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_rd(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uua,Uub)),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_489
tff(fact_9821_ATP_Olambda__490,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kz(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_490
tff(fact_9822_ATP_Olambda__491,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ky(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_491
tff(fact_9823_ATP_Olambda__492,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_sx(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_492
tff(fact_9824_ATP_Olambda__493,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_sm(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(Uu,Uub)),aa_complex_real(Uua,Uub)) ).

% ATP.lambda_493
tff(fact_9825_ATP_Olambda__494,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_sp(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_complex_rat(Uu,Uub)),aa_complex_rat(Uua,Uub)) ).

% ATP.lambda_494
tff(fact_9826_ATP_Olambda__495,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_ss(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(Uu,Uub)),aa_complex_nat(Uua,Uub)) ).

% ATP.lambda_495
tff(fact_9827_ATP_Olambda__496,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_su(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_real_complex(Uu,Uub)),aa_real_complex(Uua,Uub)) ).

% ATP.lambda_496
tff(fact_9828_ATP_Olambda__497,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sy(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_497
tff(fact_9829_ATP_Olambda__498,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_sz(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(Uu,Uub)),aa_real_rat(Uua,Uub)) ).

% ATP.lambda_498
tff(fact_9830_ATP_Olambda__499,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sv(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_499
tff(fact_9831_ATP_Olambda__500,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_sk(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_500
tff(fact_9832_ATP_Olambda__501,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_sn(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_501
tff(fact_9833_ATP_Olambda__502,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_sq(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_502
tff(fact_9834_ATP_Olambda__503,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_st(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_503
tff(fact_9835_ATP_Olambda__504,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_sw(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_int_complex(Uu,Uub)),aa_int_complex(Uua,Uub)) ).

% ATP.lambda_504
tff(fact_9836_ATP_Olambda__505,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_sl(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(Uu,Uub)),aa_int_real(Uua,Uub)) ).

% ATP.lambda_505
tff(fact_9837_ATP_Olambda__506,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_so(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(Uu,Uub)),aa_int_rat(Uua,Uub)) ).

% ATP.lambda_506
tff(fact_9838_ATP_Olambda__507,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_sr(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(Uu,Uub)),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_507
tff(fact_9839_ATP_Olambda__508,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aah(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_508
tff(fact_9840_ATP_Olambda__509,axiom,
    ! [Uu: fun_Pr7757163693017731481t_bool,Uua: fun_Pr7757163693017731481t_bool,Uub: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_fun830836278467416944t_bool(aTP_Lamm_as(Uu),Uua),Uub))
    <=> ( pp(aa_Pro2232559814647311250t_bool(Uu,Uub))
        | pp(aa_Pro2232559814647311250t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_509
tff(fact_9841_ATP_Olambda__510,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_ar(Uu),Uua),Uub))
    <=> ( pp(aa_complex_bool(Uu,Uub))
        | pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_510
tff(fact_9842_ATP_Olambda__511,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_am(Uu),Uua),Uub))
    <=> ( pp(aa_real_bool(Uu,Uub))
        | pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_511
tff(fact_9843_ATP_Olambda__512,axiom,
    ! [Uu: fun_list_nat_bool,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_an(Uu),Uua),Uub))
    <=> ( pp(aa_list_nat_bool(Uu,Uub))
        | pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_512
tff(fact_9844_ATP_Olambda__513,axiom,
    ! [Uu: fun_set_nat_bool,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_ao(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(Uu,Uub))
        | pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_513
tff(fact_9845_ATP_Olambda__514,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_ap(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        | pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_514
tff(fact_9846_ATP_Olambda__515,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_aq(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(Uu,Uub))
        | pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_515
tff(fact_9847_ATP_Olambda__516,axiom,
    ! [Uu: fun_Pr7757163693017731481t_bool,Uua: fun_Pr7757163693017731481t_bool,Uub: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_fun830836278467416944t_bool(aTP_Lamm_al(Uu),Uua),Uub))
    <=> ( pp(aa_Pro2232559814647311250t_bool(Uu,Uub))
        & pp(aa_Pro2232559814647311250t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_516
tff(fact_9848_ATP_Olambda__517,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_ak(Uu),Uua),Uub))
    <=> ( pp(aa_complex_bool(Uu,Uub))
        & pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_517
tff(fact_9849_ATP_Olambda__518,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_af(Uu),Uua),Uub))
    <=> ( pp(aa_real_bool(Uu,Uub))
        & pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_518
tff(fact_9850_ATP_Olambda__519,axiom,
    ! [Uu: fun_list_nat_bool,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_ag(Uu),Uua),Uub))
    <=> ( pp(aa_list_nat_bool(Uu,Uub))
        & pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_519
tff(fact_9851_ATP_Olambda__520,axiom,
    ! [Uu: fun_set_nat_bool,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_ah(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(Uu,Uub))
        & pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_520
tff(fact_9852_ATP_Olambda__521,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_ai(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        & pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_521
tff(fact_9853_ATP_Olambda__522,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_aj(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(Uu,Uub))
        & pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_522
tff(fact_9854_ATP_Olambda__523,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ct(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua)) ) ).

% ATP.lambda_523
tff(fact_9855_ATP_Olambda__524,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_uk(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_524
tff(fact_9856_ATP_Olambda__525,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ul(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_525
tff(fact_9857_ATP_Olambda__526,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qn(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uub)),Uu) ).

% ATP.lambda_526
tff(fact_9858_ATP_Olambda__527,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_qo(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uub)),Uu) ).

% ATP.lambda_527
tff(fact_9859_ATP_Olambda__528,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aba(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_528
tff(fact_9860_ATP_Olambda__529,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_aag(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_529
tff(fact_9861_ATP_Olambda__530,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: fun_Pr1607443439640623155t_bool,Uub: produc3843707927480180839at_nat] :
      ( pp(aa_Pro1809015098595223084t_bool(aa_fun5022828243006652836t_bool(aTP_Lamm_hn(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_Pro1809015098595223084t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_530
tff(fact_9862_ATP_Olambda__531,axiom,
    ! [Uu: product_prod_nat_nat,Uua: fun_Pr7757163693017731481t_bool,Uub: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_fun830836278467416944t_bool(aTP_Lamm_hm(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_Pro2232559814647311250t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_531
tff(fact_9863_ATP_Olambda__532,axiom,
    ! [Uu: real,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_ho(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_532
tff(fact_9864_ATP_Olambda__533,axiom,
    ! [Uu: list_nat,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_hp(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_533
tff(fact_9865_ATP_Olambda__534,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_hq(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_534
tff(fact_9866_ATP_Olambda__535,axiom,
    ! [Uu: nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_hr(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_535
tff(fact_9867_ATP_Olambda__536,axiom,
    ! [Uu: int,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_hs(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_536
tff(fact_9868_ATP_Olambda__537,axiom,
    ! [Uu: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_vc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uua,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_537
tff(fact_9869_ATP_Olambda__538,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_lz(Uu),Uua),Uub) = plus_p5714425477246183910nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Uub)),Uua)) ).

% ATP.lambda_538
tff(fact_9870_ATP_Olambda__539,axiom,
    ! [Uu: extended_enat,Uua: extended_enat,Uub: nat] : aa_nat_Extended_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_md(Uu),Uua),Uub) = plus_p3455044024723400733d_enat(Uu,times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uub),Uua)) ).

% ATP.lambda_539
tff(fact_9871_ATP_Olambda__540,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_mc(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_540
tff(fact_9872_ATP_Olambda__541,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mf(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_541
tff(fact_9873_ATP_Olambda__542,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_me(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ).

% ATP.lambda_542
tff(fact_9874_ATP_Olambda__543,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mb(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_543
tff(fact_9875_ATP_Olambda__544,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ma(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_544
tff(fact_9876_ATP_Olambda__545,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aay(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_545
tff(fact_9877_ATP_Olambda__546,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fs(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_546
tff(fact_9878_ATP_Olambda__547,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_abb(Uu),Uua),Uub))
    <=> pp(aa_set_real_bool(member_real(Uub),set_or1633881224788618240n_real(Uu,Uua))) ) ).

% ATP.lambda_547
tff(fact_9879_ATP_Olambda__548,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ql(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_548
tff(fact_9880_ATP_Olambda__549,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_qm(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_549
tff(fact_9881_ATP_Olambda__550,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_vb(Uu),Uua),Uub) = aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_550
tff(fact_9882_ATP_Olambda__551,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_va(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_551
tff(fact_9883_ATP_Olambda__552,axiom,
    ! [Uu: fun_Pr7757163693017731481t_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_fe(Uu),Uua),Uub))
    <=> pp(aa_Pro2232559814647311250t_bool(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub))) ) ).

% ATP.lambda_552
tff(fact_9884_ATP_Olambda__553,axiom,
    ! [Uu: fun_Pr2477239776880934807at_nat,Uua: nat,Uub: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_ff(Uu),Uua),Uub) = aa_Pro8023792847356852296at_nat(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_553
tff(fact_9885_ATP_Olambda__554,axiom,
    ! [Uu: fun_Pr248683899324676269t_bool,Uua: nat,Uub: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_fg(Uu),Uua),Uub) = aa_Pro595088089689268830t_bool(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_554
tff(fact_9886_ATP_Olambda__555,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_fh(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_555
tff(fact_9887_ATP_Olambda__556,axiom,
    ! [Uu: fun_Pr803113274115125473t_bool,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_fi(Uu),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub))) ) ).

% ATP.lambda_556
tff(fact_9888_ATP_Olambda__557,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qq(Uu),Uua),Uub) = aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_557
tff(fact_9889_ATP_Olambda__558,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_qr(Uu),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_558
tff(fact_9890_ATP_Olambda__559,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_up(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub))) ).

% ATP.lambda_559
tff(fact_9891_ATP_Olambda__560,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uo(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_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,Uua),Uub))) ).

% ATP.lambda_560
tff(fact_9892_ATP_Olambda__561,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ub(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_bool_bool(fconj(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_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uuc)),aa_complex_complex(times_times_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,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(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),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_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),zero_zero_complex) ).

% ATP.lambda_561
tff(fact_9893_ATP_Olambda__562,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ud(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_bool_bool(fconj(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_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uuc))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,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(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),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_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),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)))),zero_zero_real) ).

% ATP.lambda_562
tff(fact_9894_ATP_Olambda__563,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_tm(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)),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))),Uuc))),aa_complex_complex(times_times_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(uminus_uminus_real,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(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),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_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub)))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),zero_zero_complex) ).

% ATP.lambda_563
tff(fact_9895_ATP_Olambda__564,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_to(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)),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))),Uuc)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(uminus_uminus_real,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(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),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_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub)))),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)))),zero_zero_real) ).

% ATP.lambda_564
tff(fact_9896_ATP_Olambda__565,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_tx(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_complex_complex(times_times_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,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(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),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_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),zero_zero_complex) ).

% ATP.lambda_565
tff(fact_9897_ATP_Olambda__566,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_tz(Uu),Uua),Uub),Uuc) = 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_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,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(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),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_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),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)))),zero_zero_real) ).

% ATP.lambda_566
tff(fact_9898_ATP_Olambda__567,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_or(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_complex(Uua,Uuc)),aa_complex_complex(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_567
tff(fact_9899_ATP_Olambda__568,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_oz(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_568
tff(fact_9900_ATP_Olambda__569,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ot(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_569
tff(fact_9901_ATP_Olambda__570,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_ox(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_570
tff(fact_9902_ATP_Olambda__571,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ov(Uu),Uua),Uub),Uuc) = aa_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(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_571
tff(fact_9903_ATP_Olambda__572,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_os(Uu),Uua),Uub),Uuc) = aa_complex_complex(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_572
tff(fact_9904_ATP_Olambda__573,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_pa(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_573
tff(fact_9905_ATP_Olambda__574,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ou(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_574
tff(fact_9906_ATP_Olambda__575,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_oy(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_575
tff(fact_9907_ATP_Olambda__576,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ow(Uu),Uua),Uub),Uuc) = aa_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_576
tff(fact_9908_ATP_Olambda__577,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_re(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_577
tff(fact_9909_ATP_Olambda__578,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_ka(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_578
tff(fact_9910_ATP_Olambda__579,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_kb(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_579
tff(fact_9911_ATP_Olambda__580,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_kc(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu)),aa_complex_nat(Uua,Uuc)),aa_complex_nat(Uub,Uuc)) ).

% ATP.lambda_580
tff(fact_9912_ATP_Olambda__581,axiom,
    ! [Uu: complex,Uua: fun_complex_int,Uub: fun_complex_int,Uuc: complex] : aa_complex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_kd(Uu),Uua),Uub),Uuc) = aa_int_int(if_int(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu),aa_complex_int(Uua,Uuc)),aa_complex_int(Uub,Uuc)) ).

% ATP.lambda_581
tff(fact_9913_ATP_Olambda__582,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] : aa_real_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_ke(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_582
tff(fact_9914_ATP_Olambda__583,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_kf(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_583
tff(fact_9915_ATP_Olambda__584,axiom,
    ! [Uu: real,Uua: fun_real_nat,Uub: fun_real_nat,Uuc: real] : aa_real_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_kg(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_584
tff(fact_9916_ATP_Olambda__585,axiom,
    ! [Uu: real,Uua: fun_real_int,Uub: fun_real_int,Uuc: real] : aa_real_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_kh(Uu),Uua),Uub),Uuc) = aa_int_int(if_int(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uuc),Uu),aa_real_int(Uua,Uuc)),aa_real_int(Uub,Uuc)) ).

% ATP.lambda_585
tff(fact_9917_ATP_Olambda__586,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] : aa_int_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_ki(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_586
tff(fact_9918_ATP_Olambda__587,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_kj(Uu),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uuc),Uu),aa_int_rat(Uua,Uuc)),aa_int_rat(Uub,Uuc)) ).

% ATP.lambda_587
tff(fact_9919_ATP_Olambda__588,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ms(Uu),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_mr(Uu),Uua),Uub),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_588
tff(fact_9920_ATP_Olambda__589,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_mq(Uu),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_mp(Uu),Uua),Uub),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_589
tff(fact_9921_ATP_Olambda__590,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mo(Uu),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mn(Uu),Uua),Uub),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_590
tff(fact_9922_ATP_Olambda__591,axiom,
    ! [Uu: nat,Uua: code_integer,Uub: code_integer,Uuc: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_pe(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,if_Code_integer(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uub),if_Code_integer(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_Code_integer,zero_z3403309356797280102nteger))),aa_nat_Code_integer(power_8256067586552552935nteger(Uua),Uuc)) ).

% ATP.lambda_591
tff(fact_9923_ATP_Olambda__592,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_pb(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat),aa_complex_complex(uminus1482373934393186551omplex,Uub)),aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_complex),zero_zero_complex))),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_592
tff(fact_9924_ATP_Olambda__593,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_pf(Uu),Uua),Uub),Uuc) = 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,Uuc),zero_zero_nat)),aa_real_real(uminus_uminus_real,Uub)),aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),one_one_real),zero_zero_real))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uuc)) ).

% ATP.lambda_593
tff(fact_9925_ATP_Olambda__594,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_pd(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat),aa_rat_rat(uminus_uminus_rat,Uub)),aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_rat),zero_zero_rat))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_594
tff(fact_9926_ATP_Olambda__595,axiom,
    ! [Uu: nat,Uua: int,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_pc(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat),aa_int_int(uminus_uminus_int,Uub)),aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_int),zero_zero_int))),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_595
tff(fact_9927_ATP_Olambda__596,axiom,
    ! [Uu: set_nat,Uua: fun_na7671319193681171066at_nat,Uub: fun_na5614638261385347865t_bool,Uuc: list_nat] : aa_list_nat_nat(aa_fun4481408125089960074at_nat(aa_fun2629866330252507297at_nat(aTP_Lamm_ya(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_lis1299702591413342011at_nat(aTP_Lamm_xy(Uua),Uuc)),collect_nat(aa_lis6743544065013719170t_bool(aa_fun3010929636551729660t_bool(aTP_Lamm_xz(Uu),Uub),Uuc))) ).

% ATP.lambda_596
tff(fact_9928_ATP_Olambda__597,axiom,
    ! [Uu: set_set_nat,Uua: fun_se1179353343636554400at_nat,Uub: fun_se680526843285281907t_bool,Uuc: nat] : aa_nat_nat(aa_fun5592962576178777428at_nat(aa_fun3299576543290263109at_nat(aTP_Lamm_xv(Uu),Uua),Uub),Uuc) = groups4248547760180025341at_nat(aa_nat6910189468943142497at_nat(aTP_Lamm_xt(Uua),Uuc),collect_set_nat(aa_nat8857836526276733660t_bool(aa_fun6301692502395215996t_bool(aTP_Lamm_xu(Uu),Uub),Uuc))) ).

% ATP.lambda_597
tff(fact_9929_ATP_Olambda__598,axiom,
    ! [Uu: set_list_nat,Uua: fun_li2771310203728655226at_nat,Uub: fun_li4440390755235668505t_bool,Uuc: nat] : aa_nat_nat(aa_fun1837141565787665786at_nat(aa_fun6292763642686614161at_nat(aTP_Lamm_xq(Uu),Uua),Uub),Uuc) = groups2907647131375434839at_nat(aa_nat3122681260899759163at_nat(aTP_Lamm_xo(Uua),Uuc),collect_list_nat(aa_nat2079453845469346690t_bool(aa_fun2010678459463967740t_bool(aTP_Lamm_xp(Uu),Uub),Uuc))) ).

% ATP.lambda_598
tff(fact_9930_ATP_Olambda__599,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_nat,Uub: fun_nat_fun_nat_bool,Uuc: nat] : aa_nat_nat(aa_fun4644938397070871434at_nat(aa_fun7390876557192650417at_nat(aTP_Lamm_xl(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_xk(Uua),Uuc)),collect_nat(aa_nat_fun_nat_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_bu(Uu),Uub),Uuc))) ).

% ATP.lambda_599
tff(fact_9931_ATP_Olambda__600,axiom,
    ! [Uu: set_nat,Uua: fun_na8457291176974909128ex_nat,Uub: fun_na5941084329111710283x_bool,Uuc: complex] : aa_complex_nat(aa_fun8860606018881707530ex_nat(aa_fun2725303033966715859ex_nat(aTP_Lamm_xh(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_com6383720667567721865at_nat(aTP_Lamm_xg(Uua),Uuc)),collect_nat(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_bw(Uu),Uub),Uuc))) ).

% ATP.lambda_600
tff(fact_9932_ATP_Olambda__601,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_nat,Uub: fun_nat_fun_int_bool,Uuc: int] : aa_int_nat(aa_fun8682134193436059786nt_nat(aa_fun2690579329546488917nt_nat(aTP_Lamm_xe(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_int_fun_nat_nat(aTP_Lamm_xd(Uua),Uuc)),collect_nat(aa_int_fun_nat_bool(aa_fun8184653438340336124t_bool(aTP_Lamm_bv(Uu),Uub),Uuc))) ).

% ATP.lambda_601
tff(fact_9933_ATP_Olambda__602,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_real_nat,Uub: fun_na6801867027684862029l_bool,Uuc: real] : aa_real_nat(aa_fun389422839623581578al_nat(aa_fun9122955408058973909al_nat(aTP_Lamm_xa(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_real_fun_nat_nat(aTP_Lamm_jj(Uua),Uuc)),collect_nat(aa_real_fun_nat_bool(aa_fun6065609019945104124t_bool(aTP_Lamm_jk(Uu),Uub),Uuc))) ).

% ATP.lambda_602
tff(fact_9934_ATP_Olambda__603,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_wy(Uu),Uua),Uub),Uuc) = groups861055069439313189ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_ik(Uua),Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_ca(Uu),Uub),Uuc))) ).

% ATP.lambda_603
tff(fact_9935_ATP_Olambda__604,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_ww(Uu),Uua),Uub),Uuc) = groups1707563613775114915nt_nat(aa_nat_fun_int_nat(aTP_Lamm_ig(Uua),Uuc),collect_int(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_bx(Uu),Uub),Uuc))) ).

% ATP.lambda_604
tff(fact_9936_ATP_Olambda__605,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_wu(Uu),Uua),Uub),Uuc) = groups4696554848551431203al_nat(aa_nat_fun_real_nat(aTP_Lamm_ic(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_br(Uu),Uub),Uuc))) ).

% ATP.lambda_605
tff(fact_9937_ATP_Olambda__606,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_real_nat,Uub: fun_na6801867027684862029l_bool,Uuc: real] : aa_real_nat(aa_fun389422839623581578al_nat(aa_fun9122955408058973909al_nat(aTP_Lamm_jl(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_real_fun_nat_nat(aTP_Lamm_jj(Uua),Uuc)),collect_nat(aa_real_fun_nat_bool(aa_fun6065609019945104124t_bool(aTP_Lamm_jk(Uu),Uub),Uuc))) ).

% ATP.lambda_606
tff(fact_9938_ATP_Olambda__607,axiom,
    ! [Uu: set_complex,Uua: fun_co8378349014452122532t_real,Uub: fun_co2841405903858895691t_bool,Uuc: nat] : aa_nat_real(aa_fun4093520059284520584t_real(aa_fun3817074898130815221t_real(aTP_Lamm_jg(Uu),Uua),Uub),Uuc) = groups5808333547571424918x_real(aa_nat480766339200631653x_real(aTP_Lamm_jf(Uua),Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_ca(Uu),Uub),Uuc))) ).

% ATP.lambda_607
tff(fact_9939_ATP_Olambda__608,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_real,Uub: fun_int_fun_nat_bool,Uuc: nat] : aa_nat_real(aa_fun6355566433556109194t_real(aa_fun7034273878219604473t_real(aTP_Lamm_jd(Uu),Uua),Uub),Uuc) = groups8778361861064173332t_real(aa_nat_fun_int_real(aTP_Lamm_jc(Uua),Uuc),collect_int(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_bx(Uu),Uub),Uuc))) ).

% ATP.lambda_608
tff(fact_9940_ATP_Olambda__609,axiom,
    ! [Uu: set_real,Uua: fun_re4238438383274954274t_real,Uub: fun_re9121486324101951821t_bool,Uuc: nat] : aa_nat_real(aa_fun555827300007793674t_real(aa_fun8353879187736340217t_real(aTP_Lamm_ja(Uu),Uua),Uub),Uuc) = groups8097168146408367636l_real(aa_nat_fun_real_real(aTP_Lamm_iz(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_br(Uu),Uub),Uuc))) ).

% ATP.lambda_609
tff(fact_9941_ATP_Olambda__610,axiom,
    ! [Uu: set_int,Uua: fun_in1307110748760623106omplex,Uub: fun_in5750378413879446255x_bool,Uuc: complex] : aa_complex_complex(aa_fun2686559102040213132omplex(aa_fun5422379709423809691omplex(aTP_Lamm_ix(Uu),Uua),Uub),Uuc) = groups3049146728041665814omplex(aa_com9156536935073639875omplex(aTP_Lamm_iw(Uua),Uuc),collect_int(aa_com2211550117870696024t_bool(aa_fun3087709329828230012t_bool(aTP_Lamm_bz(Uu),Uub),Uuc))) ).

% ATP.lambda_610
tff(fact_9942_ATP_Olambda__611,axiom,
    ! [Uu: set_nat,Uua: fun_na1591161823130732966omplex,Uub: fun_na5941084329111710283x_bool,Uuc: complex] : aa_complex_complex(aa_fun1613615970879741416omplex(aa_fun7586814990830484307omplex(aTP_Lamm_it(Uu),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_is(Uua),Uuc),collect_nat(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_bw(Uu),Uub),Uuc))) ).

% ATP.lambda_611
tff(fact_9943_ATP_Olambda__612,axiom,
    ! [Uu: set_real,Uua: fun_re7208826350674946562omplex,Uub: fun_re3010880487650781423x_bool,Uuc: complex] : aa_complex_complex(aa_fun8128191506076533516omplex(aa_fun840110154952133019omplex(aTP_Lamm_ip(Uu),Uua),Uub),Uuc) = groups5754745047067104278omplex(aa_com4766338864777664963omplex(aTP_Lamm_io(Uua),Uuc),collect_real(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_bt(Uu),Uub),Uuc))) ).

% ATP.lambda_612
tff(fact_9944_ATP_Olambda__613,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_il(Uu),Uua),Uub),Uuc) = groups5693394587270226106ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_ik(Uua),Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_ca(Uu),Uub),Uuc))) ).

% ATP.lambda_613
tff(fact_9945_ATP_Olambda__614,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_ih(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_ig(Uua),Uuc),collect_int(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_bx(Uu),Uub),Uuc))) ).

% ATP.lambda_614
tff(fact_9946_ATP_Olambda__615,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_id(Uu),Uua),Uub),Uuc) = groups1935376822645274424al_nat(aa_nat_fun_real_nat(aTP_Lamm_ic(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_br(Uu),Uub),Uuc))) ).

% ATP.lambda_615
tff(fact_9947_ATP_Olambda__616,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_oq(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_op(Uu),Uua),Uuc)),set_ord_atMost_nat(Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uub),Uuc)) ).

% ATP.lambda_616
tff(fact_9948_ATP_Olambda__617,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_on(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_om(Uu),Uua),Uuc),set_ord_atMost_nat(Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_617
tff(fact_9949_ATP_Olambda__618,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_ol(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_ok(Uu),Uua),Uuc),set_ord_atMost_nat(Uuc))),aa_nat_int(power_power_int(Uub),Uuc)) ).

% ATP.lambda_618
tff(fact_9950_ATP_Olambda__619,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_oj(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_oi(Uu),Uua),Uuc),set_ord_atMost_nat(Uuc))),aa_nat_rat(power_power_rat(Uub),Uuc)) ).

% ATP.lambda_619
tff(fact_9951_ATP_Olambda__620,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_oh(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_og(Uu),Uua),Uuc),set_ord_atMost_nat(Uuc))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_620
tff(fact_9952_ATP_Olambda__621,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_aan(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_621
tff(fact_9953_ATP_Olambda__622,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_aal(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_622
tff(fact_9954_ATP_Olambda__623,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_aak(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_623
tff(fact_9955_ATP_Olambda__624,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_aaj(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_624
tff(fact_9956_ATP_Olambda__625,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_zq(Uu),Uua),Uub),Uuc) = 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,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uub))) ).

% ATP.lambda_625
tff(fact_9957_ATP_Olambda__626,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mn(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uua),Uu)),Uuc)),aa_nat_complex(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_626
tff(fact_9958_ATP_Olambda__627,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_mr(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_627
tff(fact_9959_ATP_Olambda__628,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_mp(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_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_628
tff(fact_9960_ATP_Olambda__629,axiom,
    ! [Uu: set_complex,Uua: fun_re3010880487650781423x_bool,Uub: real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_rea5887071448072802392x_bool(aa_fun9179849694055305084x_bool(aTP_Lamm_im(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_629
tff(fact_9961_ATP_Olambda__630,axiom,
    ! [Uu: set_complex,Uua: fun_na5941084329111710283x_bool,Uub: nat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aa_fun2551091230351774588x_bool(aTP_Lamm_iq(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_630
tff(fact_9962_ATP_Olambda__631,axiom,
    ! [Uu: set_complex,Uua: fun_in5750378413879446255x_bool,Uub: int,Uuc: complex] :
      ( pp(aa_complex_bool(aa_int5354780257279585880x_bool(aa_fun6456964791495427196x_bool(aTP_Lamm_iu(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_631
tff(fact_9963_ATP_Olambda__632,axiom,
    ! [Uu: set_real,Uua: fun_na6801867027684862029l_bool,Uub: nat,Uuc: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aa_fun3745989723528014332l_bool(aTP_Lamm_jh(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_real_bool(aa_nat_fun_real_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_632
tff(fact_9964_ATP_Olambda__633,axiom,
    ! [Uu: set_list_nat,Uua: fun_na5614638261385347865t_bool,Uub: nat,Uuc: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aa_fun4185177142701409020t_bool(aTP_Lamm_xw(Uu),Uua),Uub),Uuc))
    <=> ( member_list_nat(Uuc,Uu)
        & pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_633
tff(fact_9965_ATP_Olambda__634,axiom,
    ! [Uu: set_nat,Uua: fun_co2841405903858895691t_bool,Uub: complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_ii(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_634
tff(fact_9966_ATP_Olambda__635,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_ia(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_635
tff(fact_9967_ATP_Olambda__636,axiom,
    ! [Uu: set_nat,Uua: fun_li4440390755235668505t_bool,Uub: list_nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_lis6743544065013719170t_bool(aa_fun836430953314288380t_bool(aTP_Lamm_xm(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_lis6743544065013719170t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_636
tff(fact_9968_ATP_Olambda__637,axiom,
    ! [Uu: set_nat,Uua: fun_se680526843285281907t_bool,Uub: set_nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aa_fun1940116245756961148t_bool(aTP_Lamm_xr(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_set3744802623701456604t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_637
tff(fact_9969_ATP_Olambda__638,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_xi(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_638
tff(fact_9970_ATP_Olambda__639,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_ie(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_639
tff(fact_9971_ATP_Olambda__640,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_bool,Uub: nat,Uuc: int] :
      ( pp(aa_int_bool(aa_nat_fun_int_bool(aa_fun7617171277304647676t_bool(aTP_Lamm_xb(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_nat_fun_int_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_640
tff(fact_9972_ATP_Olambda__641,axiom,
    ! [Uu: set_complex,Uua: fun_co2841405903858895691t_bool,Uub: nat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_ca(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_641
tff(fact_9973_ATP_Olambda__642,axiom,
    ! [Uu: set_real,Uua: fun_re3010880487650781423x_bool,Uub: complex,Uuc: real] :
      ( pp(aa_real_bool(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_bt(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_642
tff(fact_9974_ATP_Olambda__643,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_br(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_643
tff(fact_9975_ATP_Olambda__644,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_bs(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_644
tff(fact_9976_ATP_Olambda__645,axiom,
    ! [Uu: set_list_nat,Uua: fun_li4440390755235668505t_bool,Uub: nat,Uuc: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aa_fun2010678459463967740t_bool(aTP_Lamm_xp(Uu),Uua),Uub),Uuc))
    <=> ( member_list_nat(Uuc,Uu)
        & pp(aa_nat_bool(aa_lis6743544065013719170t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_645
tff(fact_9977_ATP_Olambda__646,axiom,
    ! [Uu: set_set_nat,Uua: fun_se680526843285281907t_bool,Uub: nat,Uuc: set_nat] :
      ( pp(aa_set_nat_bool(aa_nat8857836526276733660t_bool(aa_fun6301692502395215996t_bool(aTP_Lamm_xu(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_set3744802623701456604t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_646
tff(fact_9978_ATP_Olambda__647,axiom,
    ! [Uu: set_nat,Uua: fun_na5941084329111710283x_bool,Uub: complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_bw(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_647
tff(fact_9979_ATP_Olambda__648,axiom,
    ! [Uu: set_nat,Uua: fun_na6801867027684862029l_bool,Uub: real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_real_fun_nat_bool(aa_fun6065609019945104124t_bool(aTP_Lamm_jk(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_real_bool(aa_nat_fun_real_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_648
tff(fact_9980_ATP_Olambda__649,axiom,
    ! [Uu: set_nat,Uua: fun_na5614638261385347865t_bool,Uub: list_nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_lis6743544065013719170t_bool(aa_fun3010929636551729660t_bool(aTP_Lamm_xz(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_649
tff(fact_9981_ATP_Olambda__650,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_bu(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_650
tff(fact_9982_ATP_Olambda__651,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_bv(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_651
tff(fact_9983_ATP_Olambda__652,axiom,
    ! [Uu: set_int,Uua: fun_in5750378413879446255x_bool,Uub: complex,Uuc: int] :
      ( pp(aa_int_bool(aa_com2211550117870696024t_bool(aa_fun3087709329828230012t_bool(aTP_Lamm_bz(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_652
tff(fact_9984_ATP_Olambda__653,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_bx(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_653
tff(fact_9985_ATP_Olambda__654,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_bool,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_fun8168818243683596540t_bool(aTP_Lamm_by(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_int_fun_int_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_654
tff(fact_9986_ATP_Olambda__655,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_zv(Uu),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_655
tff(fact_9987_ATP_Olambda__656,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_yv(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_656
tff(fact_9988_ATP_Olambda__657,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_zt(Uu),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_657
tff(fact_9989_ATP_Olambda__658,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_zx(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)) ).

% ATP.lambda_658
tff(fact_9990_ATP_Olambda__659,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_zz(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_659
tff(fact_9991_ATP_Olambda__660,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_vq(Uu),Uua),Uub),Uuc))
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_660
tff(fact_9992_ATP_Olambda__661,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_vr(Uu),Uua),Uub),Uuc))
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_661
tff(fact_9993_ATP_Olambda__662,axiom,
    ! [Uu: fun_Co6545587727236015215e_term,Uua: produc8923325533196201883nteger,Uub: fun_Co6545587727236015215e_term,Uuc: produc8923325533196201883nteger] :
      ( pp(aa_Pro5843519176478241656r_bool(aa_fun5646951941296221740r_bool(aa_Pro4785828671252635594r_bool(aTP_Lamm_vm(Uu),Uua),Uub),Uuc))
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_662
tff(fact_9994_ATP_Olambda__663,axiom,
    ! [Uu: fun_Pr272337284026993017e_term,Uua: produc8923325533196201883nteger,Uub: fun_Pr272337284026993017e_term,Uuc: produc8923325533196201883nteger] :
      ( pp(aa_Pro5843519176478241656r_bool(aa_fun5449300841655810358r_bool(aa_Pro8070820463477474004r_bool(aTP_Lamm_vn(Uu),Uua),Uub),Uuc))
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_663
tff(fact_9995_ATP_Olambda__664,axiom,
    ! [Uu: fun_Pr9164807099015279978e_term,Uua: product_prod_int_int,Uub: fun_Pr9164807099015279978e_term,Uuc: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_fun5300525129335604105t_bool(aa_Pro7791619889881332805t_bool(aTP_Lamm_vo(Uu),Uua),Uub),Uuc))
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_664
tff(fact_9996_ATP_Olambda__665,axiom,
    ! [Uu: fun_in6429348457993719614e_term,Uua: product_prod_int_int,Uub: fun_in6429348457993719614e_term,Uuc: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_fun4446166836006121821t_bool(aa_Pro4937060880834667033t_bool(aTP_Lamm_vp(Uu),Uua),Uub),Uuc))
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_665
tff(fact_9997_ATP_Olambda__666,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_dq(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_complex_complex(times_times_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_666
tff(fact_9998_ATP_Olambda__667,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_dy(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_667
tff(fact_9999_ATP_Olambda__668,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_dk(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_complex_complex(times_times_complex(aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_668
tff(fact_10000_ATP_Olambda__669,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_ds(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_669
tff(fact_10001_ATP_Olambda__670,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_ea(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_670
tff(fact_10002_ATP_Olambda__671,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_dm(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_671
tff(fact_10003_ATP_Olambda__672,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_du(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_672
tff(fact_10004_ATP_Olambda__673,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_ec(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_673
tff(fact_10005_ATP_Olambda__674,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_do(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_complex_complex(times_times_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_674
tff(fact_10006_ATP_Olambda__675,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_dw(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_675
tff(fact_10007_ATP_Olambda__676,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_ek(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_complex_complex(plus_plus_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_676
tff(fact_10008_ATP_Olambda__677,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_es(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_677
tff(fact_10009_ATP_Olambda__678,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_ee(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_complex_complex(plus_plus_complex(aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_678
tff(fact_10010_ATP_Olambda__679,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_em(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_679
tff(fact_10011_ATP_Olambda__680,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_eu(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_680
tff(fact_10012_ATP_Olambda__681,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_eg(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_complex_complex(plus_plus_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_681
tff(fact_10013_ATP_Olambda__682,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_eo(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_682
tff(fact_10014_ATP_Olambda__683,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_ew(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_683
tff(fact_10015_ATP_Olambda__684,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_ei(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_complex_complex(plus_plus_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_684
tff(fact_10016_ATP_Olambda__685,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_eq(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_685
tff(fact_10017_ATP_Olambda__686,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nj(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uuc)),aa_nat_complex(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_686
tff(fact_10018_ATP_Olambda__687,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ns(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_687
tff(fact_10019_ATP_Olambda__688,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nm(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_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_688
tff(fact_10020_ATP_Olambda__689,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_np(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uuc)),aa_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_689
tff(fact_10021_ATP_Olambda__690,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_og(Uu),Uua),Uub),Uuc) = aa_complex_complex(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_690
tff(fact_10022_ATP_Olambda__691,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_om(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_691
tff(fact_10023_ATP_Olambda__692,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_oi(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_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_692
tff(fact_10024_ATP_Olambda__693,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_op(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_693
tff(fact_10025_ATP_Olambda__694,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_ok(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_694
tff(fact_10026_ATP_Olambda__695,axiom,
    ! [Uu: set_list_nat,Uua: fun_na7671319193681171066at_nat,Uub: fun_na5614638261385347865t_bool,Uuc: nat] : aa_nat_nat(aa_fun5703926054570661498at_nat(aa_fun1685042437804453009at_nat(aTP_Lamm_xx(Uu),Uua),Uub),Uuc) = groups2907647131375434839at_nat(aa_nat3122681260899759163at_nat(Uua,Uuc),collect_list_nat(aa_nat2079453845469346690t_bool(aa_fun4185177142701409020t_bool(aTP_Lamm_xw(Uu),Uub),Uuc))) ).

% ATP.lambda_695
tff(fact_10027_ATP_Olambda__696,axiom,
    ! [Uu: set_nat,Uua: fun_se1179353343636554400at_nat,Uub: fun_se680526843285281907t_bool,Uuc: set_nat] : aa_set_nat_nat(aa_fun7954195629112176906at_nat(aa_fun2162494588655638267at_nat(aTP_Lamm_xs(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_set6537093552120046177at_nat(Uua,Uuc)),collect_nat(aa_set3744802623701456604t_bool(aa_fun1940116245756961148t_bool(aTP_Lamm_xr(Uu),Uub),Uuc))) ).

% ATP.lambda_696
tff(fact_10028_ATP_Olambda__697,axiom,
    ! [Uu: set_nat,Uua: fun_li2771310203728655226at_nat,Uub: fun_li4440390755235668505t_bool,Uuc: list_nat] : aa_list_nat_nat(aa_fun8445324095761362826at_nat(aa_fun504646108086548129at_nat(aTP_Lamm_xn(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_lis1299702591413342011at_nat(Uua,Uuc)),collect_nat(aa_lis6743544065013719170t_bool(aa_fun836430953314288380t_bool(aTP_Lamm_xm(Uu),Uub),Uuc))) ).

% ATP.lambda_697
tff(fact_10029_ATP_Olambda__698,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_nat,Uub: fun_nat_fun_nat_bool,Uuc: nat] : aa_nat_nat(aa_fun4644938397070871434at_nat(aa_fun7390876557192650417at_nat(aTP_Lamm_xj(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uua,Uuc)),collect_nat(aa_nat_fun_nat_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_xi(Uu),Uub),Uuc))) ).

% ATP.lambda_698
tff(fact_10030_ATP_Olambda__699,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_nat,Uub: fun_nat_fun_int_bool,Uuc: nat] : aa_nat_nat(aa_fun7682780481535005230at_nat(aa_fun503657754853467129at_nat(aTP_Lamm_xc(Uu),Uua),Uub),Uuc) = groups1707563613775114915nt_nat(aa_nat_fun_int_nat(Uua,Uuc),collect_int(aa_nat_fun_int_bool(aa_fun7617171277304647676t_bool(aTP_Lamm_xb(Uu),Uub),Uuc))) ).

% ATP.lambda_699
tff(fact_10031_ATP_Olambda__700,axiom,
    ! [Uu: set_real,Uua: fun_nat_fun_real_nat,Uub: fun_na6801867027684862029l_bool,Uuc: nat] : aa_nat_nat(aa_fun8064793419647352750at_nat(aa_fun581391774837177593at_nat(aTP_Lamm_wz(Uu),Uua),Uub),Uuc) = groups4696554848551431203al_nat(aa_nat_fun_real_nat(Uua,Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3745989723528014332l_bool(aTP_Lamm_jh(Uu),Uub),Uuc))) ).

% ATP.lambda_700
tff(fact_10032_ATP_Olambda__701,axiom,
    ! [Uu: set_complex,Uua: fun_na8457291176974909128ex_nat,Uub: fun_na5941084329111710283x_bool,Uuc: nat] : aa_nat_nat(aa_fun7456585239162712364at_nat(aa_fun5124956507111473653at_nat(aTP_Lamm_xf(Uu),Uua),Uub),Uuc) = groups861055069439313189ex_nat(aa_nat3391284541783913865ex_nat(Uua,Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun2551091230351774588x_bool(aTP_Lamm_iq(Uu),Uub),Uuc))) ).

% ATP.lambda_701
tff(fact_10033_ATP_Olambda__702,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_wx(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_com6383720667567721865at_nat(Uua,Uuc)),collect_nat(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_ii(Uu),Uub),Uuc))) ).

% ATP.lambda_702
tff(fact_10034_ATP_Olambda__703,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_wv(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_int_fun_nat_nat(Uua,Uuc)),collect_nat(aa_int_fun_nat_bool(aa_fun3602880909931171836t_bool(aTP_Lamm_ie(Uu),Uub),Uuc))) ).

% ATP.lambda_703
tff(fact_10035_ATP_Olambda__704,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_wt(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_real_fun_nat_nat(Uua,Uuc)),collect_nat(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_ia(Uu),Uub),Uuc))) ).

% ATP.lambda_704
tff(fact_10036_ATP_Olambda__705,axiom,
    ! [Uu: set_real,Uua: fun_nat_fun_real_nat,Uub: fun_na6801867027684862029l_bool,Uuc: nat] : aa_nat_nat(aa_fun8064793419647352750at_nat(aa_fun581391774837177593at_nat(aTP_Lamm_ji(Uu),Uua),Uub),Uuc) = groups1935376822645274424al_nat(aa_nat_fun_real_nat(Uua,Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3745989723528014332l_bool(aTP_Lamm_jh(Uu),Uub),Uuc))) ).

% ATP.lambda_705
tff(fact_10037_ATP_Olambda__706,axiom,
    ! [Uu: set_complex,Uua: fun_in1307110748760623106omplex,Uub: fun_in5750378413879446255x_bool,Uuc: int] : aa_int_complex(aa_fun751309451324587018omplex(aa_fun3438240386181387033omplex(aTP_Lamm_iv(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_int3545933452204062147omplex(Uua,Uuc),collect_complex(aa_int5354780257279585880x_bool(aa_fun6456964791495427196x_bool(aTP_Lamm_iu(Uu),Uub),Uuc))) ).

% ATP.lambda_706
tff(fact_10038_ATP_Olambda__707,axiom,
    ! [Uu: set_complex,Uua: fun_na1591161823130732966omplex,Uub: fun_na5941084329111710283x_bool,Uuc: nat] : aa_nat_complex(aa_fun2700696402887535882omplex(aa_fun8019143150332516725omplex(aTP_Lamm_ir(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(Uua,Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun2551091230351774588x_bool(aTP_Lamm_iq(Uu),Uub),Uuc))) ).

% ATP.lambda_707
tff(fact_10039_ATP_Olambda__708,axiom,
    ! [Uu: set_complex,Uua: fun_re7208826350674946562omplex,Uub: fun_re3010880487650781423x_bool,Uuc: real] : aa_real_complex(aa_fun4873726854752153866omplex(aa_fun883667864959905433omplex(aTP_Lamm_in(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_rea2618301233319580611omplex(Uua,Uuc),collect_complex(aa_rea5887071448072802392x_bool(aa_fun9179849694055305084x_bool(aTP_Lamm_im(Uu),Uub),Uuc))) ).

% ATP.lambda_708
tff(fact_10040_ATP_Olambda__709,axiom,
    ! [Uu: set_nat,Uua: fun_co8378349014452122532t_real,Uub: fun_co2841405903858895691t_bool,Uuc: complex] : aa_complex_real(aa_fun7842767745293442150x_real(aa_fun7038367274271708115x_real(aTP_Lamm_je(Uu),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_com20334259897932645t_real(Uua,Uuc),collect_nat(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_ii(Uu),Uub),Uuc))) ).

% ATP.lambda_709
tff(fact_10041_ATP_Olambda__710,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_ij(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_ii(Uu),Uub),Uuc))) ).

% ATP.lambda_710
tff(fact_10042_ATP_Olambda__711,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_real,Uub: fun_int_fun_nat_bool,Uuc: int] : aa_int_real(aa_fun8542488008249130982t_real(aa_fun1029605435529198677t_real(aTP_Lamm_jb(Uu),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_int_fun_nat_real(Uua,Uuc),collect_nat(aa_int_fun_nat_bool(aa_fun3602880909931171836t_bool(aTP_Lamm_ie(Uu),Uub),Uuc))) ).

% ATP.lambda_711
tff(fact_10043_ATP_Olambda__712,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_if(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_ie(Uu),Uub),Uuc))) ).

% ATP.lambda_712
tff(fact_10044_ATP_Olambda__713,axiom,
    ! [Uu: set_nat,Uua: fun_re4238438383274954274t_real,Uub: fun_re9121486324101951821t_bool,Uuc: real] : aa_real_real(aa_fun7981600115099051494l_real(aa_fun2847282595593119957l_real(aTP_Lamm_iy(Uu),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_real_fun_nat_real(Uua,Uuc),collect_nat(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_ia(Uu),Uub),Uuc))) ).

% ATP.lambda_713
tff(fact_10045_ATP_Olambda__714,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_ib(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_ia(Uu),Uub),Uuc))) ).

% ATP.lambda_714
tff(fact_10046_ATP_Olambda__715,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_yz(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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_715
tff(fact_10047_ATP_Olambda__716,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_yx(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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_716
tff(fact_10048_ATP_Olambda__717,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_zd(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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_717
tff(fact_10049_ATP_Olambda__718,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_zb(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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_718
tff(fact_10050_ATP_Olambda__719,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_pn(Uu),Uua),Uub),Uuc),Uud) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_pm(Uua),Uub),Uuc),Uud),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_719
tff(fact_10051_ATP_Olambda__720,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_pl(Uu),Uua),Uub),Uuc),Uud) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(aTP_Lamm_pk(Uua),Uub),Uuc),Uud),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_720
tff(fact_10052_ATP_Olambda__721,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_pj(Uu),Uua),Uub),Uuc),Uud) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aa_rat8984911954100663005at_rat(aTP_Lamm_pi(Uua),Uub),Uuc),Uud),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_721
tff(fact_10053_ATP_Olambda__722,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_ph(Uu),Uua),Uub),Uuc),Uud) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_pg(Uua),Uub),Uuc),Uud),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_722
tff(fact_10054_ATP_Olambda__723,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_aam(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_aal(Uua),Uuc),Uud),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_723
tff(fact_10055_ATP_Olambda__724,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_nt(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_ns(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_724
tff(fact_10056_ATP_Olambda__725,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_nq(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_np(Uua),Uuc),Uud),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud),Uu))),aa_nat_int(power_power_int(Uub),Uud)) ).

% ATP.lambda_725
tff(fact_10057_ATP_Olambda__726,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_nn(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nm(Uua),Uuc),Uud),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud),Uu))),aa_nat_rat(power_power_rat(Uub),Uud)) ).

% ATP.lambda_726
tff(fact_10058_ATP_Olambda__727,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_nk(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nj(Uua),Uuc),Uud),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud),Uu))),aa_nat_complex(power_power_complex(Uub),Uud)) ).

% ATP.lambda_727
tff(fact_10059_ATP_Olambda__728,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: complex,Uuc: nat,Uud: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_pg(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(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(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_728
tff(fact_10060_ATP_Olambda__729,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_pm(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_729
tff(fact_10061_ATP_Olambda__730,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_pi(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_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_730
tff(fact_10062_ATP_Olambda__731,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_pk(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(power_power_int(Uub),Uud))),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_731
tff(fact_10063_ATP_Olambda__732,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_abk(Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uuc),Uud))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uud),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu)))
        & ! [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(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))
           => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(Uua,I4)),X_13))
            <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Uub),I4)) ) )
        & ( ( Uuc = Uud )
         => ! [X3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Uua)))
             => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_13)) ) )
        & ( ( Uuc != Uud )
         => ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),Uud))
            & ! [X3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu)))
               => ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),X3))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),X3))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Uud)) ) ) ) ) ) ) ) ).

% ATP.lambda_732
tff(fact_10064_ATP_Olambda__733,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_um(Uu),Uua) = Uu ).

% ATP.lambda_733
tff(fact_10065_ATP_Olambda__734,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_un,Uu),Uua) = Uu ).

% ATP.lambda_734
tff(fact_10066_ATP_Olambda__735,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_mg,Uu) = Uu ).

% ATP.lambda_735
tff(fact_10067_ATP_Olambda__736,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ly,Uu) = Uu ).

% ATP.lambda_736
tff(fact_10068_ATP_Olambda__737,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_acm,Uu) = Uu ).

% ATP.lambda_737
tff(fact_10069_ATP_Olambda__738,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_ch,Uu) = zero_zero_complex ).

% ATP.lambda_738
tff(fact_10070_ATP_Olambda__739,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ci,Uu) = zero_zero_real ).

% ATP.lambda_739
tff(fact_10071_ATP_Olambda__740,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_cj,Uu) = zero_zero_rat ).

% ATP.lambda_740
tff(fact_10072_ATP_Olambda__741,axiom,
    ! [Uu: nat] : aa_nat_complex(aTP_Lamm_qg,Uu) = zero_zero_complex ).

% ATP.lambda_741
tff(fact_10073_ATP_Olambda__742,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ft,Uu) = zero_zero_real ).

% ATP.lambda_742
tff(fact_10074_ATP_Olambda__743,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ck,Uu) = zero_zero_nat ).

% ATP.lambda_743
tff(fact_10075_ATP_Olambda__744,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_qh,Uu) = zero_zero_int ).

% ATP.lambda_744
tff(fact_10076_ATP_Olambda__745,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_cl,Uu) = zero_zero_int ).

% ATP.lambda_745
tff(fact_10077_ATP_Olambda__746,axiom,
    ! [Uu: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aTP_Lamm_cd,Uu))
    <=> $false ) ).

% ATP.lambda_746
tff(fact_10078_ATP_Olambda__747,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_ce,Uu))
    <=> $false ) ).

% ATP.lambda_747
tff(fact_10079_ATP_Olambda__748,axiom,
    ! [Uu: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_cb,Uu))
    <=> $false ) ).

% ATP.lambda_748
tff(fact_10080_ATP_Olambda__749,axiom,
    ! [Uu: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_cc,Uu))
    <=> $false ) ).

% ATP.lambda_749
tff(fact_10081_ATP_Olambda__750,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_cf,Uu))
    <=> $false ) ).

% ATP.lambda_750
tff(fact_10082_ATP_Olambda__751,axiom,
    ! [Uu: int] :
      ( pp(aa_int_bool(aTP_Lamm_cg,Uu))
    <=> $false ) ).

% ATP.lambda_751
tff(fact_10083_ATP_Olambda__752,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_abt,Uu))
    <=> $true ) ).

% ATP.lambda_752

% Helper facts (72)
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(aa_boo8970175072276269406l_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(aa_boo8970175072276269406l_bool(fimplies,P),Q)) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( pp(P)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q)) ) ).

tff(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] : aa_int_int(if_int(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] : aa_int_int(if_int(fTrue,X),Y) = X ).

tff(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fTrue),X),Y) = X ).

tff(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] : aa_rat_rat(if_rat(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] : aa_rat_rat(if_rat(fTrue,X),Y) = X ).

tff(help_fEx_1_1_fEx_001t__Nat__Onat_U,axiom,
    ! [P: fun_nat_bool,X: nat] :
      ( ~ pp(aa_nat_bool(P,X))
      | pp(aa_fun_nat_bool_bool(fEx_nat,P)) ) ).

tff(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(if_real(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(if_real(fTrue),X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      | pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(fequal_rat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(fequal_rat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X: bool,Y: bool] :
      ( ( X != Y )
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X: bool,Y: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X),Y))
      | ( X = Y ) ) ).

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__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(if_complex(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(if_complex(fTrue,X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( X != Y )
      | pp(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] : if_Extended_enat(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] : if_Extended_enat(fTrue,X,Y) = X ).

tff(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] : if_Code_integer(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] : if_Code_integer(fTrue,X,Y) = X ).

tff(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] : if_VEBT_VEBT(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] : if_VEBT_VEBT(fTrue,X,Y) = X ).

tff(help_fAll_1_1_fAll_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( ~ pp(fAll_VEBT_VEBT(P))
      | pp(aa_VEBT_VEBT_bool(P,X)) ) ).

tff(help_fequal_2_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( X != Y )
      | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X != Y )
      | pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(fequal_set_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(fequal_set_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] : if_option_nat(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] : if_option_nat(fTrue,X,Y) = X ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( X != Y )
      | pp(aa_list_nat_bool(fequal_list_nat(X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ~ pp(aa_list_nat_bool(fequal_list_nat(X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__fun_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: fun_int_int,Y: fun_int_int] : if_fun_int_int(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__fun_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: fun_int_int,Y: fun_int_int] : if_fun_int_int(fTrue,X,Y) = X ).

tff(help_fequal_2_1_fequal_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( X != Y )
      | pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] : aa_Pro6970504127699319548nt_int(if_Pro3027730157355071871nt_int(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] : aa_Pro6970504127699319548nt_int(if_Pro3027730157355071871nt_int(fTrue,X),Y) = X ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(fTrue),X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( X != Y )
      | pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(fequal9188845342289841878at_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ~ pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(fequal9188845342289841878at_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_COMBB_1_1_COMBB_001t__HOL__Obool_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_bool_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_3745307990312268622T_VEBT(P,Q),R) = aa_bool_bool(P,aa_VEBT_VEBT_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool_001t__HOL__Obool_U,axiom,
    ! [P: fun_VE2066785035971159626l_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBS_6222329278419335549l_bool(P,Q),R) = aa_bool_bool(aa_VEB8047906998784800507l_bool(P,R),aa_VEBT_VEBT_bool(Q,R)) ).

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,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fTrue,X,Y) = X ).

tff(help_COMBB_1_1_COMBB_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_fu7168253649841883962l_bool,Q: fun_VE3763956218935105199t_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_2885587777813823575T_VEBT(P,Q),R) = aa_fun_nat_bool_bool(P,aa_VEB4832260192069861672t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_001t__HOL__Obool_001t__fun_It__HOL__Obool_Mt__HOL__Obool_J_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_bo1841576699660343469l_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEB8047906998784800507l_bool(cOMBB_3438363824644082366T_VEBT(P,Q),R) = aa_boo8970175072276269406l_bool(P,aa_VEBT_VEBT_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_001t__VEBT____Definitions__OVEBT_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_001t__HOL__Obool_U,axiom,
    ! [P: fun_VE7346385682144900493T_bool,Q: set_VEBT_VEBT,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBC_771164880915498576T_bool(P,Q),R) = aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(P,R),Q) ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( X != Y )
      | pp(aa_Pro1809015098595223084t_bool(fequal7949547063978738364at_nat(X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ~ pp(aa_Pro1809015098595223084t_bool(fequal7949547063978738364at_nat(X),Y))
      | ( X = Y ) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    pp(aa_nat_bool(vEBT_invar_vebt(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na))),na)) ).

%------------------------------------------------------------------------------