%------------------------------------------------------------------------------
% File     : ITP242_1 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Succ 00204_010200
% 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    : 0070_VEBT_Succ_00204_010200 [Des22]

% Status   : Theorem
% Rating   : 0.89 v8.2.0, 0.90 v8.1.0
% Syntax   : Number of formulae    : 13177 (4068 unt;2598 typ;   0 def)
%            Number of atoms       : 25952 (9886 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 17923 (2550   ~; 510   |;1829   &)
%                                         (2727 <=>;10307  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   23 (   2 avg)
%            Number of types       :  541 ( 540 usr)
%            Number of type conns  : 2439 (1643   >; 796   *;   0   +;   0  <<)
%            Number of predicates  :   83 (  80 usr;   2 prp; 0-4 aty)
%            Number of functors    : 1978 (1978 usr; 415 con; 0-8 aty)
%            Number of variables   : 25373 (24606   !; 767   ?;25373   :)
% 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 00:02:48.728
%------------------------------------------------------------------------------
% Could-be-implicit typings (540)
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Code____Numeral__Ointeger_J,type,
    fun_re6978948838062957830nteger: $tType ).

tff(ty_n_t__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J,type,
    set_list_complex: $tType ).

tff(ty_n_t__fun_It__VEBT____Definitions__OVEBT_Mt__Rat__Orat_J,type,
    fun_VEBT_VEBT_rat: $tType ).

tff(ty_n_t__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    fun_VEBT_VEBT_nat: $tType ).

tff(ty_n_t__fun_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Int__Oint_J_Mt__HOL__Obool_J,type,
    fun_set_int_bool: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
    fun_int_VEBT_VEBT: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__HOL__Obool_J,type,
    fun_Co8822979462182099027r_bool: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Code____Numeral__Ointeger_J,type,
    fun_num_Code_integer: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    fun_nat_set_nat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J,type,
    fun_nat_Code_integer: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
    fun_complex_complex: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J,type,
    fun_Code_integer_nat: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__Int__Oint_J,type,
    fun_Code_integer_int: $tType ).

tff(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
    set_set_complex: $tType ).

tff(ty_n_t__fun_It__Extended____Nat__Oenat_Mt__HOL__Obool_J,type,
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tff(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    list_list_nat: $tType ).

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,
    fun_real_complex: $tType ).

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,
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tff(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
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tff(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(ty_n_t__List__Olist_It__Code____Numeral__Ointeger_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__HOL__Obool_J,type,
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tff(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    set_VEBT_VEBT: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J,type,
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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,
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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,
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tff(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    set_set_nat: $tType ).

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__fun_It__String__Ochar_Mt__Nat__Onat_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__Nat__Onat_Mt__String__Ochar_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__HOL__Obool_J,type,
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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__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Int__Oint_J,type,
    fun_real_int: $tType ).

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

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

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

tff(ty_n_t__fun_It__HOL__Obool_Mt__HOL__Obool_J,type,
    fun_bool_bool: $tType ).

tff(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
    list_complex: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__HOL__Obool_J,type,
    fun_rat_bool: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__HOL__Obool_J,type,
    fun_num_bool: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__HOL__Obool_J,type,
    fun_nat_bool: $tType ).

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__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__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,
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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__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__Filter__Ofilter_It__Real__Oreal_J,type,
    filter_real: $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__Filter__Ofilter_It__Nat__Onat_J,type,
    filter_nat: $tType ).

tff(ty_n_t__Set__Oset_It__String__Ochar_J,type,
    set_char: $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__String__Ochar,type,
    char: $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 (2058)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_VEBT_VEBT > fun_VEBT_VEBT_bool ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: set_complex > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__aaa____,type,
    aTP_Lamm_aaa: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__aab____,type,
    aTP_Lamm_aab: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__aac____,type,
    aTP_Lamm_aac: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__aad____,type,
    aTP_Lamm_aad: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__aae____,type,
    aTP_Lamm_aae: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__aaf____,type,
    aTP_Lamm_aaf: fun_real_real ).

tff(sy_c_ATP_058Lamm__aag____,type,
    aTP_Lamm_aag: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aah____,type,
    aTP_Lamm_aah: fun_real_real ).

tff(sy_c_ATP_058Lamm__aai____,type,
    aTP_Lamm_aai: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aaj____,type,
    aTP_Lamm_aaj: fun_nat_real ).

tff(sy_c_ATP_058Lamm__aak____,type,
    aTP_Lamm_aak: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aal____,type,
    aTP_Lamm_aal: fun_nat_real ).

tff(sy_c_ATP_058Lamm__aam____,type,
    aTP_Lamm_aam: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aan____,type,
    aTP_Lamm_aan: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aao____,type,
    aTP_Lamm_aao: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aap____,type,
    aTP_Lamm_aap: fun_re4238438383274954274t_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_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aas____,type,
    aTP_Lamm_aas: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aat____,type,
    aTP_Lamm_aat: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aau____,type,
    aTP_Lamm_aau: fun_nat_real > fun_fu6119738925743952495t_real ).

tff(sy_c_ATP_058Lamm__aav____,type,
    aTP_Lamm_aav: fun_nat_real > fun_nat_real ).

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_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aay____,type,
    aTP_Lamm_aay: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__aaz____,type,
    aTP_Lamm_aaz: fun_real_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__aba____,type,
    aTP_Lamm_aba: fun_real_real > 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_real_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__abd____,type,
    aTP_Lamm_abd: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__abe____,type,
    aTP_Lamm_abe: fun_real_real ).

tff(sy_c_ATP_058Lamm__abf____,type,
    aTP_Lamm_abf: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__abg____,type,
    aTP_Lamm_abg: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__abh____,type,
    aTP_Lamm_abh: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__abi____,type,
    aTP_Lamm_abi: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__abj____,type,
    aTP_Lamm_abj: fun_real_bool > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__abk____,type,
    aTP_Lamm_abk: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__abl____,type,
    aTP_Lamm_abl: real > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__abm____,type,
    aTP_Lamm_abm: fun_re3844104906058862192omplex ).

tff(sy_c_ATP_058Lamm__abn____,type,
    aTP_Lamm_abn: real > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__abo____,type,
    aTP_Lamm_abo: fun_re5512370375968561900l_real ).

tff(sy_c_ATP_058Lamm__abp____,type,
    aTP_Lamm_abp: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abq____,type,
    aTP_Lamm_abq: fun_nat_real > fun_fu3106432010075488275t_real ).

tff(sy_c_ATP_058Lamm__abr____,type,
    aTP_Lamm_abr: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abs____,type,
    aTP_Lamm_abs: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abt____,type,
    aTP_Lamm_abt: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__abu____,type,
    aTP_Lamm_abu: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__abv____,type,
    aTP_Lamm_abv: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__abw____,type,
    aTP_Lamm_abw: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__abx____,type,
    aTP_Lamm_abx: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__aby____,type,
    aTP_Lamm_aby: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__abz____,type,
    aTP_Lamm_abz: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: set_list_nat > fun_list_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: nat > fun_na5614638261385347865t_bool ).

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__ad____,type,
    aTP_Lamm_ad: set_set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: set_nat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: set_int > fun_int_bool ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: rat > fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: int > fun_int_int ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_real_real ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_int_int ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: list_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: list_list_nat > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: list_set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: list_VEBT_VEBT > fun_VEBT_VEBT_bool ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: list_bool > fun_bool_bool ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: list_nat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: list_int > fun_int_bool ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: set_bool > fun_na205931550456536010l_bool ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: set_int > fun_na7919044170365968317t_bool ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: set_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: set_bool > fun_na205931550456536010l_bool ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: set_int > fun_na7919044170365968317t_bool ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: set_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: set_int > fun_set_int_bool ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: set_complex > fun_set_complex_bool ).

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

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: fun_real_bool > fun_fu1184404422467473750l_bool ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: fun_re1255401033344146673l_bool > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: fun_real_bool > fun_fu910350400258205270t_bool ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: fun_na6801867027684862029l_bool > fun_re9121486324101951821t_bool ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: fun_real_bool > fun_fu5476287734010629974t_bool ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: fun_in3970322703178350321l_bool > fun_re8661203372455305201t_bool ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: fun_real_bool > fun_fu4972670022695973590x_bool ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: fun_co2286642472385812463l_bool > fun_re3010880487650781423x_bool ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: fun_nat_bool > fun_fu6956578997697959034l_bool ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: fun_re9121486324101951821t_bool > fun_na6801867027684862029l_bool ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: fun_nat_bool > fun_fu7750253452911709562t_bool ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: fun_nat_fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: fun_nat_bool > fun_fu8499154328239760506t_bool ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_int_fun_nat_bool > fun_nat_fun_int_bool ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_nat_bool > fun_fu9035823186496086266x_bool ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_co2841405903858895691t_bool > fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_int_bool > fun_fu158763272997538006l_bool ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: fun_re8661203372455305201t_bool > fun_in3970322703178350321l_bool ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_int_bool > fun_fu6976923931338485718t_bool ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: fun_nat_fun_int_bool > fun_int_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_na6801867027684862029l_bool ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: fun_real_bool > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: fun_list_nat_bool > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: fun_set_nat_bool > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_nat_bool > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_int_bool > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: fun_complex_bool > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_real_bool > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_list_nat_bool > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_set_nat_bool > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_nat_bool > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_int_bool > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: fun_complex_bool > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: set_VEBT_VEBT > fun_fu2721280555464079775T_bool ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: set_real > fun_fu4470565429337120259l_bool ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: set_VEBT_VEBT > fun_fu7178057608955062431T_bool ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: set_real > fun_fu66453571216240387l_bool ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: set_VEBT_VEBT > fun_fu748158482767045919T_bool ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: set_real > fun_fu6440467031773438595l_bool ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: set_nat > fun_fu4970875099359191043t_bool ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: set_nat > fun_fu7165551060251750659t_bool ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: set_nat > fun_fu2548079958729413763t_bool ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: set_int > fun_fu7719728765329718531t_bool ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: fun_real_bool > fun_fu1123323498867902878l_bool ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_real_bool > fun_fu3805282464119925846t_bool ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_real_bool > fun_fu4682539544734137246t_bool ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_real_bool > fun_fu7849178876846444698x_bool ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: fun_real_bool > fun_fu2626987694442567070l_bool ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: fun_real_bool > fun_fu7448995929138640982t_bool ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: fun_real_bool > fun_fu8326253009752852382t_bool ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: fun_real_bool > fun_fu8257491226618371226x_bool ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: fun_real_bool > fun_fu4079054602259297438l_bool ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: fun_real_bool > fun_fu891509079466760022t_bool ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: fun_real_bool > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: fun_real_bool > fun_fu293283182871086144t_bool ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: fun_real_bool > fun_fu6224827943572703624t_bool ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: fun_real_bool > fun_fu948393671388380548x_bool ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: fun_nat_bool > fun_fu4882312447150602980l_bool ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: fun_nat_bool > fun_fu8422113094057456412t_bool ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: fun_nat_bool > fun_fu5130285817904298084t_bool ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: fun_nat_bool > fun_fu1728481436528360544x_bool ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: fun_int_bool > fun_fu3785229466506604424l_bool ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: fun_int_bool > fun_fu6167199935867882432t_bool ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: set_VEBT_VEBT > fun_se6960202813843830349T_bool ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: set_complex > fun_se2973560062716774839x_bool ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: set_real > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: set_list_nat > fun_se7035039998181355731t_bool ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: set_set_nat > fun_se1925730536474940935t_bool ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: set_int > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: set_nat > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: set_VEBT_VEBT > fun_fu8277615433164260398T_bool ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: set_VEBT_VEBT > fun_fu1247336914219032636T_bool ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: set_real > fun_fu914841811607101702l_bool ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: set_real > fun_fu7100649987335273288l_bool ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: set_nat > fun_fu4346542433904371390t_bool ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: set_nat > fun_fu8382797250367883996t_bool ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: set_int > fun_fu7152481525987436038t_bool ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: set_int > fun_fu5129045444369776328t_bool ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: set_complex > fun_fu5519326811792979714x_bool ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: set_complex > fun_fu3191020621175015362x_bool ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: set_VEBT_VEBT > fun_fu4158003351038161580T_bool ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: set_VEBT_VEBT > fun_fu5001625740209071164T_bool ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: set_int > fun_fu549321512867696136t_bool ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: set_int > fun_fu3401574108658714444t_bool ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: set_complex > fun_fu6356453956411242500x_bool ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: set_complex > fun_fu5995210064384465734x_bool ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: fun_real_real ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_int_int ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: set_VEBT_VEBT > fun_fu8277615433164260398T_bool ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: set_VEBT_VEBT > fun_fu1247336914219032636T_bool ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: set_real > fun_fu914841811607101702l_bool ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: set_real > fun_fu7100649987335273288l_bool ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: set_nat > fun_fu4346542433904371390t_bool ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: set_nat > fun_fu8382797250367883996t_bool ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: set_int > fun_fu7152481525987436038t_bool ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: set_int > fun_fu5129045444369776328t_bool ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: set_complex > fun_fu5519326811792979714x_bool ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: set_complex > fun_fu3191020621175015362x_bool ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: set_VEBT_VEBT > fun_fu4158003351038161580T_bool ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: set_VEBT_VEBT > fun_fu5001625740209071164T_bool ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: set_int > fun_fu549321512867696136t_bool ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: set_int > fun_fu3401574108658714444t_bool ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: set_complex > fun_fu6356453956411242500x_bool ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: set_complex > fun_fu5995210064384465734x_bool ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: nat > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: fun_nat_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_Co8310303586393106575r_bool ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: vEBT_VEBT > fun_fu8659448316070137091omplex ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: vEBT_VEBT > fun_fu2188483506383969923BT_rat ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: vEBT_VEBT > fun_fu8659448316070137091omplex ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: vEBT_VEBT > fun_fu2188483506383969923BT_rat ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_int_int > fun_fu923337204662071120nt_int ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: fun_int_int > fun_fu3739269683040691718nt_int ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: fun_complex_complex > fun_fu8998703528969179354omplex ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_complex_complex > fun_fu8084223252616112272omplex ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: fun_nat_nat > fun_fu2454420574033104442at_nat ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: fun_nat_real > fun_fu3306995184869731826t_real ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: fun_complex_complex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: fun_complex_complex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: set_int > fun_fu4313891142084863779t_bool ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: set_int > fun_fu3599772511451410272BT_int ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_VE294566951569886236nt_int > fun_in1463024724634152352BT_int ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: set_VEBT_VEBT > fun_fu3522331824740900180nt_int ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: set_int > fun_fu3139726878529933315t_bool ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: set_int > fun_fu4310402290150085684al_int ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: fun_real_fun_int_int > fun_int_fun_real_int ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: set_real > fun_fu5129330665597888180nt_int ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: set_int > fun_fu3078129256884107011t_bool ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: set_int > fun_fu2946920718754786320at_int ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: fun_nat_fun_int_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: set_nat > fun_fu6165624312919156332nt_int ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: set_int > fun_fu4844541694308389507t_bool ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: set_int > fun_fu2836861327644265266ex_int ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: fun_co333780937270390912nt_int > fun_in3657390099950231424ex_int ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: set_complex > fun_fu8510821397121033091x_bool ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: set_complex > fun_fu6400036368084001456nt_int ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: set_complex > fun_fu5159174019657214751x_bool ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: set_complex > fun_fu8973321760839499616omplex ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_VE2006399705683203104omplex > fun_co3541073130412562976omplex ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: set_VEBT_VEBT > fun_fu3138954370166276442omplex ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: set_complex > fun_fu4189668387880940419x_bool ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: set_complex > fun_fu3241646296630235056omplex ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_re7208826350674946562omplex > fun_co286204877037288194omplex ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: set_real > fun_fu7272995576155203506omplex ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: set_complex > fun_fu8281188428872102787x_bool ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: set_complex > fun_fu5276431085538847372omplex ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_na1591161823130732966omplex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: set_nat > fun_fu5286008751397070186omplex ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: set_complex > fun_fu3391973010043818627x_bool ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: set_complex > fun_fu2147946303232500784omplex ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: fun_in1307110748760623106omplex > fun_co7673900812888889602omplex ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: set_int > fun_fu8949065344085950851t_bool ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: set_int > fun_fu2298032621795249330omplex ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: set_nat > fun_fu6002750694894812635t_bool ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: set_nat > fun_fu6772184894847113056BT_nat ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_VE597728521808591204at_nat > fun_na6208021027755453088BT_nat ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: set_VEBT_VEBT > fun_fu4406026827936152384at_nat ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: set_nat > fun_fu9073671928554219267t_bool ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: set_nat > fun_fu6463170612142064108al_nat ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_real_fun_nat_nat > fun_nat_fun_real_nat ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: set_real > fun_fu3370445115705697296at_nat ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_int_int > fun_int_fun_int_int ).

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

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: set_VEBT_VEBT > fun_fu5250730037759307465T_bool ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: fun_VE5569070535936029429omplex > fun_fu6701721618212012456omplex ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: set_real > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: fun_real_complex > fun_fu1393723446858488622omplex ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: set_nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_nat_complex > fun_fu7905158015251895726omplex ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: set_int > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_int_complex > fun_fu6363802315473983150omplex ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_VEBT_VEBT_real > fun_fu5928985204125839270T_real ).

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_VEBT_VEBT_rat > fun_fu8873448384659826898BT_rat ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_real_rat > fun_fu7249360421009165912al_rat ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: fun_real_complex > fun_real_bool ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_int_complex > fun_int_bool ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_real_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_int_real > fun_int_bool ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_complex_real > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_real_rat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_int_rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_complex_rat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: fun_real_nat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_int_nat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: fun_VE9219535225822561060nteger > fun_fu3480337415675744067nteger ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: fun_co8153928105989009288nteger > fun_fu1400560744564786701nteger ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: fun_VEBT_VEBT_real > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: fun_complex_real > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: fun_na4138050610385705875at_nat ).

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

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

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_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: fun_VEBT_VEBT_nat > fun_fu5238640373165125763BT_nat ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: fun_complex_nat > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: num > fun_num_bool ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: fun_re1255401033344146673l_bool ).

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

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: fun_nat_real > fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: fun_complex_complex > fun_complex_real ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: fun_nat_real ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: fun_nat_real ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: fun_nat_bool > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: fun_nat_bool > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: fun_nat_bool > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: fun_nat_bool > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: set_nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: set_nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: set_nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: fun_nat_complex > fun_co8378349014452122532t_real ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: fun_nat_complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: fun_nat_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: nat > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: nat > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: nat > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: fun_nat_real > fun_se2100927751953696370t_real ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: real > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: real > fun_fu3477889739401271834t_real ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: real > fun_fu4806068833880117316t_real ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: real > fun_fu2115079063411676732t_real ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: real > fun_fu4540635375566031512t_real ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: fun_int_int ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: fun_real_real ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: rat > fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: vEBT_VEBT > fun_fu8659448316070137091omplex ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: vEBT_VEBT > fun_fu8659448316070137091omplex ).

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

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

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

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

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: fun_int_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: set_nat > fun_fu6772184894847113056BT_nat ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: set_VEBT_VEBT > fun_fu4406026827936152384at_nat ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: set_nat > fun_fu6463170612142064108al_nat ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: set_real > fun_fu3370445115705697296at_nat ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: set_nat > fun_fu2583778531842586371t_bool ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: set_nat > fun_fu8221675813083619692nt_nat ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: fun_int_fun_nat_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: set_int > fun_fu5002972218919249680at_nat ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: set_nat > fun_fu5088574307552523139t_bool ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: set_nat > fun_fu4637029388598527722ex_nat ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: fun_co636942507509095880at_nat > fun_na8457291176974909128ex_nat ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: set_complex > fun_fu1598310740840436355x_bool ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: set_complex > fun_fu1817986637694864396at_nat ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: set_nat > fun_fu3446192182781468256BT_int ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: fun_VE7331017413731640256at_int > fun_na3717937882823726332BT_int ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: set_VEBT_VEBT > fun_fu9054112635728933880at_int ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: set_nat > fun_fu6487107696295728876al_int ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_real_fun_nat_int > fun_nat_fun_real_int ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: set_real > fun_fu1164691120062057232at_int ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: set_nat > fun_fu1666499901826068458ex_int ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: fun_co7370231399432144932at_int > fun_na5967208032043182372ex_int ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: set_complex > fun_fu3438847080485518604at_int ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: set_int > fun_fu3599772511451410272BT_int ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: set_VEBT_VEBT > fun_fu3522331824740900180nt_int ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: set_int > fun_fu4310402290150085684al_int ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: set_real > fun_fu5129330665597888180nt_int ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: set_int > fun_fu2836861327644265266ex_int ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: set_complex > fun_fu6400036368084001456nt_int ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: num > fun_num_bool ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: fun_VE5569070535936029429omplex > fun_fu6701721618212012456omplex ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: fun_real_complex > fun_fu1393723446858488622omplex ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: fun_nat_complex > fun_fu7905158015251895726omplex ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: fun_int_complex > fun_fu6363802315473983150omplex ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: fun_complex_complex > fun_fu248857536038387758omplex ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: fun_VEBT_VEBT_real > fun_fu5928985204125839270T_real ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: fun_real_real > fun_fu1760894646116856364l_real ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: fun_nat_real > fun_fu9152424228179946668t_real ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: fun_int_real > fun_fu6185597869953674668t_real ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: fun_complex_real > fun_fu1911030855436031788x_real ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: real > fun_fu3106432010075488275t_real ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: complex > fun_fu4781654129426620053omplex ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: int > fun_fu4512855384587262995at_int ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: int > fun_fu6415725884910174923nt_int ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: fun_real_complex > fun_real_bool ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: fun_int_complex > fun_int_bool ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: fun_complex_complex > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: fun_real_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: fun_int_real > fun_int_bool ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: fun_complex_real > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: fun_real_rat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: fun_int_rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: fun_complex_rat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: fun_real_nat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: fun_int_real > fun_int_real ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: fun_complex_real > fun_complex_real ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: fun_int_complex > fun_int_complex ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: fun_complex_complex > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: fun_nat_int > fun_na7654559027617327379at_int ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: fun_real_bool ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: fun_real_bool ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: complex > fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: fun_co2286642472385812463l_bool ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: code_integer > fun_Co3025358997185726883r_bool ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: set_nat > fun_nat_fun_nat_bool ).

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

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: fun_re8661203372455305201t_bool ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__xz____,type,
    aTP_Lamm_xz: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__yb____,type,
    aTP_Lamm_yb: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__yc____,type,
    aTP_Lamm_yc: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__yd____,type,
    aTP_Lamm_yd: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__yf____,type,
    aTP_Lamm_yf: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__yh____,type,
    aTP_Lamm_yh: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__yi____,type,
    aTP_Lamm_yi: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__yj____,type,
    aTP_Lamm_yj: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__yk____,type,
    aTP_Lamm_yk: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__ym____,type,
    aTP_Lamm_ym: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__yn____,type,
    aTP_Lamm_yn: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__yo____,type,
    aTP_Lamm_yo: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__yp____,type,
    aTP_Lamm_yp: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yq____,type,
    aTP_Lamm_yq: nat > fun_li8232715659020003717t_bool ).

tff(sy_c_ATP_058Lamm__yr____,type,
    aTP_Lamm_yr: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__ys____,type,
    aTP_Lamm_ys: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__yt____,type,
    aTP_Lamm_yt: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__yu____,type,
    aTP_Lamm_yu: fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__yv____,type,
    aTP_Lamm_yv: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__yw____,type,
    aTP_Lamm_yw: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__yx____,type,
    aTP_Lamm_yx: fun_real_bool ).

tff(sy_c_ATP_058Lamm__yy____,type,
    aTP_Lamm_yy: fun_real_bool ).

tff(sy_c_ATP_058Lamm__yz____,type,
    aTP_Lamm_yz: int > fun_in6148020733644367646t_bool ).

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

tff(sy_c_ATP_058Lamm__zb____,type,
    aTP_Lamm_zb: int > fun_in6148020733644367646t_bool ).

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

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

tff(sy_c_ATP_058Lamm__ze____,type,
    aTP_Lamm_ze: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__zf____,type,
    aTP_Lamm_zf: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__zg____,type,
    aTP_Lamm_zg: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__zh____,type,
    aTP_Lamm_zh: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__zi____,type,
    aTP_Lamm_zi: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__zj____,type,
    aTP_Lamm_zj: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__zk____,type,
    aTP_Lamm_zk: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__zl____,type,
    aTP_Lamm_zl: int > fun_int_int ).

tff(sy_c_ATP_058Lamm__zm____,type,
    aTP_Lamm_zm: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__zn____,type,
    aTP_Lamm_zn: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__zo____,type,
    aTP_Lamm_zo: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__zp____,type,
    aTP_Lamm_zp: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__zq____,type,
    aTP_Lamm_zq: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__zr____,type,
    aTP_Lamm_zr: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__zs____,type,
    aTP_Lamm_zs: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__zt____,type,
    aTP_Lamm_zt: fun_re4238438383274954274t_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__zu____,type,
    aTP_Lamm_zu: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__zv____,type,
    aTP_Lamm_zv: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__zw____,type,
    aTP_Lamm_zw: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zx____,type,
    aTP_Lamm_zx: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zy____,type,
    aTP_Lamm_zy: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__zz____,type,
    aTP_Lamm_zz: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

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: 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__Real__Oreal,type,
    archim2898591450579166408c_real: fun_real_real ).

tff(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
    archim7778729529865785530nd_rat: rat > int ).

tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: fun_real_int ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: fun_nat_fun_nat_nat ).

tff(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
    gbinomial_complex: complex > fun_nat_complex ).

tff(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
    gbinomial_rat: rat > fun_nat_rat ).

tff(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
    gbinomial_real: fun_re4238438383274954274t_real ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun_Pr3099820731930151597t_bool ).

tff(sy_c_Bit__Operations_Oand__not__num,type,
    bit_and_not_num: ( num * num ) > option_num ).

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: ( nat * int ) > fun_int_int ).

tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: fun_num_fun_num_num ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
    bit_ri7919022796975470100ot_int: fun_int_int ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
    bit_ri6519982836138164636nteger: ( nat * code_integer ) > code_integer ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
    bit_ri631733984087533419it_int: fun_nat_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
    bit_se725231765392027082nd_int: int > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
    bit_se727722235901077358nd_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
    bit_se8568078237143864401it_int: fun_nat_fun_int_int ).

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tff(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
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tff(sy_c_Int_ORep__Integ,type,
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tff(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Odistinct_001t__Int__Oint,type,
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tff(sy_c_List_Odrop_001t__Nat__Onat,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,
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tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Oset_001t__HOL__Obool,type,
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tff(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
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tff(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oupt,type,
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tff(sy_c_Nat_OSuc,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__HOL__Obool_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__HOL__Obool_Mt__HOL__Obool_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
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tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
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tff(sy_c_Nat__Bijection_Oprod__encode,type,
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tff(sy_c_Nat__Bijection_Oset__decode,type,
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tff(sy_c_Nat__Bijection_Oset__encode,type,
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tff(sy_c_Nat__Bijection_Otriangle,type,
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tff(sy_c_NthRoot_Oroot,type,
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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,
    inc: num > num ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,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_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
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tff(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,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_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,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__Code____Numeral__Ointeger,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,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_Opow,type,
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tff(sy_c_Num_Opred__numeral,type,
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tff(sy_c_Num_Osqr,type,
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tff(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
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tff(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
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tff(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_Osize__option_001t__Nat__Onat,type,
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tff(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > fun_set_nat_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
    ord_less_eq_set_num: ( set_num * set_num ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
    ord_less_eq_set_rat: ( set_rat * set_rat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_eq_set_real: ( set_real * set_real ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_le6893508408891458716et_nat: ( set_set_nat * set_set_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
    ord_max_int: int > fun_int_int ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
    ord_max_num: fun_num_fun_num_num ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
    ord_max_rat: ( rat * rat ) > rat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
    ord_max_real: fun_re7681423839302404990l_real ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_max_set_nat: ( set_nat * set_nat ) > set_nat ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
    ord_mi8085742599997312461d_enat: fun_Ex8341929632020906428d_enat ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
    order_Greatest_nat: fun_nat_bool > nat ).

tff(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal,type,
    order_9091379641038594480t_real: fun_nat_real > $o ).

tff(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_mono_nat_nat: fun_nat_nat > $o ).

tff(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Real__Oreal,type,
    order_mono_nat_real: fun_nat_real > $o ).

tff(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_5726023648592871131at_nat: fun_nat_nat > $o ).

tff(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal,type,
    order_7092887310737990675l_real: fun_real_real > $o ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__HOL__Obool_J,type,
    top_top_set_bool: set_bool ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

tff(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
    power_8256067586552552935nteger: code_integer > fun_nat_Code_integer ).

tff(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
    power_power_complex: complex > fun_nat_complex ).

tff(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
    power_power_int: int > fun_nat_int ).

tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
    power_power_rat: rat > fun_nat_rat ).

tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: fun_re4238438383274954274t_real ).

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__HOL__Obool,type,
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tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__HOL__Obool,type,
    produc8297744024737662807l_bool: ( bool * bool ) > produc2804947525362035943l_bool ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__Int__Oint,type,
    produc8871513391732212460ol_int: ( bool * int ) > produc5612625445546783218ol_int ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__Nat__Onat,type,
    produc8874003862241262736ol_nat: ( bool * nat ) > produc567104428201204118ol_nat ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
    produc9074556814250467700T_VEBT: ( bool * vEBT_VEBT ) > produc8408751633324799108T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: ( int * int ) > product_prod_int_int ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: nat > fun_na1476630412504255114at_nat ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
    product_Pair_nat_num: ( nat * num ) > product_prod_nat_num ).

tff(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
    product_Pair_num_num: ( num * num ) > product_prod_num_num ).

tff(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
    produc5098337634421038937on_nat: ( option_nat * option_nat ) > produc4953844613479565601on_nat ).

tff(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
    produc8585076106096196333on_num: ( option_num * option_num ) > produc3447558737645232053on_num ).

tff(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc488173922507101015at_nat: ( option4927543243414619207at_nat * option4927543243414619207at_nat ) > produc6121120109295599847at_nat ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
    produc3035007008238225530T_bool: ( vEBT_VEBT * bool ) > produc6448083994118892042T_bool ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc738532404422230701BT_nat: ( vEBT_VEBT * nat ) > produc9072475918466114483BT_nat ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc537772716801021591T_VEBT: ( vEBT_VEBT * vEBT_VEBT ) > produc8243902056947475879T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    produc2004789812759622991on_nat: ( fun_nat_fun_nat_bool * produc4953844613479565601on_nat ) > produc3700678068921409119on_nat ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    produc6699827698519632094on_nat: ( fun_nat_fun_nat_nat * produc4953844613479565601on_nat ) > produc7004362771917097900on_nat ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Num__Onum_Mt__fun_It__Num__Onum_Mt__HOL__Obool_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    produc8433281988922931063on_num: ( fun_num_fun_num_bool * produc3447558737645232053on_num ) > produc6167219322466932359on_num ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Num__Onum_Mt__fun_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    produc6360833703471310868on_num: ( fun_num_fun_num_num * produc3447558737645232053on_num ) > produc2703417854250991842on_num ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__HOL__Obool_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc5961576194287291153at_nat: ( fun_Pr248683899324676269t_bool * produc6121120109295599847at_nat ) > produc533602882674032673at_nat ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc7814092408141220007at_nat: ( fun_Pr2477239776880934807at_nat * produc6121120109295599847at_nat ) > produc1260151622200656695at_nat ).

tff(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc6916734918728496179nteger: ( fun_Co2470855966376689427nteger * produc8923325533196201883nteger ) > produc8923325533196201883nteger ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__HOL__Obool_J,type,
    produc6117575242473779287r_bool: ( fun_Co3025358997185726883r_bool * produc8923325533196201883nteger ) > produc7417142470700560111r_bool ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__HOL__Obool,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__HOL__Obool,type,
    produc8560705615207772305t_bool: fun_int_fun_int_bool > fun_Pr803113274115125473t_bool ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4245557441103728435nt_int: ( fun_in1696471567231710611nt_int * product_prod_int_int ) > product_prod_int_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
    produc1917071388513777916omplex: ( fun_na1142814060743658952omplex * product_prod_nat_nat ) > complex ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__HOL__Obool,type,
    produc8955392012077580105t_bool: fun_nat_fun_nat_bool > fun_Pr7757163693017731481t_bool ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
    produc6840382203811409530at_int: ( fun_nat_fun_nat_int * product_prod_nat_nat ) > int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    produc6842872674320459806at_nat: fun_nat_fun_nat_nat > fun_Pr2602117396471556196at_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc2626176000494625587at_nat: fun_na4138050610385705875at_nat > fun_Pr2118491707810114563at_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
    produc6207742614233964070at_rat: ( fun_nat_fun_nat_rat * product_prod_nat_nat ) > rat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
    produc1703576794950452218t_real: ( fun_nat_fun_nat_real * product_prod_nat_nat ) > real ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__HOL__Obool_J,type,
    produc5023667602222594023t_bool: ( fun_na4786964484885125043t_bool * product_prod_nat_nat ) > fun_Pr7757163693017731481t_bool ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_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,
    produc4528391901023682385at_nat: fun_na1668100129766719005at_nat > fun_Pr2477239776880934807at_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
    produc478579273971653890on_num: ( fun_na7595719129879163854on_num * product_prod_nat_num ) > option_num ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001t__HOL__Obool,type,
    produc5257810830037637649l_bool: fun_re1255401033344146673l_bool > fun_Pr9030630356418945121l_bool ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc8508995932063986495nteger: produc8923325533196201883nteger > code_integer ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
    product_fst_int_int: product_prod_int_int > int ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
    product_fst_nat_nat: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
    product_snd_int_int: product_prod_int_int > int ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Rat_OFract,type,
    fract: int > fun_int_rat ).

tff(sy_c_Rat_OFrct,type,
    frct: product_prod_int_int > rat ).

tff(sy_c_Rat_ORep__Rat,type,
    rep_Rat: rat > product_prod_int_int ).

tff(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
    field_5140801741446780682s_real: set_real ).

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod_int_int > product_prod_int_int ).

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_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
    real_V2521375963428798218omplex: set_complex ).

tff(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
    real_V5970128139526366754l_real: fun_real_real > $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_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: fun_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 * rat ) > rat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
    divide_divide_real: fun_re7681423839302404990l_real ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
    dvd_dvd_Code_integer: fun_Co8310303586393106575r_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
    dvd_dvd_complex: fun_co1378794544173401837x_bool ).

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,
    dvd_dvd_real: fun_re1255401033344146673l_bool ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
    modulo364778990260209775nteger: ( code_integer * code_integer ) > code_integer ).

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,
    modulo_modulo_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
    zero_n356916108424825756nteger: bool > code_integer ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
    zero_n1201886186963655149omplex: bool > complex ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
    zero_n2684676970156552555ol_int: fun_bool_int ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
    zero_n2687167440665602831ol_nat: bool > nat ).

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,
    zero_n3304061248610475627l_real: bool > real ).

tff(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
    suminf_complex: fun_nat_complex > complex ).

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

tff(sy_c_Series_Osuminf_001t__Nat__Onat,type,
    suminf_nat: fun_nat_nat > nat ).

tff(sy_c_Series_Osuminf_001t__Real__Oreal,type,
    suminf_real: fun_nat_real > real ).

tff(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
    summable_complex: fun_nat_complex > $o ).

tff(sy_c_Series_Osummable_001t__Int__Oint,type,
    summable_int: fun_nat_int > $o ).

tff(sy_c_Series_Osummable_001t__Nat__Onat,type,
    summable_nat: fun_nat_nat > $o ).

tff(sy_c_Series_Osummable_001t__Real__Oreal,type,
    summable_real: fun_nat_real > $o ).

tff(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
    sums_complex: fun_nat_complex > fun_complex_bool ).

tff(sy_c_Series_Osums_001t__Int__Oint,type,
    sums_int: fun_nat_int > fun_int_bool ).

tff(sy_c_Series_Osums_001t__Nat__Onat,type,
    sums_nat: fun_nat_nat > fun_nat_bool ).

tff(sy_c_Series_Osums_001t__Real__Oreal,type,
    sums_real: fun_nat_real > fun_real_bool ).

tff(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
    collect_Code_integer: fun_Co8822979462182099027r_bool > set_Code_integer ).

tff(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
    collect_complex: fun_complex_bool > set_complex ).

tff(sy_c_Set_OCollect_001t__HOL__Obool,type,
    collect_bool: fun_bool_bool > set_bool ).

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 ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    collec5608196760682091941T_VEBT: fun_li6006399982103069984T_bool > set_list_VEBT_VEBT ).

tff(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: fun_nat_bool > set_nat ).

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,
    collec8663557070575231912omplex: fun_Pr8073036449086768093x_bool > set_Pr5085853215250843933omplex ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    collec213857154873943460nt_int: fun_Pr803113274115125473t_bool > set_Pr958786334691620121nt_int ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    collec3799799289383736868l_real: fun_Pr9030630356418945121l_bool > set_Pr6218003697084177305l_real ).

tff(sy_c_Set_OCollect_001t__Rat__Orat,type,
    collect_rat: fun_rat_bool > set_rat ).

tff(sy_c_Set_OCollect_001t__Real__Oreal,type,
    collect_real: fun_real_bool > set_real ).

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__VEBT____Definitions__OVEBT,type,
    collect_VEBT_VEBT: fun_VEBT_VEBT_bool > set_VEBT_VEBT ).

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__Real__Oreal,type,
    image_nat_real: ( fun_nat_real * set_nat ) > set_real ).

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__Nat__Onat_001t__String__Ochar,type,
    image_nat_char: ( fun_nat_char * set_nat ) > set_char ).

tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
    image_5971271580939081552omplex: ( fun_re3844104906058862192omplex * set_real ) > set_fi4554929511873752355omplex ).

tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    image_2178119161166701260l_real: ( fun_re5512370375968561900l_real * set_real ) > set_fi7789364187291644575l_real ).

tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
    image_real_real: ( fun_real_real * set_real ) > set_real ).

tff(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
    image_char_nat: ( fun_char_nat * set_char ) > set_nat ).

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

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

tff(sy_c_Set_Oinsert_001t__Real__Oreal,type,
    insert_real: ( real * set_real ) > set_real ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
    set_fo1517530859248394432omplex: ( fun_na1591161823130732966omplex * nat ) > fun_na1591161823130732966omplex ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
    set_fo2581907887559384638at_int: ( fun_nat_fun_int_int * nat ) > fun_nat_fun_int_int ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
    set_fo2584398358068434914at_nat: ( fun_nat_fun_nat_nat * nat ) > fun_nat_fun_nat_nat ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
    set_fo1949268297981939178at_rat: ( fun_nat_fun_rat_rat * nat * nat * rat ) > rat ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
    set_fo3111899725591712190t_real: ( fun_na6724076748692969250l_real * nat ) > fun_na6724076748692969250l_real ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
    set_or1266510415728281911st_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
    set_or1269000886237332187st_nat: nat > fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
    set_or7049704709247886629st_num: ( num * num ) > set_num ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
    set_or633870826150836451st_rat: ( rat * rat ) > set_rat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
    set_or1222579329274155063t_real: ( real * real ) > set_real ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or4548717258645045905et_nat: ( set_nat * set_nat ) > set_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
    set_or4662586982721622107an_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
    set_or4665077453230672383an_nat: nat > fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
    set_ord_atLeast_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
    set_ord_atMost_int: int > set_int ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
    set_ord_atMost_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Num__Onum,type,
    set_ord_atMost_num: num > set_num ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Rat__Orat,type,
    set_ord_atMost_rat: rat > set_rat ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
    set_ord_atMost_real: real > set_real ).

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: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
    set_or6659071591806873216st_nat: nat > fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
    set_or5832277885323065728an_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
    set_or5834768355832116004an_nat: nat > fun_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: fun_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_String_Oascii__of,type,
    ascii_of: char > char ).

tff(sy_c_String_Ochar_OChar,type,
    char2: ( bool * bool * bool * bool * bool * bool * bool * bool ) > char ).

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: fun_char_nat ).

tff(sy_c_String_Ointeger__of__char,type,
    integer_of_char: char > code_integer ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: fun_nat_char ).

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__Int__Oint,type,
    topolo4899668324122417113eq_int: fun_nat_int > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
    topolo4902158794631467389eq_nat: fun_nat_nat > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
    topolo1459490580787246023eq_num: fun_nat_num > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
    topolo4267028734544971653eq_rat: fun_nat_rat > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
    topolo6980174941875973593q_real: fun_nat_real > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Nat__Onat_J,type,
    topolo7278393974255667507et_nat: fun_nat_set_nat > $o ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
    topolo2177554685111907308n_real: ( real * set_real ) > filter_real ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
    topolo2815343760600316023s_real: real > filter_real ).

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

tff(sy_c_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__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

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

tff(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
    diffs_int: fun_nat_int > fun_nat_int ).

tff(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
    diffs_rat: fun_nat_rat > fun_nat_rat ).

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__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_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 ) > fun_VE3165209603783583043T_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: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set_nat ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_bool ).

tff(sy_c_VEBT__Insert_Ovebt__insert,type,
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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: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: fun_op7792301606653096457t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: fun_op7792301606653096457t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: fun_op7792301606653096457t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: set_nat > fun_nat_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: set_nat > fun_nat_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
    vEBT_V4262088993061758097ft_nat: fun_nat_fun_nat_nat > fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
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tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
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tff(sy_c_VEBT__MinMax_Ovebt__maxt,type,
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tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__MinMax_Ovebt__mint,type,
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tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Succ_Ois__succ__in__set,type,
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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,
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tff(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
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tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Complex__Ocomplex_Mt__HOL__Obool_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Complex__Ocomplex_Mt__HOL__Obool_J_J,type,
    aa_fun5298830385910147808x_bool: ( fun_fu5159174019657214751x_bool * fun_VE2128610149277289041x_bool ) > fun_VE2128610149277289041x_bool ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Int__Oint_Mt__HOL__Obool_J_J_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_fun5422038952152218214nt_int: ( fun_fu1076773466057892517nt_int * fun_VE3303673267288458579t_bool ) > fun_int_int ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Int__Oint_Mt__HOL__Obool_J_J_001t__fun_It__Int__Oint_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J_J,type,
    aa_fun4417922564241321824T_bool: ( fun_fu7178057608955062431T_bool * fun_VE3303673267288458579t_bool ) > fun_in6743502246894437583T_bool ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Int__Oint_Mt__HOL__Obool_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
    aa_fun4615758601980977314BT_int: ( fun_fu8394515377879832761BT_int * fun_VE3303673267288458579t_bool ) > fun_VEBT_VEBT_int ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Int__Oint_Mt__HOL__Obool_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Int__Oint_Mt__HOL__Obool_J_J,type,
    aa_fun978093584635342820t_bool: ( fun_fu4313891142084863779t_bool * fun_VE3303673267288458579t_bool ) > fun_VE3303673267288458579t_bool ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Int__Oint_Mt__HOL__Obool_J_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_fun3638225402958239493nt_int: ( fun_fu3522331824740900180nt_int * fun_VE294566951569886236nt_int ) > fun_fu1076773466057892517nt_int ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Int__Oint_Mt__HOL__Obool_J_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
    aa_fun627869447348354137BT_int: ( fun_fu3599772511451410272BT_int * fun_VE294566951569886236nt_int ) > fun_fu8394515377879832761BT_int ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    aa_fun1384843155787029862at_int: ( fun_fu5075381843755512741at_int * fun_VE3763956218935105199t_bool ) > fun_nat_int ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_fun5562694175296226570at_nat: ( fun_fu7565464988687239497at_nat * fun_VE3763956218935105199t_bool ) > fun_nat_nat ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__Nat__Onat_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J_J,type,
    aa_fun4018725027000996448T_bool: ( fun_fu2721280555464079775T_bool * fun_VE3763956218935105199t_bool ) > fun_na2351039247863542899T_bool ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
    aa_fun6141810409722670662BT_int: ( fun_fu697195148766750301BT_int * fun_VE3763956218935105199t_bool ) > fun_VEBT_VEBT_int ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    aa_fun1096289392377091562BT_nat: ( fun_fu3187278293698477057BT_nat * fun_VE3763956218935105199t_bool ) > fun_VEBT_VEBT_nat ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J,type,
    aa_fun5431641998072558748t_bool: ( fun_fu6002750694894812635t_bool * fun_VE3763956218935105199t_bool ) > fun_VE3763956218935105199t_bool ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    aa_fun6066863168309489065at_int: ( fun_fu9054112635728933880at_int * fun_VE7331017413731640256at_int ) > fun_fu5075381843755512741at_int ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
    aa_fun6474440919679335769BT_int: ( fun_fu3446192182781468256BT_int * fun_VE7331017413731640256at_int ) > fun_fu697195148766750301BT_int ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_fun8318684123515291377at_nat: ( fun_fu4406026827936152384at_nat * fun_VE597728521808591204at_nat ) > fun_fu7565464988687239497at_nat ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    aa_fun7476968357888788569BT_nat: ( fun_fu6772184894847113056BT_nat * fun_VE597728521808591204at_nat ) > fun_fu3187278293698477057BT_nat ).

tff(sy_c_fAll_001t__VEBT____Definitions__OVEBT,type,
    fAll_VEBT_VEBT: fun_VEBT_VEBT_bool > bool ).

tff(sy_c_fChoice_001t__Real__Oreal,type,
    fChoice_real: fun_real_bool > real ).

tff(sy_c_fEx_001t__Nat__Onat,type,
    fEx_nat: fun_fu7168253649841883962l_bool ).

tff(sy_c_fEx_001t__Real__Oreal,type,
    fEx_real: fun_fu7511176317168184086l_bool ).

tff(sy_c_fFalse,type,
    fFalse: bool ).

tff(sy_c_fNot,type,
    fNot: fun_bool_bool ).

tff(sy_c_fTrue,type,
    fTrue: bool ).

tff(sy_c_fconj,type,
    fconj: bool > fun_bool_bool ).

tff(sy_c_fdisj,type,
    fdisj: bool > fun_bool_bool ).

tff(sy_c_fequal_001t__Code____Numeral__Ointeger,type,
    fequal_Code_integer: fun_Co8310303586393106575r_bool ).

tff(sy_c_fequal_001t__Complex__Ocomplex,type,
    fequal_complex: fun_co1378794544173401837x_bool ).

tff(sy_c_fequal_001t__HOL__Obool,type,
    fequal_bool: bool > fun_bool_bool ).

tff(sy_c_fequal_001t__Int__Oint,type,
    fequal_int: fun_int_fun_int_bool ).

tff(sy_c_fequal_001t__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_bool ).

tff(sy_c_fequal_001t__Option__Ooption_It__Nat__Onat_J,type,
    fequal_option_nat: fun_op7792301606653096457t_bool ).

tff(sy_c_fequal_001t__Real__Oreal,type,
    fequal_real: fun_re1255401033344146673l_bool ).

tff(sy_c_fequal_001t__VEBT____Definitions__OVEBT,type,
    fequal_VEBT_VEBT: fun_VE5877677516038742509T_bool ).

tff(sy_c_fimplies,type,
    fimplies: fun_bo1841576699660343469l_bool ).

tff(sy_c_member_001t__Complex__Ocomplex,type,
    member_complex: complex > fun_set_complex_bool ).

tff(sy_c_member_001t__HOL__Obool,type,
    member_bool: ( bool * set_bool ) > $o ).

tff(sy_c_member_001t__Int__Oint,type,
    member_int: int > fun_set_int_bool ).

tff(sy_c_member_001t__List__Olist_It__HOL__Obool_J,type,
    member_list_bool: ( list_bool * set_list_bool ) > $o ).

tff(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
    member_list_int: ( list_int * set_list_int ) > $o ).

tff(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
    member_list_nat: ( list_nat * set_list_nat ) > $o ).

tff(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    member2936631157270082147T_VEBT: ( list_VEBT_VEBT * set_list_VEBT_VEBT ) > $o ).

tff(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > fun_set_nat_bool ).

tff(sy_c_member_001t__Num__Onum,type,
    member_num: ( num * set_num ) > $o ).

tff(sy_c_member_001t__Rat__Orat,type,
    member_rat: ( rat * set_rat ) > $o ).

tff(sy_c_member_001t__Real__Oreal,type,
    member_real: ( real * set_real ) > $o ).

tff(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: ( set_nat * set_set_nat ) > $o ).

tff(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
    member_VEBT_VEBT: fun_VE7346385682144900493T_bool ).

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_maxl____,type,
    maxl: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_succy____,type,
    succy: 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 ).

% Relevant facts (10516)
tff(fact_0_False,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)) ).

% False
tff(fact_1_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_2_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_3_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_4__C06_C,axiom,
    pp(aa_set_nat_bool(member_nat(succy),vEBT_VEBT_set_vebt(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).

% "06"
tff(fact_5__C4_Ohyps_C_I10_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"(10)
tff(fact_6_add__self__div__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = M ).

% add_self_div_2
tff(fact_7_div__exp__eq,axiom,
    ! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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(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(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_8_div__exp__eq,axiom,
    ! [A: int,M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,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(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(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_9_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)),numeral_numeral_real(aa_num_num(bit0,one))))) ) ).

% field_less_half_sum
tff(fact_10_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),divide_divide_rat(plus_plus_rat(X,Y),numeral_numeral_rat(aa_num_num(bit0,one))))) ) ).

% field_less_half_sum
tff(fact_11_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_12_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_13_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_14__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_15_field__sum__of__halves,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),numeral_numeral_real(aa_num_num(bit0,one)))) = X ).

% field_sum_of_halves
tff(fact_16_field__sum__of__halves,axiom,
    ! [X: rat] : plus_plus_rat(divide_divide_rat(X,numeral_numeral_rat(aa_num_num(bit0,one))),divide_divide_rat(X,numeral_numeral_rat(aa_num_num(bit0,one)))) = X ).

% field_sum_of_halves
tff(fact_17_semiring__norm_I85_J,axiom,
    ! [M: num] : aa_num_num(bit0,M) != one ).

% semiring_norm(85)
tff(fact_18_semiring__norm_I83_J,axiom,
    ! [N: num] : one != aa_num_num(bit0,N) ).

% semiring_norm(83)
tff(fact_19_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% numeral_plus_numeral
tff(fact_20_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(M)),numeral_numeral_real(N)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% numeral_plus_numeral
tff(fact_21_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : plus_plus_rat(numeral_numeral_rat(M),numeral_numeral_rat(N)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% numeral_plus_numeral
tff(fact_22_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% numeral_plus_numeral
tff(fact_23_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% numeral_plus_numeral
tff(fact_24_add__numeral__left,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_25_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_26_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] : plus_plus_rat(numeral_numeral_rat(V),plus_plus_rat(numeral_numeral_rat(W),Z)) = plus_plus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)),Z) ).

% add_numeral_left
tff(fact_27_add__numeral__left,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_28_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_29__C4_Ohyps_C_I5_J,axiom,
    m = na ).

% "4.hyps"(5)
tff(fact_30_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( numera6690914467698888265omplex(M) = numera6690914467698888265omplex(N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_31_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_real(M) = numeral_numeral_real(N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_32_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_rat(M) = numeral_numeral_rat(N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_33_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_nat(numeral_numeral_nat,M) = aa_num_nat(numeral_numeral_nat,N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_34_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_int(numeral_numeral_int,M) = aa_num_int(numeral_numeral_int,N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_35_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(ord_less_num(aa_num_num(bit0,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(ord_less_num(M),N)) ) ).

% semiring_norm(78)
tff(fact_36_semiring__norm_I87_J,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_num(bit0,M) = aa_num_num(bit0,N) )
    <=> ( M = N ) ) ).

% semiring_norm(87)
tff(fact_37_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(ord_less_num(M),one)) ).

% semiring_norm(75)
tff(fact_38__C4_Ohyps_C_I6_J,axiom,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

% "4.hyps"(6)
tff(fact_39_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(M)),numeral_numeral_real(N)))
    <=> pp(aa_num_bool(ord_less_num(M),N)) ) ).

% numeral_less_iff
tff(fact_40_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(M)),numeral_numeral_rat(N)))
    <=> pp(aa_num_bool(ord_less_num(M),N)) ) ).

% numeral_less_iff
tff(fact_41_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(ord_less_num(M),N)) ) ).

% numeral_less_iff
tff(fact_42_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(ord_less_num(M),N)) ) ).

% numeral_less_iff
tff(fact_43_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% numeral_times_numeral
tff(fact_44_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(M)),numeral_numeral_real(N)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% numeral_times_numeral
tff(fact_45_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(numeral_numeral_rat(M)),numeral_numeral_rat(N)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% numeral_times_numeral
tff(fact_46_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% numeral_times_numeral
tff(fact_47_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% numeral_times_numeral
tff(fact_48_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_49_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_50_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),Z)) = aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_51_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_52_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_53_semiring__norm_I76_J,axiom,
    ! [N: num] : pp(aa_num_bool(ord_less_num(one),aa_num_num(bit0,N))) ).

% semiring_norm(76)
tff(fact_54_semiring__norm_I6_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% semiring_norm(6)
tff(fact_55_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_56_distrib__right__numeral,axiom,
    ! [A: complex,B: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V))) ).

% distrib_right_numeral
tff(fact_57_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),numeral_numeral_real(V))) ).

% distrib_right_numeral
tff(fact_58_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),numeral_numeral_rat(V)) = plus_plus_rat(aa_rat_rat(times_times_rat(A),numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(B),numeral_numeral_rat(V))) ).

% distrib_right_numeral
tff(fact_59_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,V))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_num_nat(numeral_numeral_nat,V))) ).

% distrib_right_numeral
tff(fact_60_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),aa_num_int(numeral_numeral_int,V))),aa_int_int(times_times_int(B),aa_num_int(numeral_numeral_int,V))) ).

% distrib_right_numeral
tff(fact_61_distrib__left__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(B),C)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C)) ).

% distrib_left_numeral
tff(fact_62_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),C)) ).

% distrib_left_numeral
tff(fact_63_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),plus_plus_rat(B,C)) = plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),B),aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),C)) ).

% distrib_left_numeral
tff(fact_64_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),C)) ).

% distrib_left_numeral
tff(fact_65_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(plus_plus_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),C)) ).

% distrib_left_numeral
tff(fact_66_semiring__norm_I2_J,axiom,
    aa_num_num(aa_num_fun_num_num(plus_plus_num,one),one) = aa_num_num(bit0,one) ).

% semiring_norm(2)
tff(fact_67__C4_Ohyps_C_I9_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,mi),ma)) ).

% "4.hyps"(9)
tff(fact_68_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))),B)) ) ).

% less_divide_eq_numeral1(1)
tff(fact_69_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),divide_divide_rat(B,numeral_numeral_rat(W))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W))),B)) ) ).

% less_divide_eq_numeral1(1)
tff(fact_70_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W)))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_71_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,numeral_numeral_rat(W))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W)))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_72_power__add__numeral2,axiom,
    ! [A: complex,M: num,N: num,B: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,M))),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B) ).

% power_add_numeral2
tff(fact_73_power__add__numeral2,axiom,
    ! [A: real,M: num,N: num,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,M))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B) ).

% power_add_numeral2
tff(fact_74_power__add__numeral2,axiom,
    ! [A: rat,M: num,N: num,B: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,M))),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B) ).

% power_add_numeral2
tff(fact_75_power__add__numeral2,axiom,
    ! [A: nat,M: num,N: num,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B) ).

% power_add_numeral2
tff(fact_76_power__add__numeral2,axiom,
    ! [A: int,M: num,N: num,B: int] : aa_int_int(times_times_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,M))),aa_int_int(times_times_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_int_int(times_times_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B) ).

% power_add_numeral2
tff(fact_77_power__add__numeral,axiom,
    ! [A: complex,M: num,N: num] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% power_add_numeral
tff(fact_78_power__add__numeral,axiom,
    ! [A: real,M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% power_add_numeral
tff(fact_79_power__add__numeral,axiom,
    ! [A: rat,M: num,N: num] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% power_add_numeral
tff(fact_80_power__add__numeral,axiom,
    ! [A: nat,M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% power_add_numeral
tff(fact_81_power__add__numeral,axiom,
    ! [A: int,M: num,N: num] : aa_int_int(times_times_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% power_add_numeral
tff(fact_82_add__One__commute,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N) = aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one) ).

% add_One_commute
tff(fact_83_power__commuting__commutes,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( aa_complex_complex(times_times_complex(X),Y) = aa_complex_complex(times_times_complex(Y),X) )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),N)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(power_power_complex(X),N)) ) ) ).

% power_commuting_commutes
tff(fact_84_power__commuting__commutes,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) ) ) ).

% power_commuting_commutes
tff(fact_85_power__commuting__commutes,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( aa_rat_rat(times_times_rat(X),Y) = aa_rat_rat(times_times_rat(Y),X) )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),N)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(power_power_rat(X),N)) ) ) ).

% power_commuting_commutes
tff(fact_86_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),X) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)) ) ) ).

% power_commuting_commutes
tff(fact_87_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( aa_int_int(times_times_int(X),Y) = aa_int_int(times_times_int(Y),X) )
     => ( aa_int_int(times_times_int(aa_nat_int(power_power_int(X),N)),Y) = aa_int_int(times_times_int(Y),aa_nat_int(power_power_int(X),N)) ) ) ).

% power_commuting_commutes
tff(fact_88_power__mult__distrib,axiom,
    ! [A: complex,B: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(times_times_complex(A),B)),N) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),N)),aa_nat_complex(power_power_complex(B),N)) ).

% power_mult_distrib
tff(fact_89_power__mult__distrib,axiom,
    ! [A: real,B: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_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_mult_distrib
tff(fact_90_power__mult__distrib,axiom,
    ! [A: rat,B: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(times_times_rat(A),B)),N) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)) ).

% power_mult_distrib
tff(fact_91_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_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_mult_distrib
tff(fact_92_power__mult__distrib,axiom,
    ! [A: int,B: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(times_times_int(A),B)),N) = aa_int_int(times_times_int(aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)) ).

% power_mult_distrib
tff(fact_93_power__commutes,axiom,
    ! [A: complex,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),N)),A) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),N)) ).

% power_commutes
tff(fact_94_power__commutes,axiom,
    ! [A: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).

% power_commutes
tff(fact_95_power__commutes,axiom,
    ! [A: rat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),N)),A) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N)) ).

% power_commutes
tff(fact_96_power__commutes,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),A) = 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_commutes
tff(fact_97_power__commutes,axiom,
    ! [A: int,N: nat] : aa_int_int(times_times_int(aa_nat_int(power_power_int(A),N)),A) = aa_int_int(times_times_int(A),aa_nat_int(power_power_int(A),N)) ).

% power_commutes
tff(fact_98_power__mult,axiom,
    ! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),N) ).

% power_mult
tff(fact_99_power__mult,axiom,
    ! [A: real,M: nat,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),N) ).

% power_mult
tff(fact_100_power__mult,axiom,
    ! [A: complex,M: nat,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),M)),N) ).

% power_mult
tff(fact_101_power__mult,axiom,
    ! [A: int,M: nat,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),M)),N) ).

% power_mult
tff(fact_102_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_103_mem__Collect__eq,axiom,
    ! [A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),collect_VEBT_VEBT(P)))
    <=> pp(aa_VEBT_VEBT_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_104_mem__Collect__eq,axiom,
    ! [A: complex,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(member_complex(A),collect_complex(P)))
    <=> pp(aa_complex_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_105_mem__Collect__eq,axiom,
    ! [A: real,P: fun_real_bool] :
      ( member_real(A,collect_real(P))
    <=> pp(aa_real_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_106_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_107_mem__Collect__eq,axiom,
    ! [A: set_nat,P: fun_set_nat_bool] :
      ( member_set_nat(A,collect_set_nat(P))
    <=> pp(aa_set_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_108_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_109_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_110_Collect__mem__eq,axiom,
    ! [A2: set_VEBT_VEBT] : collect_VEBT_VEBT(aTP_Lamm_a(A2)) = A2 ).

% Collect_mem_eq
tff(fact_111_Collect__mem__eq,axiom,
    ! [A2: set_complex] : collect_complex(aTP_Lamm_aa(A2)) = A2 ).

% Collect_mem_eq
tff(fact_112_Collect__mem__eq,axiom,
    ! [A2: set_real] : collect_real(aTP_Lamm_ab(A2)) = A2 ).

% Collect_mem_eq
tff(fact_113_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] : collect_list_nat(aTP_Lamm_ac(A2)) = A2 ).

% Collect_mem_eq
tff(fact_114_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] : collect_set_nat(aTP_Lamm_ad(A2)) = A2 ).

% Collect_mem_eq
tff(fact_115_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aTP_Lamm_ae(A2)) = A2 ).

% Collect_mem_eq
tff(fact_116_Collect__mem__eq,axiom,
    ! [A2: set_int] : collect_int(aTP_Lamm_af(A2)) = A2 ).

% Collect_mem_eq
tff(fact_117_Collect__cong,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( ! [X2: real] :
          ( pp(aa_real_bool(P,X2))
        <=> pp(aa_real_bool(Q,X2)) )
     => ( collect_real(P) = collect_real(Q) ) ) ).

% Collect_cong
tff(fact_118_Collect__cong,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ! [X2: list_nat] :
          ( pp(aa_list_nat_bool(P,X2))
        <=> pp(aa_list_nat_bool(Q,X2)) )
     => ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_119_Collect__cong,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ! [X2: set_nat] :
          ( pp(aa_set_nat_bool(P,X2))
        <=> pp(aa_set_nat_bool(Q,X2)) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_120_Collect__cong,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ! [X2: nat] :
          ( pp(aa_nat_bool(P,X2))
        <=> pp(aa_nat_bool(Q,X2)) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_121_Collect__cong,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X2: int] :
          ( pp(aa_int_bool(P,X2))
        <=> pp(aa_int_bool(Q,X2)) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_122_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_123_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_124_div__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),Q2) ).

% div_mult2_eq
tff(fact_125_mult__numeral__1__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(one)) = A ).

% mult_numeral_1_right
tff(fact_126_mult__numeral__1__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(one)) = A ).

% mult_numeral_1_right
tff(fact_127_mult__numeral__1__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),numeral_numeral_rat(one)) = A ).

% mult_numeral_1_right
tff(fact_128_mult__numeral__1__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,one)) = A ).

% mult_numeral_1_right
tff(fact_129_mult__numeral__1__right,axiom,
    ! [A: int] : aa_int_int(times_times_int(A),aa_num_int(numeral_numeral_int,one)) = A ).

% mult_numeral_1_right
tff(fact_130_mult__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(one)),A) = A ).

% mult_numeral_1
tff(fact_131_mult__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(one)),A) = A ).

% mult_numeral_1
tff(fact_132_mult__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(one)),A) = A ).

% mult_numeral_1
tff(fact_133_mult__numeral__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,one)),A) = A ).

% mult_numeral_1
tff(fact_134_mult__numeral__1,axiom,
    ! [A: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,one)),A) = A ).

% mult_numeral_1
tff(fact_135_power__add,axiom,
    ! [A: complex,M: nat,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N)) ).

% power_add
tff(fact_136_power__add,axiom,
    ! [A: real,M: nat,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).

% power_add
tff(fact_137_power__add,axiom,
    ! [A: rat,M: nat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)) ).

% power_add
tff(fact_138_power__add,axiom,
    ! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_add
tff(fact_139_power__add,axiom,
    ! [A: int,M: nat,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(times_times_int(aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) ).

% power_add
tff(fact_140_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_141_left__add__twice,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_142_left__add__twice,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_143_left__add__twice,axiom,
    ! [A: rat,B: rat] : plus_plus_rat(A,plus_plus_rat(A,B)) = plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),A),B) ).

% left_add_twice
tff(fact_144_left__add__twice,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),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))),A)),B) ).

% left_add_twice
tff(fact_145_left__add__twice,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(A),B)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_146_mult__2__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),numera6690914467698888265omplex(aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2_right
tff(fact_147_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),numeral_numeral_real(aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2_right
tff(fact_148_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),numeral_numeral_rat(aa_num_num(bit0,one))) = plus_plus_rat(Z,Z) ).

% mult_2_right
tff(fact_149_mult__2__right,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2_right
tff(fact_150_mult__2__right,axiom,
    ! [Z: int] : aa_int_int(times_times_int(Z),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(plus_plus_int(Z),Z) ).

% mult_2_right
tff(fact_151_mult__2,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2
tff(fact_152_mult__2,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Z) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2
tff(fact_153_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Z) = plus_plus_rat(Z,Z) ).

% mult_2
tff(fact_154_mult__2,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Z) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2
tff(fact_155_mult__2,axiom,
    ! [Z: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Z) = aa_int_int(plus_plus_int(Z),Z) ).

% mult_2
tff(fact_156_power2__eq__square,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(times_times_complex(A),A) ).

% power2_eq_square
tff(fact_157_power2__eq__square,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))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) ).

% power2_eq_square
tff(fact_158_power2__eq__square,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(A),A) ).

% power2_eq_square
tff(fact_159_power2__eq__square,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))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A) ).

% power2_eq_square
tff(fact_160_power2__eq__square,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(times_times_int(A),A) ).

% power2_eq_square
tff(fact_161_power4__eq__xxxx,axiom,
    ! [X: complex] : aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_162_power4__eq__xxxx,axiom,
    ! [X: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_163_power4__eq__xxxx,axiom,
    ! [X: rat] : aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_164_power4__eq__xxxx,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_165_power4__eq__xxxx,axiom,
    ! [X: int] : aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_int_int(times_times_int(aa_int_int(times_times_int(aa_int_int(times_times_int(X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_166_power__even__eq,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_167_power__even__eq,axiom,
    ! [A: real,N: nat] : 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)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_168_power__even__eq,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_169_power__even__eq,axiom,
    ! [A: int,N: nat] : 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)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_170_is__num__normalize_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)) ).

% is_num_normalize(1)
tff(fact_171_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : plus_plus_rat(plus_plus_rat(A,B),C) = plus_plus_rat(A,plus_plus_rat(B,C)) ).

% is_num_normalize(1)
tff(fact_172_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(B),C)) ).

% is_num_normalize(1)
tff(fact_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_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_181_power2__sum,axiom,
    ! [X: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_182_power2__sum,axiom,
    ! [X: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,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))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_183_power2__sum,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(plus_plus_rat(X,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = plus_plus_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)))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_184_power2__sum,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,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))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_185_power2__sum,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(plus_plus_int(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(plus_plus_int(aa_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))))),aa_int_int(times_times_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_186_power__divide,axiom,
    ! [A: complex,B: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(A),B)),N) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),N)),aa_nat_complex(power_power_complex(B),N)) ).

% power_divide
tff(fact_187_power__divide,axiom,
    ! [A: real,B: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),N) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_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_divide
tff(fact_188_power__divide,axiom,
    ! [A: rat,B: rat,N: nat] : aa_nat_rat(power_power_rat(divide_divide_rat(A,B)),N) = divide_divide_rat(aa_nat_rat(power_power_rat(A),N),aa_nat_rat(power_power_rat(B),N)) ).

% power_divide
tff(fact_189_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_190_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_191_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_192_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_193_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_194_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_195_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_196_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_197_numeral__Bit0,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit0,N)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N)) ).

% numeral_Bit0
tff(fact_198_numeral__Bit0,axiom,
    ! [N: num] : numeral_numeral_real(aa_num_num(bit0,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(N)),numeral_numeral_real(N)) ).

% numeral_Bit0
tff(fact_199_numeral__Bit0,axiom,
    ! [N: num] : numeral_numeral_rat(aa_num_num(bit0,N)) = plus_plus_rat(numeral_numeral_rat(N),numeral_numeral_rat(N)) ).

% numeral_Bit0
tff(fact_200_numeral__Bit0,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_Bit0
tff(fact_201_numeral__Bit0,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N)) ).

% numeral_Bit0
tff(fact_202_divide__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(one)) = A ).

% divide_numeral_1
tff(fact_203_divide__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(one)) = A ).

% divide_numeral_1
tff(fact_204_divide__numeral__1,axiom,
    ! [A: rat] : divide_divide_rat(A,numeral_numeral_rat(one)) = A ).

% divide_numeral_1
tff(fact_205_numeral__Bit0__div__2,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,N) ).

% numeral_Bit0_div_2
tff(fact_206_numeral__Bit0__div__2,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,N) ).

% numeral_Bit0_div_2
tff(fact_207__C04_C,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),succy)) ).

% "04"
tff(fact_208_afinite,axiom,
    pp(aa_set_nat_bool(finite_finite_nat,vEBT_VEBT_set_vebt(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).

% afinite
tff(fact_209__092_060open_062high_Ax_An_A_060_A2_A_094_An_A_092_060and_062_Alow_Ax_An_A_060_A2_A_094_An_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)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na)))
    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))) ) ).

% \<open>high x n < 2 ^ n \<and> low x n < 2 ^ n\<close>
tff(fact_210__092_060open_062length_AtreeList_A_061_A2_A_094_An_092_060close_062,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))),na) ).

% \<open>length treeList = 2 ^ n\<close>
tff(fact_211_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_212_times__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),C) ).

% times_divide_eq_right
tff(fact_213_times__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) ).

% times_divide_eq_right
tff(fact_214_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),divide_divide_rat(B,C)) = divide_divide_rat(aa_rat_rat(times_times_rat(A),B),C) ).

% times_divide_eq_right
tff(fact_215_divide__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(divide1717551699836669952omplex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),B) ).

% divide_divide_eq_right
tff(fact_216_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ).

% divide_divide_eq_right
tff(fact_217_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(A,divide_divide_rat(B,C)) = divide_divide_rat(aa_rat_rat(times_times_rat(A),C),B) ).

% divide_divide_eq_right
tff(fact_218_divide__divide__eq__left,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(B),C)) ).

% divide_divide_eq_left
tff(fact_219_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% divide_divide_eq_left
tff(fact_220_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(divide_divide_rat(A,B),C) = divide_divide_rat(A,aa_rat_rat(times_times_rat(B),C)) ).

% divide_divide_eq_left
tff(fact_221_times__divide__eq__left,axiom,
    ! [B: complex,C: complex,A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(B),C)),A) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(B),A)),C) ).

% times_divide_eq_left
tff(fact_222_times__divide__eq__left,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),C) ).

% times_divide_eq_left
tff(fact_223_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(divide_divide_rat(B,C)),A) = divide_divide_rat(aa_rat_rat(times_times_rat(B),A),C) ).

% times_divide_eq_left
tff(fact_224_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_225_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,C)),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_226_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_227_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(plus_plus_int(A),C)),aa_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_228_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_229_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(C,A)),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_230_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_231_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(plus_plus_int(C),A)),aa_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_232_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_233_min__in__set__def,axiom,
    ! [Xs: set_nat,X: nat] :
      ( pp(aa_nat_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_234_max__in__set__def,axiom,
    ! [Xs: set_nat,X: nat] :
      ( pp(aa_nat_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_235_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))
       => ~ ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),Xs))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),X4)) ) ) ) ).

% succ_none_empty
tff(fact_236_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_237_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_238_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat(A,B) = plus_plus_rat(A,C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_239_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_240_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = aa_int_int(plus_plus_int(A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_241_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_242_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( plus_plus_rat(B,A) = plus_plus_rat(C,A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_243_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_244_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(plus_plus_int(B),A) = aa_int_int(plus_plus_int(C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_245__C4_Ohyps_C_I4_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"(4)
tff(fact_246__092_060open_062mi_A_092_060le_062_Ax_092_060close_062,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,mi),xa)) ).

% \<open>mi \<le> x\<close>
tff(fact_247__092_060open_0622_A_092_060le_062_Adeg_092_060close_062,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)) ).

% \<open>2 \<le> deg\<close>
tff(fact_248_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(M)),numeral_numeral_real(N)))
    <=> pp(aa_num_bool(ord_less_eq_num(M),N)) ) ).

% numeral_le_iff
tff(fact_249_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(M)),numeral_numeral_rat(N)))
    <=> pp(aa_num_bool(ord_less_eq_num(M),N)) ) ).

% numeral_le_iff
tff(fact_250_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(ord_less_eq_num(M),N)) ) ).

% numeral_le_iff
tff(fact_251_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(ord_less_eq_num(M),N)) ) ).

% numeral_le_iff
tff(fact_252_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_253_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(C,A)),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_254_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_255_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(plus_plus_int(C),A)),aa_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_256_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_257_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,C)),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_258_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_259_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(plus_plus_int(A),C)),aa_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_260_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_261_semiring__norm_I13_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% semiring_norm(13)
tff(fact_262_semiring__norm_I12_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,one),N) = N ).

% semiring_norm(12)
tff(fact_263_semiring__norm_I11_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(times_times_num,M),one) = M ).

% semiring_norm(11)
tff(fact_264_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(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_265_num__double,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,one)),N) = aa_num_num(bit0,N) ).

% num_double
tff(fact_266_power__mult__numeral,axiom,
    ! [A: nat,M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% power_mult_numeral
tff(fact_267_power__mult__numeral,axiom,
    ! [A: real,M: num,N: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% power_mult_numeral
tff(fact_268_power__mult__numeral,axiom,
    ! [A: complex,M: num,N: num] : aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% power_mult_numeral
tff(fact_269_power__mult__numeral,axiom,
    ! [A: int,M: num,N: num] : aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% power_mult_numeral
tff(fact_270_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W)))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_271_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,numeral_numeral_rat(W))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W)))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_272_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))),B)) ) ).

% le_divide_eq_numeral1(1)
tff(fact_273_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),divide_divide_rat(B,numeral_numeral_rat(W))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W))),B)) ) ).

% le_divide_eq_numeral1(1)
tff(fact_274__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062succy_O_Ais__succ__in__set_A_Iset__vebt_H_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_J_A_Ilow_Ax_A_Ideg_Adiv_A2_J_J_Asuccy_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Succy: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Succy)) ).

% \<open>\<And>thesis. (\<And>succy. is_succ_in_set (set_vebt' (treeList ! high x (deg div 2))) (low x (deg div 2)) succy \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_275__C4_Ohyps_C_I11_J,axiom,
    ( ( 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),na) = 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),na))) )
          & ! [X4: nat] :
              ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),na) = 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),na))) )
             => ( 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)) ) ) ) ) ) ).

% "4.hyps"(11)
tff(fact_276_le__refl,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N)) ).

% le_refl
tff(fact_277_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_278_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_279_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_280_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_281_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)) )
       => ? [X2: nat] :
            ( pp(aa_nat_bool(P,X2))
            & ! [Y3: nat] :
                ( pp(aa_nat_bool(P,Y3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X2)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_282_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_283_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_284_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,plus_plus_rat(I,K)),plus_plus_rat(J,L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_285_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_286_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(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_287_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_288_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,plus_plus_rat(I,K)),plus_plus_rat(J,L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_289_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_290_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(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_291_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_292_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,plus_plus_rat(I,K)),plus_plus_rat(J,L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_293_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_294_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(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_295_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_296_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,plus_plus_rat(A,C)),plus_plus_rat(B,D))) ) ) ).

% add_mono
tff(fact_297_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_298_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(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),D))) ) ) ).

% add_mono
tff(fact_299_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_300_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,plus_plus_rat(C,A)),plus_plus_rat(C,B))) ) ).

% add_left_mono
tff(fact_301_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_302_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(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B))) ) ).

% add_left_mono
tff(fact_303_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ~ ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ) ).

% less_eqE
tff(fact_304_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_305_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,plus_plus_rat(A,C)),plus_plus_rat(B,C))) ) ).

% add_right_mono
tff(fact_306_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_307_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(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C))) ) ).

% add_right_mono
tff(fact_308_le__iff__add,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) ) ).

% le_iff_add
tff(fact_309_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_310_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,plus_plus_rat(C,A)),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_311_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_312_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(plus_plus_int(C),A)),aa_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_313_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_314_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,plus_plus_rat(A,C)),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_315_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_316_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(plus_plus_int(A),C)),aa_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_317_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_318_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_319_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_320_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_321_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_322_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_323_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_324_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_325_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_326_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_327_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_328_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,I3)),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_329_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_330_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_331_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_332_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_333_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_334_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_335_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_336_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_337_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_338_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_339_nat__le__iff__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> ? [K2: nat] : N = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2) ) ).

% nat_le_iff_add
tff(fact_340_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_341_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_342_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_343_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_344_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_345_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_346_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_347_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_348_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,plus_plus_rat(I,K)),plus_plus_rat(J,L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_349_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_350_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(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_351_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_352_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,plus_plus_rat(I,K)),plus_plus_rat(J,L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_353_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_354_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(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_355_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_356_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,plus_plus_rat(A,C)),plus_plus_rat(B,D))) ) ) ).

% add_le_less_mono
tff(fact_357_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_358_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(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),D))) ) ) ).

% add_le_less_mono
tff(fact_359_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_360_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,plus_plus_rat(A,C)),plus_plus_rat(B,D))) ) ) ).

% add_less_le_mono
tff(fact_361_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_362_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(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),D))) ) ) ).

% add_less_le_mono
tff(fact_363_div__mult2__numeral__eq,axiom,
    ! [A: nat,K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,L)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,K),L))) ).

% div_mult2_numeral_eq
tff(fact_364_div__mult2__numeral__eq,axiom,
    ! [A: int,K: num,L: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,L)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,K),L))) ).

% div_mult2_numeral_eq
tff(fact_365_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,M3)),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_366_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_367_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_368_linordered__field__no__lb,axiom,
    ! [X4: real] :
    ? [Y2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),X4)) ).

% linordered_field_no_lb
tff(fact_369_linordered__field__no__lb,axiom,
    ! [X4: rat] :
    ? [Y2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y2),X4)) ).

% linordered_field_no_lb
tff(fact_370_linordered__field__no__ub,axiom,
    ! [X4: real] :
    ? [X_1: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),X_1)) ).

% linordered_field_no_ub
tff(fact_371_linordered__field__no__ub,axiom,
    ! [X4: rat] :
    ? [X_1: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),X_1)) ).

% linordered_field_no_ub
tff(fact_372_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_373_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(times_times_rat(A),C)) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% mult.left_commute
tff(fact_374_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_375_mult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(times_times_int(B),aa_int_int(times_times_int(A),C)) = aa_int_int(times_times_int(A),aa_int_int(times_times_int(B),C)) ).

% mult.left_commute
tff(fact_376_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_377_mult_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),B) = aa_rat_rat(times_times_rat(B),A) ).

% mult.commute
tff(fact_378_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_379_mult_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(A),B) = aa_int_int(times_times_int(B),A) ).

% mult.commute
tff(fact_380_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_381_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% mult.assoc
tff(fact_382_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_383_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(times_times_int(A),B)),C) = aa_int_int(times_times_int(A),aa_int_int(times_times_int(B),C)) ).

% mult.assoc
tff(fact_384_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_385_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_386_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_387_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(times_times_int(A),B)),C) = aa_int_int(times_times_int(A),aa_int_int(times_times_int(B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_388_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_389_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : plus_plus_rat(plus_plus_rat(A,B),C) = plus_plus_rat(A,plus_plus_rat(B,C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_390_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_391_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_392_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_393_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( plus_plus_rat(I,K) = plus_plus_rat(J,L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_394_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_395_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_int_int(plus_plus_int(I),K) = aa_int_int(plus_plus_int(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_396_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_397_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = plus_plus_rat(K,A) )
     => ( plus_plus_rat(A2,B) = plus_plus_rat(K,plus_plus_rat(A,B)) ) ) ).

% group_cancel.add1
tff(fact_398_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_399_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(plus_plus_int(K),A) )
     => ( aa_int_int(plus_plus_int(A2),B) = aa_int_int(plus_plus_int(K),aa_int_int(plus_plus_int(A),B)) ) ) ).

% group_cancel.add1
tff(fact_400_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_401_group__cancel_Oadd2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = plus_plus_rat(K,B) )
     => ( plus_plus_rat(A,B2) = plus_plus_rat(K,plus_plus_rat(A,B)) ) ) ).

% group_cancel.add2
tff(fact_402_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_403_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(plus_plus_int(K),B) )
     => ( aa_int_int(plus_plus_int(A),B2) = aa_int_int(plus_plus_int(K),aa_int_int(plus_plus_int(A),B)) ) ) ).

% group_cancel.add2
tff(fact_404_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_405_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : plus_plus_rat(plus_plus_rat(A,B),C) = plus_plus_rat(A,plus_plus_rat(B,C)) ).

% add.assoc
tff(fact_406_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_407_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(B),C)) ).

% add.assoc
tff(fact_408_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_409_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat(A,B) = plus_plus_rat(A,C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_410_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = aa_int_int(plus_plus_int(A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_411_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_412_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( plus_plus_rat(B,A) = plus_plus_rat(C,A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_413_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(plus_plus_int(B),A) = aa_int_int(plus_plus_int(C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_414_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_415_add_Ocommute,axiom,
    ! [A: rat,B: rat] : plus_plus_rat(A,B) = plus_plus_rat(B,A) ).

% add.commute
tff(fact_416_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_417_add_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(A),B) = aa_int_int(plus_plus_int(B),A) ).

% add.commute
tff(fact_418_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_419_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : plus_plus_rat(B,plus_plus_rat(A,C)) = plus_plus_rat(A,plus_plus_rat(B,C)) ).

% add.left_commute
tff(fact_420_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_421_add_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(plus_plus_int(B),aa_int_int(plus_plus_int(A),C)) = aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(B),C)) ).

% add.left_commute
tff(fact_422_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_423_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat(A,B) = plus_plus_rat(A,C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_424_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_425_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = aa_int_int(plus_plus_int(A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_426_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_427_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( plus_plus_rat(B,A) = plus_plus_rat(C,A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_428_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_429_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(plus_plus_int(B),A) = aa_int_int(plus_plus_int(C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_430_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_431_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_432_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_433_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_434_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,plus_plus_rat(I,K)),plus_plus_rat(J,L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_435_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_436_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(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_437_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_438_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,plus_plus_rat(I,K)),plus_plus_rat(J,L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_439_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_440_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(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_441_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_442_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,plus_plus_rat(I,K)),plus_plus_rat(J,L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_443_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_444_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(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_445_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_446_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,plus_plus_rat(A,C)),plus_plus_rat(B,D))) ) ) ).

% add_strict_mono
tff(fact_447_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_448_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(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),D))) ) ) ).

% add_strict_mono
tff(fact_449_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_450_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,plus_plus_rat(C,A)),plus_plus_rat(C,B))) ) ).

% add_strict_left_mono
tff(fact_451_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_452_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(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B))) ) ).

% add_strict_left_mono
tff(fact_453_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_454_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,plus_plus_rat(A,C)),plus_plus_rat(B,C))) ) ).

% add_strict_right_mono
tff(fact_455_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_456_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(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C))) ) ).

% add_strict_right_mono
tff(fact_457_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_458_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(C,A)),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_459_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_460_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(plus_plus_int(C),A)),aa_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_461_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_462_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,C)),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_463_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_464_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(plus_plus_int(A),C)),aa_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_465_times__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(X),Z)),aa_complex_complex(times_times_complex(Y),W)) ).

% times_divide_times_eq
tff(fact_466_times__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),W)) ).

% times_divide_times_eq
tff(fact_467_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] : aa_rat_rat(times_times_rat(divide_divide_rat(X,Y)),divide_divide_rat(Z,W)) = divide_divide_rat(aa_rat_rat(times_times_rat(X),Z),aa_rat_rat(times_times_rat(Y),W)) ).

% times_divide_times_eq
tff(fact_468_divide__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(X),W)),aa_complex_complex(times_times_complex(Y),Z)) ).

% divide_divide_times_eq
tff(fact_469_divide__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),W)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ).

% divide_divide_times_eq
tff(fact_470_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] : divide_divide_rat(divide_divide_rat(X,Y),divide_divide_rat(Z,W)) = divide_divide_rat(aa_rat_rat(times_times_rat(X),W),aa_rat_rat(times_times_rat(Y),Z)) ).

% divide_divide_times_eq
tff(fact_471_divide__divide__eq__left_H,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(C),B)) ).

% divide_divide_eq_left'
tff(fact_472_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ).

% divide_divide_eq_left'
tff(fact_473_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(divide_divide_rat(A,B),C) = divide_divide_rat(A,aa_rat_rat(times_times_rat(C),B)) ).

% divide_divide_eq_left'
tff(fact_474_add__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),B)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B),C)) ).

% add_divide_distrib
tff(fact_475_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ).

% add_divide_distrib
tff(fact_476_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(plus_plus_rat(A,B),C) = plus_plus_rat(divide_divide_rat(A,C),divide_divide_rat(B,C)) ).

% add_divide_distrib
tff(fact_477__C03_C,axiom,
    ? [Y2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Y2))
      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Y2)) ) ).

% "03"
tff(fact_478_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,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_479_sum__squares__bound,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),X)),Y)),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_480_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(aa_VEB4832260192069861672t_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(aa_VEB4832260192069861672t_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_481_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_482__C02_C,axiom,
    pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),maxl)) ).

% "02"
tff(fact_483_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_484__C05_C,axiom,
    some_nat(succy) = vEBT_vebt_succ(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% "05"
tff(fact_485__C00_C,axiom,
    ( ( some_nat(maxl) = vEBT_vebt_maxt(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),maxl)) ) ).

% "00"
tff(fact_486__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062maxl_O_ASome_Amaxl_A_061_Avebt__maxt_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_A_092_060and_062_Alow_Ax_A_Ideg_Adiv_A2_J_A_060_Amaxl_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Maxl: nat] :
        ~ ( ( some_nat(Maxl) = vEBT_vebt_maxt(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Maxl)) ) ).

% \<open>\<And>thesis. (\<And>maxl. Some maxl = vebt_maxt (treeList ! high x (deg div 2)) \<and> low x (deg div 2) < maxl \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_487_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
           => ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Ys,I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_488_nth__equalityI,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys) )
     => ( ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_bool(Xs)))
           => ( pp(aa_nat_bool(nth_bool(Xs),I3))
            <=> pp(aa_nat_bool(nth_bool(Ys),I3)) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_489_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_nat(Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_490_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_int(Xs)))
           => ( aa_nat_int(nth_int(Xs),I3) = aa_nat_int(nth_int(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_491_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_12: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I4),X_12)) )
    <=> ? [Xs2: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs2) = 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(Xs2,I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_492_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_12: bool] : pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I4),X_12)) )
    <=> ? [Xs2: list_bool] :
          ( ( size_size_list_bool(Xs2) = 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(Xs2),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_493_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_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I4),X_12)) )
    <=> ? [Xs2: list_nat] :
          ( ( size_size_list_nat(Xs2) = K )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I4),aa_nat_nat(nth_nat(Xs2),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_494_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_12: int] : pp(aa_int_bool(aa_nat_fun_int_bool(P,I4),X_12)) )
    <=> ? [Xs2: list_int] :
          ( ( size_size_list_int(Xs2) = 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(Xs2),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_495_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_496_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_497_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I4) = aa_nat_nat(nth_nat(Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_498_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_499_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_500_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_501_mod__mod__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mod_trivial
tff(fact_502_mod__mod__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(modulo_modulo_int(A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mod_trivial
tff(fact_503_mod__mod__trivial,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(modulo364778990260209775nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% mod_mod_trivial
tff(fact_504_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_505_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(plus_plus_rat(A,B),B) = A ).

% add_diff_cancel_right'
tff(fact_506_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_507_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_508_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_509_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] : minus_minus_rat(plus_plus_rat(A,C),plus_plus_rat(B,C)) = minus_minus_rat(A,B) ).

% add_diff_cancel_right
tff(fact_510_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_511_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C)) = aa_int_int(minus_minus_int(A),B) ).

% add_diff_cancel_right
tff(fact_512_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_513_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(plus_plus_rat(A,B),A) = B ).

% add_diff_cancel_left'
tff(fact_514_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_515_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_516_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_517_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] : minus_minus_rat(plus_plus_rat(C,A),plus_plus_rat(C,B)) = minus_minus_rat(A,B) ).

% add_diff_cancel_left
tff(fact_518_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_519_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B)) = aa_int_int(minus_minus_int(A),B) ).

% add_diff_cancel_left
tff(fact_520_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_521_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : plus_plus_rat(minus_minus_rat(A,B),B) = A ).

% diff_add_cancel
tff(fact_522_diff__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A),B)),B) = A ).

% diff_add_cancel
tff(fact_523_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_524_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(plus_plus_rat(A,B),B) = A ).

% add_diff_cancel
tff(fact_525_add__diff__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),B)),B) = A ).

% add_diff_cancel
tff(fact_526_mod__add__self2,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self2
tff(fact_527_mod__add__self2,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_add_self2
tff(fact_528_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self2
tff(fact_529_mod__add__self1,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self1
tff(fact_530_mod__add__self1,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(B),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_add_self1
tff(fact_531_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(B,A),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self1
tff(fact_532_minus__mod__self2,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_mod_self2
tff(fact_533_minus__mod__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% minus_mod_self2
tff(fact_534_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_535_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_536_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_537_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(ord_less_eq_num(aa_num_num(bit0,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(ord_less_eq_num(M),N)) ) ).

% semiring_norm(71)
tff(fact_538_semiring__norm_I68_J,axiom,
    ! [N: num] : pp(aa_num_bool(ord_less_eq_num(one),N)) ).

% semiring_norm(68)
tff(fact_539_left__diff__distrib__numeral,axiom,
    ! [A: complex,B: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V))) ).

% left_diff_distrib_numeral
tff(fact_540_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),numeral_numeral_real(V)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),numeral_numeral_real(V))) ).

% left_diff_distrib_numeral
tff(fact_541_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),numeral_numeral_rat(V)) = minus_minus_rat(aa_rat_rat(times_times_rat(A),numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(B),numeral_numeral_rat(V))) ).

% left_diff_distrib_numeral
tff(fact_542_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(times_times_int(aa_int_int(minus_minus_int(A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(A),aa_num_int(numeral_numeral_int,V))),aa_int_int(times_times_int(B),aa_num_int(numeral_numeral_int,V))) ).

% left_diff_distrib_numeral
tff(fact_543_right__diff__distrib__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(minus_minus_complex(B),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_544_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),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(times_times_real,numeral_numeral_real(V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_545_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),B),aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_546_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_547_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_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(modulo_modulo_nat,A),B) ).

% mod_mult_self1
tff(fact_548_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(C),B))),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self1
tff(fact_549_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self1
tff(fact_550_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_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(modulo_modulo_nat,A),B) ).

% mod_mult_self2
tff(fact_551_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(B),C))),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self2
tff(fact_552_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self2
tff(fact_553_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self3
tff(fact_554_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(C),B)),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self3
tff(fact_555_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self3
tff(fact_556_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self4
tff(fact_557_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),C)),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self4
tff(fact_558_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self4
tff(fact_559_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_560_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_561_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_562_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(ord_less_eq_num(aa_num_num(bit0,M)),one)) ).

% semiring_norm(69)
tff(fact_563_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_564_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_565_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_566_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_567_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] : minus_minus_rat(minus_minus_rat(A,C),B) = minus_minus_rat(minus_minus_rat(A,B),C) ).

% diff_right_commute
tff(fact_568_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_569_diff__right__commute,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A),C)),B) = aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A),B)),C) ).

% diff_right_commute
tff(fact_570_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_571_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A,B) = minus_minus_rat(C,D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_572_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(minus_minus_int(A),B) = aa_int_int(minus_minus_int(C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_573_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_574_mod__diff__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),B)),C) ).

% mod_diff_right_eq
tff(fact_575_mod__diff__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_right_eq
tff(fact_576_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),B)),C) ).

% mod_diff_left_eq
tff(fact_577_mod__diff__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A,C),B),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_left_eq
tff(fact_578_mod__diff__cong,axiom,
    ! [A: int,C: int,A3: int,B: int,B3: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A3),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B3),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A3),B3)),C) ) ) ) ).

% mod_diff_cong
tff(fact_579_mod__diff__cong,axiom,
    ! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B3,C) )
       => ( modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A3,B3),C) ) ) ) ).

% mod_diff_cong
tff(fact_580_mod__diff__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),B)),C) ).

% mod_diff_eq
tff(fact_581_mod__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_eq
tff(fact_582_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_583_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_584_modulo__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)) ).

% modulo_nat_def
tff(fact_585_mod__mult__eq,axiom,
    ! [A: nat,C: 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,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_eq
tff(fact_586_mod__mult__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),B)),C) ).

% mod_mult_eq
tff(fact_587_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,C)),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).

% mod_mult_eq
tff(fact_588_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A3: nat,B: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),B3)),C) ) ) ) ).

% mod_mult_cong
tff(fact_589_mod__mult__cong,axiom,
    ! [A: int,C: int,A3: int,B: int,B3: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A3),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B3),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A3),B3)),C) ) ) ) ).

% mod_mult_cong
tff(fact_590_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B3,C) )
       => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A3),B3),C) ) ) ) ).

% mod_mult_cong
tff(fact_591_mod__mult__mult2,axiom,
    ! [A: nat,C: 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),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,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) ).

% mod_mult_mult2
tff(fact_592_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) = aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(A),B)),C) ).

% mod_mult_mult2
tff(fact_593_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,B)),C) ).

% mod_mult_mult2
tff(fact_594_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_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_mod_right
tff(fact_595_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(times_times_int(C),aa_int_int(modulo_modulo_int(A),B)) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) ).

% mult_mod_right
tff(fact_596_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),modulo364778990260209775nteger(A,B)) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ).

% mult_mod_right
tff(fact_597_mod__mult__left__eq,axiom,
    ! [A: nat,C: 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,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_left_eq
tff(fact_598_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),B)),C) ).

% mod_mult_left_eq
tff(fact_599_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,C)),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).

% mod_mult_left_eq
tff(fact_600_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : 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(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_right_eq
tff(fact_601_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),B)),C) ).

% mod_mult_right_eq
tff(fact_602_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).

% mod_mult_right_eq
tff(fact_603_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_right_eq
tff(fact_604_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),B)),C) ).

% mod_add_right_eq
tff(fact_605_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_right_eq
tff(fact_606_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: 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,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_left_eq
tff(fact_607_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),B)),C) ).

% mod_add_left_eq
tff(fact_608_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_left_eq
tff(fact_609_mod__add__cong,axiom,
    ! [A: nat,C: nat,A3: nat,B: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B3)),C) ) ) ) ).

% mod_add_cong
tff(fact_610_mod__add__cong,axiom,
    ! [A: int,C: int,A3: int,B: int,B3: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A3),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B3),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A3),B3)),C) ) ) ) ).

% mod_add_cong
tff(fact_611_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B3,C) )
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A3,B3),C) ) ) ) ).

% mod_add_cong
tff(fact_612_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: 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,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_eq
tff(fact_613_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),B)),C) ).

% mod_add_eq
tff(fact_614_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_eq
tff(fact_615_power__mod,axiom,
    ! [A: nat,B: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),N)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),B) ).

% power_mod
tff(fact_616_power__mod,axiom,
    ! [A: int,B: int,N: nat] : aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(aa_int_int(modulo_modulo_int(A),B)),N)),B) = aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(A),N)),B) ).

% power_mod
tff(fact_617_power__mod,axiom,
    ! [A: code_integer,B: code_integer,N: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(modulo364778990260209775nteger(A,B)),N),B) = modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),N),B) ).

% power_mod
tff(fact_618_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_619_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,minus_minus_rat(A,C)),minus_minus_rat(B,D))) ) ) ).

% diff_mono
tff(fact_620_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(minus_minus_int(A),C)),aa_int_int(minus_minus_int(B),D))) ) ) ).

% diff_mono
tff(fact_621_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_622_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,minus_minus_rat(C,A)),minus_minus_rat(C,B))) ) ).

% diff_left_mono
tff(fact_623_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(minus_minus_int(C),A)),aa_int_int(minus_minus_int(C),B))) ) ).

% diff_left_mono
tff(fact_624_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_625_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,minus_minus_rat(A,C)),minus_minus_rat(B,C))) ) ).

% diff_right_mono
tff(fact_626_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(minus_minus_int(A),C)),aa_int_int(minus_minus_int(B),C))) ) ).

% diff_right_mono
tff(fact_627_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_628_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A,B) = 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_629_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(minus_minus_int(A),B) = aa_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_630_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_631_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,minus_minus_rat(A,C)),minus_minus_rat(B,C))) ) ).

% diff_strict_right_mono
tff(fact_632_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(minus_minus_int(A),C)),aa_int_int(minus_minus_int(B),C))) ) ).

% diff_strict_right_mono
tff(fact_633_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_634_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,minus_minus_rat(C,A)),minus_minus_rat(C,B))) ) ).

% diff_strict_left_mono
tff(fact_635_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(minus_minus_int(C),A)),aa_int_int(minus_minus_int(C),B))) ) ).

% diff_strict_left_mono
tff(fact_636_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_637_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A,B) = 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_638_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(minus_minus_int(A),B) = aa_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_639_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_640_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,minus_minus_rat(A,C)),minus_minus_rat(B,D))) ) ) ).

% diff_strict_mono
tff(fact_641_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(minus_minus_int(A),C)),aa_int_int(minus_minus_int(B),D))) ) ) ).

% diff_strict_mono
tff(fact_642_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: 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),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D)) ).

% add_diff_add
tff(fact_643_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] : minus_minus_rat(plus_plus_rat(A,C),plus_plus_rat(B,D)) = plus_plus_rat(minus_minus_rat(A,B),minus_minus_rat(C,D)) ).

% add_diff_add
tff(fact_644_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),D)) = aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A),B)),aa_int_int(minus_minus_int(C),D)) ).

% add_diff_add
tff(fact_645_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_646_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : minus_minus_rat(minus_minus_rat(A,B),C) = minus_minus_rat(A,plus_plus_rat(B,C)) ).

% diff_diff_eq
tff(fact_647_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_648_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A),B)),C) = aa_int_int(minus_minus_int(A),aa_int_int(plus_plus_int(B),C)) ).

% diff_diff_eq
tff(fact_649_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_650_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( plus_plus_rat(C,B) = A )
     => ( C = minus_minus_rat(A,B) ) ) ).

% add_implies_diff
tff(fact_651_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_652_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( aa_int_int(plus_plus_int(C),B) = A )
     => ( C = aa_int_int(minus_minus_int(A),B) ) ) ).

% add_implies_diff
tff(fact_653_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_654_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] : minus_minus_rat(A,plus_plus_rat(B,C)) = minus_minus_rat(minus_minus_rat(A,C),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_655_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(minus_minus_int(A),aa_int_int(plus_plus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_656_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_657_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] : plus_plus_rat(minus_minus_rat(A,B),C) = minus_minus_rat(plus_plus_rat(A,C),B) ).

% diff_add_eq
tff(fact_658_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A),B)),C) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),C)),B) ).

% diff_add_eq
tff(fact_659_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_660_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] : minus_minus_rat(A,minus_minus_rat(B,C)) = minus_minus_rat(plus_plus_rat(A,C),B) ).

% diff_diff_eq2
tff(fact_661_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(minus_minus_int(A),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),C)),B) ).

% diff_diff_eq2
tff(fact_662_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_663_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : plus_plus_rat(A,minus_minus_rat(B,C)) = minus_minus_rat(plus_plus_rat(A,B),C) ).

% add_diff_eq
tff(fact_664_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(A),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),B)),C) ).

% add_diff_eq
tff(fact_665_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_666_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A = minus_minus_rat(C,B) )
    <=> ( plus_plus_rat(A,B) = C ) ) ).

% eq_diff_eq
tff(fact_667_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A = aa_int_int(minus_minus_int(C),B) )
    <=> ( aa_int_int(plus_plus_int(A),B) = C ) ) ).

% eq_diff_eq
tff(fact_668_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_669_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat(A,B) = C )
    <=> ( A = plus_plus_rat(C,B) ) ) ).

% diff_eq_eq
tff(fact_670_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(minus_minus_int(A),B) = C )
    <=> ( A = aa_int_int(plus_plus_int(C),B) ) ) ).

% diff_eq_eq
tff(fact_671_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_672_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = plus_plus_rat(K,A) )
     => ( minus_minus_rat(A2,B) = plus_plus_rat(K,minus_minus_rat(A,B)) ) ) ).

% group_cancel.sub1
tff(fact_673_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(plus_plus_int(K),A) )
     => ( aa_int_int(minus_minus_int(A2),B) = aa_int_int(plus_plus_int(K),aa_int_int(minus_minus_int(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_674_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_675_diff__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),B)),C) = aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B),C)) ).

% diff_divide_distrib
tff(fact_676_diff__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_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(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ).

% diff_divide_distrib
tff(fact_677_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(minus_minus_rat(A,B),C) = minus_minus_rat(divide_divide_rat(A,C),divide_divide_rat(B,C)) ).

% diff_divide_distrib
tff(fact_678_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_679_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_680_le__num__One__iff,axiom,
    ! [X: num] :
      ( pp(aa_num_bool(ord_less_eq_num(X),one))
    <=> ( X = one ) ) ).

% le_num_One_iff
tff(fact_681_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_682_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_683_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_684_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_685_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_686_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_687_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_688_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_689_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_690_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_691_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_692_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_693_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_694_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(B),C) )
     => ~ ! [D2: int] : B != aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(C),D2)) ) ).

% mod_eqE
tff(fact_695_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
     => ~ ! [D2: code_integer] : B != plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D2)) ) ).

% mod_eqE
tff(fact_696_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: 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)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_add1_eq
tff(fact_697_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C)) ).

% div_add1_eq
tff(fact_698_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C)) ).

% div_add1_eq
tff(fact_699_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_700_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_701_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_702_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_703_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_704_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_705_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_706_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_707_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_708_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_709_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_710_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),minus_minus_rat(C,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,B)),C)) ) ).

% le_diff_eq
tff(fact_711_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(minus_minus_int(C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(A),B)),C)) ) ).

% le_diff_eq
tff(fact_712_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_713_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(A,B)),C))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),plus_plus_rat(C,B))) ) ).

% diff_le_eq
tff(fact_714_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(minus_minus_int(A),B)),C))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(plus_plus_int(C),B))) ) ).

% diff_le_eq
tff(fact_715_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_716_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),minus_minus_rat(C,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,B)),C)) ) ).

% less_diff_eq
tff(fact_717_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(minus_minus_int(C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(A),B)),C)) ) ).

% less_diff_eq
tff(fact_718_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_719_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(A,B)),C))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),plus_plus_rat(C,B))) ) ).

% diff_less_eq
tff(fact_720_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(minus_minus_int(A),B)),C))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(plus_plus_int(C),B))) ) ).

% diff_less_eq
tff(fact_721_mult__diff__mult,axiom,
    ! [X: real,Y: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A)),B)) ).

% mult_diff_mult
tff(fact_722_mult__diff__mult,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat] : minus_minus_rat(aa_rat_rat(times_times_rat(X),Y),aa_rat_rat(times_times_rat(A),B)) = plus_plus_rat(aa_rat_rat(times_times_rat(X),minus_minus_rat(Y,B)),aa_rat_rat(times_times_rat(minus_minus_rat(X,A)),B)) ).

% mult_diff_mult
tff(fact_723_mult__diff__mult,axiom,
    ! [X: int,Y: int,A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(times_times_int(X),Y)),aa_int_int(times_times_int(A),B)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(X),aa_int_int(minus_minus_int(Y),B))),aa_int_int(times_times_int(aa_int_int(minus_minus_int(X),A)),B)) ).

% mult_diff_mult
tff(fact_724_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: 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,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% L2_set_mult_ineq_lemma
tff(fact_725_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_726_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_727_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_728_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_729_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_730_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_731_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_732_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_733_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_734_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_735_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(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(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_736_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(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(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(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_737_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: 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)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),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(modulo_modulo_nat,B),C))),C)) ).

% div_mult1_eq
tff(fact_738_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A),aa_int_int(modulo_modulo_int(B),C))),C)) ).

% div_mult1_eq
tff(fact_739_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),modulo364778990260209775nteger(B,C)),C)) ).

% div_mult1_eq
tff(fact_740_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),Q2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ).

% mod_mult2_eq
tff(fact_741_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_742_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_743_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_744_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_745_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_746_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_747_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_748_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(Xs3) = N ).

% Ex_list_of_length
tff(fact_749_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_bool] : size_size_list_bool(Xs3) = N ).

% Ex_list_of_length
tff(fact_750_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_nat] : size_size_list_nat(Xs3) = N ).

% Ex_list_of_length
tff(fact_751_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_int] : size_size_list_int(Xs3) = N ).

% Ex_list_of_length
tff(fact_752_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_753_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_754_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_755_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_756_power2__commute,axiom,
    ! [X: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Y),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_757_power2__commute,axiom,
    ! [X: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_758_power2__commute,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(minus_minus_rat(X,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(minus_minus_rat(Y,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_759_power2__commute,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(minus_minus_int(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(aa_int_int(minus_minus_int(Y),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_760_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_761_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_762_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_763_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,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(plus_plus_nat,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_764_div__exp__mod__exp__eq,axiom,
    ! [A: int,N: nat,M: nat] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),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))),M)) = aa_int_int(aa_int_fun_int_int(divide_divide_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(plus_plus_nat,N),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_765_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N: nat,M: nat] : modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) = divide6298287555418463151nteger(modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_766_power2__diff,axiom,
    ! [X: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_767_power2__diff,axiom,
    ! [X: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,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))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_768_power2__diff,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(minus_minus_rat(X,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = minus_minus_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)))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_769_power2__diff,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(minus_minus_int(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(minus_minus_int(aa_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))))),aa_int_int(times_times_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_770_length__induct,axiom,
    ! [P: fun_li6006399982103069984T_bool,Xs: list_VEBT_VEBT] :
      ( ! [Xs3: 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(Xs3)))
             => pp(aa_lis2822252094023462681T_bool(P,Ys2)) )
         => pp(aa_lis2822252094023462681T_bool(P,Xs3)) )
     => pp(aa_lis2822252094023462681T_bool(P,Xs)) ) ).

% length_induct
tff(fact_771_length__induct,axiom,
    ! [P: fun_list_bool_bool,Xs: list_bool] :
      ( ! [Xs3: 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(Xs3)))
             => pp(aa_list_bool_bool(P,Ys2)) )
         => pp(aa_list_bool_bool(P,Xs3)) )
     => pp(aa_list_bool_bool(P,Xs)) ) ).

% length_induct
tff(fact_772_length__induct,axiom,
    ! [P: fun_list_nat_bool,Xs: list_nat] :
      ( ! [Xs3: 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(Xs3)))
             => pp(aa_list_nat_bool(P,Ys2)) )
         => pp(aa_list_nat_bool(P,Xs3)) )
     => pp(aa_list_nat_bool(P,Xs)) ) ).

% length_induct
tff(fact_773_length__induct,axiom,
    ! [P: fun_list_int_bool,Xs: list_int] :
      ( ! [Xs3: 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(Xs3)))
             => pp(aa_list_int_bool(P,Ys2)) )
         => pp(aa_list_int_bool(P,Xs3)) )
     => pp(aa_list_int_bool(P,Xs)) ) ).

% length_induct
tff(fact_774_four__x__squared,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),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,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% four_x_squared
tff(fact_775_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_776_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_777_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_778__C4_Ohyps_C_I3_J,axiom,
    ! [X: nat,Sx: nat] :
      ( ( vEBT_vebt_succ(summary,X) = some_nat(Sx) )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(summary)),X),Sx)) ) ).

% "4.hyps"(3)
tff(fact_779_True,axiom,
    ( ( vEBT_vebt_maxt(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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 )
    & pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ) ).

% True
tff(fact_780_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_781_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_782_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( plus_plus_rat(minus_minus_rat(A,B),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_783_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_784_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(plus_plus_int(aa_int_int(minus_minus_int(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_785_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_786_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( plus_plus_rat(B,minus_minus_rat(A,B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_787_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_788_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(plus_plus_int(B),aa_int_int(minus_minus_int(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_789__C4_Ohyps_C_I7_J,axiom,
    ! [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_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I2)),X_12))
      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,summary),I2)) ) ) ).

% "4.hyps"(7)
tff(fact_790_div__mod__decomp,axiom,
    ! [A2: nat,N: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),N)),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N)) ).

% div_mod_decomp
tff(fact_791_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ~ ! [S2: 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,Q2),S2)) ) ) ).

% mod_eq_nat1E
tff(fact_792_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y22: num] :
      ( ( aa_num_num(bit0,X22) = aa_num_num(bit0,Y22) )
    <=> ( X22 = Y22 ) ) ).

% verit_eq_simplify(8)
tff(fact_793_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit0
tff(fact_794_real__divide__square__eq,axiom,
    ! [R: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),R)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),R) ).

% real_divide_square_eq
tff(fact_795_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W))) ).

% zmod_numeral_Bit0
tff(fact_796_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_797_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_798__C4_Ohyps_C_I2_J,axiom,
    pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,summary),m)) ).

% "4.hyps"(2)
tff(fact_799__C4_Ohyps_C_I8_J,axiom,
    ( ( mi = ma )
   => ! [X4: vEBT_VEBT] :
        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(treeList)))
       => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_13)) ) ) ).

% "4.hyps"(8)
tff(fact_800_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_801_complete__real,axiom,
    ! [S3: set_real] :
      ( ? [X4: real] : member_real(X4,S3)
     => ( ? [Z3: real] :
          ! [X2: real] :
            ( member_real(X2,S3)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Z3)) )
       => ? [Y2: real] :
            ( ! [X4: real] :
                ( member_real(X4,S3)
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y2)) )
            & ! [Z3: real] :
                ( ! [X2: real] :
                    ( member_real(X2,S3)
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Z3)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),Z3)) ) ) ) ) ).

% complete_real
tff(fact_802_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_803_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Z),Y))
     => ( plus_p3455044024723400733d_enat(X,minus_3235023915231533773d_enat(Y,Z)) = minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(X,Y),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_804_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_805_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ pp(aa_num_bool(ord_less_eq_num(A),B))
      | ~ pp(aa_num_bool(ord_less_eq_num(B),A)) ) ).

% verit_la_disequality
tff(fact_806_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_807_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_808_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(A),A)) ).

% verit_comp_simplify1(2)
tff(fact_809_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_810_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : pp(aa_num_bool(ord_less_eq_num(A),A)) ).

% verit_comp_simplify1(2)
tff(fact_811_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_812_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_813_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_814_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_815_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ pp(aa_num_bool(ord_less_num(A),A)) ).

% verit_comp_simplify1(1)
tff(fact_816_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_817_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_818_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_819_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_820_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_821_verit__comp__simplify1_I3_J,axiom,
    ! [B3: real,A3: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A3))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A3),B3)) ) ).

% verit_comp_simplify1(3)
tff(fact_822_verit__comp__simplify1_I3_J,axiom,
    ! [B3: rat,A3: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A3))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A3),B3)) ) ).

% verit_comp_simplify1(3)
tff(fact_823_verit__comp__simplify1_I3_J,axiom,
    ! [B3: num,A3: num] :
      ( ~ pp(aa_num_bool(ord_less_eq_num(B3),A3))
    <=> pp(aa_num_bool(ord_less_num(A3),B3)) ) ).

% verit_comp_simplify1(3)
tff(fact_824_verit__comp__simplify1_I3_J,axiom,
    ! [B3: nat,A3: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A3))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A3),B3)) ) ).

% verit_comp_simplify1(3)
tff(fact_825_verit__comp__simplify1_I3_J,axiom,
    ! [B3: int,A3: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A3))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A3),B3)) ) ).

% verit_comp_simplify1(3)
tff(fact_826_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_827_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),C) = plus_plus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_828_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_829_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_830_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),plus_plus_rat(B,C)) = plus_plus_rat(aa_rat_rat(times_times_rat(A),B),aa_rat_rat(times_times_rat(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_831_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(A),aa_int_int(plus_plus_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),B)),aa_int_int(times_times_int(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_832_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% comm_semiring_class.distrib
tff(fact_833_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),C) = plus_plus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) ).

% comm_semiring_class.distrib
tff(fact_834_comm__semiring__class_Odistrib,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(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_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)) ).

% comm_semiring_class.distrib
tff(fact_835_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ).

% comm_semiring_class.distrib
tff(fact_836_distrib__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% distrib_left
tff(fact_837_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),plus_plus_rat(B,C)) = plus_plus_rat(aa_rat_rat(times_times_rat(A),B),aa_rat_rat(times_times_rat(A),C)) ).

% distrib_left
tff(fact_838_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_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,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% distrib_left
tff(fact_839_distrib__left,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(A),aa_int_int(plus_plus_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),B)),aa_int_int(times_times_int(A),C)) ).

% distrib_left
tff(fact_840_distrib__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% distrib_right
tff(fact_841_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),C) = plus_plus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) ).

% distrib_right
tff(fact_842_distrib__right,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(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_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)) ).

% distrib_right
tff(fact_843_distrib__right,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ).

% distrib_right
tff(fact_844_combine__common__factor,axiom,
    ! [A: real,E: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),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(plus_plus_real,A),B)),E)),C) ).

% combine_common_factor
tff(fact_845_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C: rat] : plus_plus_rat(aa_rat_rat(times_times_rat(A),E),plus_plus_rat(aa_rat_rat(times_times_rat(B),E),C)) = plus_plus_rat(aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),E),C) ).

% combine_common_factor
tff(fact_846_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E)),C) ).

% combine_common_factor
tff(fact_847_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(plus_plus_int(A),B)),E)),C) ).

% combine_common_factor
tff(fact_848_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] : one != aa_num_num(bit0,X22) ).

% verit_eq_simplify(10)
tff(fact_849_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_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(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib'
tff(fact_850_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(times_times_rat(A),B),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib'
tff(fact_851_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_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)) ).

% right_diff_distrib'
tff(fact_852_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(A),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(A),B)),aa_int_int(times_times_int(A),C)) ).

% right_diff_distrib'
tff(fact_853_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)) ).

% left_diff_distrib'
tff(fact_854_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(minus_minus_rat(B,C)),A) = minus_minus_rat(aa_rat_rat(times_times_rat(B),A),aa_rat_rat(times_times_rat(C),A)) ).

% left_diff_distrib'
tff(fact_855_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_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)) ).

% left_diff_distrib'
tff(fact_856_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(times_times_int(aa_int_int(minus_minus_int(B),C)),A) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(B),A)),aa_int_int(times_times_int(C),A)) ).

% left_diff_distrib'
tff(fact_857_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_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(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib
tff(fact_858_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(times_times_rat(A),B),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib
tff(fact_859_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(A),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(A),B)),aa_int_int(times_times_int(A),C)) ).

% right_diff_distrib
tff(fact_860_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_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(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% left_diff_distrib
tff(fact_861_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),C) = minus_minus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) ).

% left_diff_distrib
tff(fact_862_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(minus_minus_int(A),B)),C) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ).

% left_diff_distrib
tff(fact_863_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_864_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,plus_plus_rat(I,K)),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),minus_minus_rat(N,K))) ) ).

% add_le_imp_le_diff
tff(fact_865_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_866_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(plus_plus_int(I),K)),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),aa_int_int(minus_minus_int(N),K))) ) ).

% add_le_imp_le_diff
tff(fact_867_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_868_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,plus_plus_rat(I,K)),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),plus_plus_rat(J,K)))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(I,K)),N))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),plus_plus_rat(J,K)))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(N,K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_869_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_870_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(plus_plus_int(I),K)),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(plus_plus_int(J),K)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_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(plus_plus_int(J),K)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(minus_minus_int(N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_871_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_872_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))
     => ( plus_plus_rat(B,minus_minus_rat(A,B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_873_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_874_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(plus_plus_int(B),aa_int_int(minus_minus_int(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_875_square__diff__square__factored,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) ).

% square_diff_square_factored
tff(fact_876_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] : minus_minus_rat(aa_rat_rat(times_times_rat(X),X),aa_rat_rat(times_times_rat(Y),Y)) = aa_rat_rat(times_times_rat(plus_plus_rat(X,Y)),minus_minus_rat(X,Y)) ).

% square_diff_square_factored
tff(fact_877_square__diff__square__factored,axiom,
    ! [X: int,Y: int] : aa_int_int(minus_minus_int(aa_int_int(times_times_int(X),X)),aa_int_int(times_times_int(Y),Y)) = aa_int_int(times_times_int(aa_int_int(plus_plus_int(X),Y)),aa_int_int(minus_minus_int(X),Y)) ).

% square_diff_square_factored
tff(fact_878_eq__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D) )
    <=> ( 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)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_879_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C) = plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D) )
    <=> ( C = plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(B,A)),E),D) ) ) ).

% eq_add_iff2
tff(fact_880_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),D) )
    <=> ( C = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_881_eq__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D) )
    <=> ( 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)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_882_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C) = plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D) )
    <=> ( plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),E),C) = D ) ) ).

% eq_add_iff1
tff(fact_883_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),D) )
    <=> ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_884_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = 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,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_885_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = 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,Q2))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_886_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,M)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,N)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,N),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_887_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,one)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,one)) ).

% cong_exp_iff_simps(4)
tff(fact_888_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,one)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,one)) ).

% cong_exp_iff_simps(4)
tff(fact_889_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_num_Code_integer(numera6620942414471956472nteger,one)) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,N),aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% cong_exp_iff_simps(4)
tff(fact_890_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_891_nat__mod__eq__iff,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) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_892_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: 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),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),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)),E)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_893_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C)),plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(B,A)),E),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_894_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),C)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(B),A)),E)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_895_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: 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),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),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)),E)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_896_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C)),plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),E),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_897_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),C)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(A),B)),E)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_898_less__add__iff1,axiom,
    ! [A: real,E: 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),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),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)),E)),C)),D)) ) ).

% less_add_iff1
tff(fact_899_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C)),plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),E),C)),D)) ) ).

% less_add_iff1
tff(fact_900_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),C)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(A),B)),E)),C)),D)) ) ).

% less_add_iff1
tff(fact_901_less__add__iff2,axiom,
    ! [A: real,E: 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),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),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)),E)),D))) ) ).

% less_add_iff2
tff(fact_902_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C)),plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(B,A)),E),D))) ) ).

% less_add_iff2
tff(fact_903_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),C)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(B),A)),E)),D))) ) ).

% less_add_iff2
tff(fact_904_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != 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,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_905_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_906_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,M)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_907_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: 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,Q2))) != 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,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_908_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: 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,Q2))) != 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,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_909_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,N)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_910_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: 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,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_911_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(modulo_modulo_int(A),B))),C) = aa_int_int(plus_plus_int(A),C) ).

% cancel_div_mod_rules(2)
tff(fact_912_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(2)
tff(fact_913_cancel__div__mod__rules_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,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_914_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(modulo_modulo_int(A),B))),C) = aa_int_int(plus_plus_int(A),C) ).

% cancel_div_mod_rules(1)
tff(fact_915_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(1)
tff(fact_916_mod__div__decomp,axiom,
    ! [A: nat,B: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ).

% mod_div_decomp
tff(fact_917_mod__div__decomp,axiom,
    ! [A: int,B: int] : A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(modulo_modulo_int(A),B)) ).

% mod_div_decomp
tff(fact_918_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] : A = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) ).

% mod_div_decomp
tff(fact_919_div__mult__mod__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% div_mult_mod_eq
tff(fact_920_div__mult__mod__eq,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(modulo_modulo_int(A),B)) = A ).

% div_mult_mod_eq
tff(fact_921_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) = A ).

% div_mult_mod_eq
tff(fact_922_mod__div__mult__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_923_mod__div__mult__eq,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),B)),aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_924_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = A ).

% mod_div_mult_eq
tff(fact_925_mod__mult__div__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = A ).

% mod_mult_div_eq
tff(fact_926_mod__mult__div__eq,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),B)),aa_int_int(times_times_int(B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = A ).

% mod_mult_div_eq
tff(fact_927_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = A ).

% mod_mult_div_eq
tff(fact_928_mult__div__mod__eq,axiom,
    ! [B: nat,A: nat] : 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(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% mult_div_mod_eq
tff(fact_929_mult__div__mod__eq,axiom,
    ! [B: int,A: int] : aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(modulo_modulo_int(A),B)) = A ).

% mult_div_mod_eq
tff(fact_930_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)) = A ).

% mult_div_mod_eq
tff(fact_931_minus__mult__div__eq__mod,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(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% minus_mult_div_eq_mod
tff(fact_932_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),aa_int_int(times_times_int(B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_mult_div_eq_mod
tff(fact_933_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = modulo364778990260209775nteger(A,B) ).

% minus_mult_div_eq_mod
tff(fact_934_minus__mod__eq__mult__div,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(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_935_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),aa_int_int(modulo_modulo_int(A),B)) = aa_int_int(times_times_int(B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_936_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)) ).

% minus_mod_eq_mult_div
tff(fact_937_minus__mod__eq__div__mult,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(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_938_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),aa_int_int(modulo_modulo_int(A),B)) = aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_939_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B) ).

% minus_mod_eq_div_mult
tff(fact_940_minus__div__mult__eq__mod,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(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% minus_div_mult_eq_mod
tff(fact_941_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_div_mult_eq_mod
tff(fact_942_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = modulo364778990260209775nteger(A,B) ).

% minus_div_mult_eq_mod
tff(fact_943_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_944_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ~ ! [S2: 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,Q2),S2)) ) ) ).

% mod_eq_nat2E
tff(fact_945_real__average__minus__second,axiom,
    ! [B: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),numeral_numeral_real(aa_num_num(bit0,one)))),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),numeral_numeral_real(aa_num_num(bit0,one))) ).

% real_average_minus_second
tff(fact_946_real__average__minus__first,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(aa_num_num(bit0,one)))),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),numeral_numeral_real(aa_num_num(bit0,one))) ).

% real_average_minus_first
tff(fact_947_not__Some__eq,axiom,
    ! [X: option_nat] :
      ( ! [Y4: nat] : X != some_nat(Y4)
    <=> ( X = none_nat ) ) ).

% not_Some_eq
tff(fact_948_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_949_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ! [Y4: num] : X != some_num(Y4)
    <=> ( X = none_num ) ) ).

% not_Some_eq
tff(fact_950_not__None__eq,axiom,
    ! [X: option_nat] :
      ( ( X != none_nat )
    <=> ? [Y4: nat] : X = some_nat(Y4) ) ).

% not_None_eq
tff(fact_951_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_952_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
    <=> ? [Y4: num] : X = some_num(Y4) ) ).

% not_None_eq
tff(fact_953_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,Tree),X))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_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_954_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T)),X))
       => ( vEBT_vebt_maxt(T) = some_nat(X) ) ) ) ).

% maxt_sound
tff(fact_955_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_maxt(T) = some_nat(X) )
       => pp(aa_nat_bool(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T)),X)) ) ) ).

% maxt_corr
tff(fact_956_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X)) ) ).

% min_Null_member
tff(fact_957_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_958_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Maxi)) ) ) ) ).

% maxt_corr_help
tff(fact_959_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X)) ) ) ).

% both_member_options_equiv_member
tff(fact_960_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X)) ) ) ).

% valid_member_both_member_options
tff(fact_961_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_set_nat_bool(finite_finite_nat,vEBT_VEBT_set_vebt(T))) ) ).

% set_vebt_finite
tff(fact_962_inthall,axiom,
    ! [Xs: list_real,P: fun_real_bool,N: nat] :
      ( ! [X2: real] :
          ( member_real(X2,set_real2(Xs))
         => pp(aa_real_bool(P,X2)) )
     => ( 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_963_inthall,axiom,
    ! [Xs: list_complex,P: fun_complex_bool,N: nat] :
      ( ! [X2: complex] :
          ( pp(aa_set_complex_bool(member_complex(X2),set_complex2(Xs)))
         => pp(aa_complex_bool(P,X2)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3451745648224563538omplex(Xs)))
       => pp(aa_complex_bool(P,aa_nat_complex(nth_complex(Xs),N))) ) ) ).

% inthall
tff(fact_964_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,N: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,X2)) )
     => ( 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_965_inthall,axiom,
    ! [Xs: list_bool,P: fun_bool_bool,N: nat] :
      ( ! [X2: bool] :
          ( member_bool(X2,set_bool2(Xs))
         => pp(aa_bool_bool(P,X2)) )
     => ( 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_966_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,N: nat] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
         => pp(aa_nat_bool(P,X2)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% inthall
tff(fact_967_inthall,axiom,
    ! [Xs: list_int,P: fun_int_bool,N: nat] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),set_int2(Xs)))
         => pp(aa_int_bool(P,X2)) )
     => ( 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_968_maxt__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Maxi)) ) ) ).

% maxt_member
tff(fact_969_option_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( some_nat(X22) = some_nat(Y22) )
    <=> ( X22 = Y22 ) ) ).

% option.inject
tff(fact_970_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_971_option_Oinject,axiom,
    ! [X22: num,Y22: num] :
      ( ( some_num(X22) = some_num(Y22) )
    <=> ( X22 = Y22 ) ) ).

% option.inject
tff(fact_972_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X))
      <=> pp(aa_set_nat_bool(member_nat(X),vEBT_set_vebt(T))) ) ) ).

% member_correct
tff(fact_973__C4_Ohyps_C_I1_J,axiom,
    ! [X4: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(treeList)))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X4),na))
        & ! [Xa: nat,Xb: nat] :
            ( ( vEBT_vebt_succ(X4,Xa) = some_nat(Xb) )
          <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(X4)),Xa),Xb)) ) ) ) ).

% "4.hyps"(1)
tff(fact_974_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( vEBT_set_vebt(T) = vEBT_VEBT_set_vebt(T) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_975_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : finite5795047828879050333T_VEBT(set_VEBT_VEBT2(Xs)) ).

% List.finite_set
tff(fact_976_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : pp(aa_set_nat_bool(finite_finite_nat,set_nat2(Xs))) ).

% List.finite_set
tff(fact_977_List_Ofinite__set,axiom,
    ! [Xs: list_int] : pp(aa_set_int_bool(finite_finite_int,set_int2(Xs))) ).

% List.finite_set
tff(fact_978_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : pp(aa_set_complex_bool(finite3207457112153483333omplex,set_complex2(Xs))) ).

% List.finite_set
tff(fact_979__C01_C,axiom,
    ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),na))
    & pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),set_VEBT_VEBT2(treeList))) ) ).

% "01"
tff(fact_980_div__mod__decomp__int,axiom,
    ! [A2: int,N: int] : A2 = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),N)),N)),aa_int_int(modulo_modulo_int(A2),N)) ).

% div_mod_decomp_int
tff(fact_981_subset__code_I1_J,axiom,
    ! [Xs: list_real,B2: set_real] :
      ( ord_less_eq_set_real(set_real2(Xs),B2)
    <=> ! [X3: real] :
          ( member_real(X3,set_real2(Xs))
         => member_real(X3,B2) ) ) ).

% subset_code(1)
tff(fact_982_subset__code_I1_J,axiom,
    ! [Xs: list_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Xs)),B2))
    <=> ! [X3: complex] :
          ( pp(aa_set_complex_bool(member_complex(X3),set_complex2(Xs)))
         => pp(aa_set_complex_bool(member_complex(X3),B2)) ) ) ).

% subset_code(1)
tff(fact_983_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_984_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_985_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B2: set_nat] :
      ( pp(aa_set_nat_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_986_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A2)
     => ? [Xs3: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs3) = A2 ) ).

% finite_list
tff(fact_987_finite__list,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ? [Xs3: list_nat] : set_nat2(Xs3) = A2 ) ).

% finite_list
tff(fact_988_finite__list,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ? [Xs3: list_int] : set_int2(Xs3) = A2 ) ).

% finite_list
tff(fact_989_finite__list,axiom,
    ! [A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ? [Xs3: list_complex] : set_complex2(Xs3) = A2 ) ).

% finite_list
tff(fact_990_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)))
     => member_real(aa_nat_real(nth_real(Xs),N),set_real2(Xs)) ) ).

% nth_mem
tff(fact_991_nth__mem,axiom,
    ! [N: nat,Xs: list_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3451745648224563538omplex(Xs)))
     => pp(aa_set_complex_bool(member_complex(aa_nat_complex(nth_complex(Xs),N)),set_complex2(Xs))) ) ).

% nth_mem
tff(fact_992_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_993_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_994_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
     => pp(aa_set_nat_bool(member_nat(aa_nat_nat(nth_nat(Xs),N)),set_nat2(Xs))) ) ).

% nth_mem
tff(fact_995_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_996_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)))
     => ( ! [X2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(Xs)))
           => pp(aa_VEBT_VEBT_bool(P,X2)) )
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,N))) ) ) ).

% list_ball_nth
tff(fact_997_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)))
     => ( ! [X2: bool] :
            ( member_bool(X2,set_bool2(Xs))
           => pp(aa_bool_bool(P,X2)) )
       => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_998_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)))
     => ( ! [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
           => pp(aa_nat_bool(P,X2)) )
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_999_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)))
     => ( ! [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),set_int2(Xs)))
           => pp(aa_int_bool(P,X2)) )
       => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_1000_in__set__conv__nth,axiom,
    ! [X: real,Xs: list_real] :
      ( 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_1001_in__set__conv__nth,axiom,
    ! [X: complex,Xs: list_complex] :
      ( pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s3451745648224563538omplex(Xs)))
          & ( aa_nat_complex(nth_complex(Xs),I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_1002_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_1003_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_1004_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(nth_nat(Xs),I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_1005_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_1006_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_bool,X: real] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_real(Xs)))
         => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),I3))) )
     => ( member_real(X,set_real2(Xs))
       => pp(aa_real_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_1007_all__nth__imp__all__set,axiom,
    ! [Xs: list_complex,P: fun_complex_bool,X: complex] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s3451745648224563538omplex(Xs)))
         => pp(aa_complex_bool(P,aa_nat_complex(nth_complex(Xs),I3))) )
     => ( pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
       => pp(aa_complex_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_1008_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,I3))) )
     => ( 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_1009_all__nth__imp__all__set,axiom,
    ! [Xs: list_bool,P: fun_bool_bool,X: bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_bool(Xs)))
         => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),I3))) )
     => ( member_bool(X,set_bool2(Xs))
       => pp(aa_bool_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_1010_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,X: nat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I3))) )
     => ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
       => pp(aa_nat_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_1011_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_bool,X: int] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_int(Xs)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I3))) )
     => ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
       => pp(aa_int_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_1012_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_1013_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_1014_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_bool] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
         => pp(aa_nat_bool(P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_1015_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_1016_option_Odistinct_I1_J,axiom,
    ! [X22: nat] : none_nat != some_nat(X22) ).

% option.distinct(1)
tff(fact_1017_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X22) ).

% option.distinct(1)
tff(fact_1018_option_Odistinct_I1_J,axiom,
    ! [X22: num] : none_num != some_num(X22) ).

% option.distinct(1)
tff(fact_1019_option_OdiscI,axiom,
    ! [Option: option_nat,X22: nat] :
      ( ( Option = some_nat(X22) )
     => ( Option != none_nat ) ) ).

% option.discI
tff(fact_1020_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X22) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_1021_option_OdiscI,axiom,
    ! [Option: option_num,X22: num] :
      ( ( Option = some_num(X22) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_1022_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X23: nat] : Y != some_nat(X23) ) ).

% option.exhaust
tff(fact_1023_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] : Y != some_P7363390416028606310at_nat(X23) ) ).

% option.exhaust
tff(fact_1024_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X23: num] : Y != some_num(X23) ) ).

% option.exhaust
tff(fact_1025_split__option__ex,axiom,
    ! [P: fun_option_nat_bool] :
      ( ? [X_12: option_nat] : pp(aa_option_nat_bool(P,X_12))
    <=> ( pp(aa_option_nat_bool(P,none_nat))
        | ? [X3: nat] : pp(aa_option_nat_bool(P,some_nat(X3))) ) ) ).

% split_option_ex
tff(fact_1026_split__option__ex,axiom,
    ! [P: fun_op2627489687811228755t_bool] :
      ( ? [X_12: option4927543243414619207at_nat] : pp(aa_opt6800557945374683724t_bool(P,X_12))
    <=> ( 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_1027_split__option__ex,axiom,
    ! [P: fun_option_num_bool] :
      ( ? [X_12: option_num] : pp(aa_option_num_bool(P,X_12))
    <=> ( pp(aa_option_num_bool(P,none_num))
        | ? [X3: num] : pp(aa_option_num_bool(P,some_num(X3))) ) ) ).

% split_option_ex
tff(fact_1028_split__option__all,axiom,
    ! [P: fun_option_nat_bool] :
      ( ! [X_12: option_nat] : pp(aa_option_nat_bool(P,X_12))
    <=> ( pp(aa_option_nat_bool(P,none_nat))
        & ! [X3: nat] : pp(aa_option_nat_bool(P,some_nat(X3))) ) ) ).

% split_option_all
tff(fact_1029_split__option__all,axiom,
    ! [P: fun_op2627489687811228755t_bool] :
      ( ! [X_12: option4927543243414619207at_nat] : pp(aa_opt6800557945374683724t_bool(P,X_12))
    <=> ( 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_1030_split__option__all,axiom,
    ! [P: fun_option_num_bool] :
      ( ! [X_12: option_num] : pp(aa_option_num_bool(P,X_12))
    <=> ( pp(aa_option_num_bool(P,none_num))
        & ! [X3: num] : pp(aa_option_num_bool(P,some_num(X3))) ) ) ).

% split_option_all
tff(fact_1031_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)) )
       => ( ! [A4: nat,B4: nat] :
              ( ( X = some_nat(A4) )
             => ( ( Y = some_nat(B4) )
               => 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_1032_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)) )
       => ( ! [A4: nat,B4: product_prod_nat_nat] :
              ( ( X = some_nat(A4) )
             => ( ( Y = some_P7363390416028606310at_nat(B4) )
               => 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_1033_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)) )
       => ( ! [A4: nat,B4: num] :
              ( ( X = some_nat(A4) )
             => ( ( Y = some_num(B4) )
               => 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_1034_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)) )
       => ( ! [A4: product_prod_nat_nat,B4: nat] :
              ( ( X = some_P7363390416028606310at_nat(A4) )
             => ( ( Y = some_nat(B4) )
               => 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_1035_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)) )
       => ( ! [A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
              ( ( X = some_P7363390416028606310at_nat(A4) )
             => ( ( Y = some_P7363390416028606310at_nat(B4) )
               => 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_1036_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)) )
       => ( ! [A4: product_prod_nat_nat,B4: num] :
              ( ( X = some_P7363390416028606310at_nat(A4) )
             => ( ( Y = some_num(B4) )
               => 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_1037_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)) )
       => ( ! [A4: num,B4: nat] :
              ( ( X = some_num(A4) )
             => ( ( Y = some_nat(B4) )
               => 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_1038_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)) )
       => ( ! [A4: num,B4: product_prod_nat_nat] :
              ( ( X = some_num(A4) )
             => ( ( Y = some_P7363390416028606310at_nat(B4) )
               => 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_1039_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)) )
       => ( ! [A4: num,B4: num] :
              ( ( X = some_num(A4) )
             => ( ( Y = some_num(B4) )
               => 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_1040_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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_vebt_member,vEBT_vebt_insert(T,X)),Y))
           => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Y))
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_1041_misiz,axiom,
    ! [T: vEBT_VEBT,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( some_nat(M) = vEBT_vebt_mint(T) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ).

% misiz
tff(fact_1042_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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_1043_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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_1044_mint__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_mint(T) = some_nat(X) )
       => pp(aa_nat_bool(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T)),X)) ) ) ).

% mint_corr
tff(fact_1045_mint__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T)),X))
       => ( vEBT_vebt_mint(T) = some_nat(X) ) ) ) ).

% mint_sound
tff(fact_1046_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_mint(T) = some_nat(Mini) )
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mini),X)) ) ) ) ).

% mint_corr_help
tff(fact_1047_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(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),X)) ) ) ) ).

% both_member_options_ding
tff(fact_1048_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X2),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_1049_mint__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_mint(T) = some_nat(Maxi) )
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Maxi)) ) ) ).

% mint_member
tff(fact_1050_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ( vEBT_vebt_mint(T) = none_nat ) ) ).

% minNullmin
tff(fact_1051_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),N))
     => ( Deg = N ) ) ).

% deg_deg_n
tff(fact_1052_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(T) = none_nat )
     => pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T)) ) ).

% minminNull
tff(fact_1053__092_060open_0621_A_092_060le_062_An_092_060close_062,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),na)) ).

% \<open>1 \<le> n\<close>
tff(fact_1054_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_1055_mult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% mult_1
tff(fact_1056_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% mult_1
tff(fact_1057_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_1058_mult__1,axiom,
    ! [A: int] : aa_int_int(times_times_int(one_one_int),A) = A ).

% mult_1
tff(fact_1059_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_1060_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_1061_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_1062_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_1063_mult_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(times_times_int(A),one_one_int) = A ).

% mult.right_neutral
tff(fact_1064_div__by__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),one_one_complex) = A ).

% div_by_1
tff(fact_1065_div__by__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),one_one_real) = A ).

% div_by_1
tff(fact_1066_div__by__1,axiom,
    ! [A: rat] : divide_divide_rat(A,one_one_rat) = A ).

% div_by_1
tff(fact_1067_div__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).

% div_by_1
tff(fact_1068_div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% div_by_1
tff(fact_1069_bits__div__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).

% bits_div_by_1
tff(fact_1070_bits__div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% bits_div_by_1
tff(fact_1071_power__one,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(one_one_rat),N) = one_one_rat ).

% power_one
tff(fact_1072_power__one,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),N) = one_one_nat ).

% power_one
tff(fact_1073_power__one,axiom,
    ! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),N) = one_one_real ).

% power_one
tff(fact_1074_power__one,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(one_one_complex),N) = one_one_complex ).

% power_one
tff(fact_1075_power__one,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(one_one_int),N) = one_one_int ).

% power_one
tff(fact_1076_power__one__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),one_one_nat) = A ).

% power_one_right
tff(fact_1077_power__one__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),one_one_nat) = A ).

% power_one_right
tff(fact_1078_power__one__right,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),one_one_nat) = A ).

% power_one_right
tff(fact_1079_power__one__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),one_one_nat) = A ).

% power_one_right
tff(fact_1080_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_1081_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_1082_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex(N) = one_one_complex )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_1083_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real(N) = one_one_real )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_1084_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat(N) = one_one_rat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_1085_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_nat(numeral_numeral_nat,N) = one_one_nat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_1086_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_int(numeral_numeral_int,N) = one_one_int )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_1087_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_1088_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_real = numeral_numeral_real(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_1089_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat = numeral_numeral_rat(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_1090_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_1091_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_1092_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_1093_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_1094_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_1095_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_1096_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_1097_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_1098_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_1099_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_1100_one__add__one,axiom,
    aa_complex_complex(plus_plus_complex(one_one_complex),one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_1101_one__add__one,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_1102_one__add__one,axiom,
    plus_plus_rat(one_one_rat,one_one_rat) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_1103_one__add__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_1104_one__add__one,axiom,
    aa_int_int(plus_plus_int(one_one_int),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_1105_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_1106_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_1107_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_1108_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_1109_bits__one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% bits_one_mod_two_eq_one
tff(fact_1110_bits__one__mod__two__eq__one,axiom,
    aa_int_int(modulo_modulo_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_one_mod_two_eq_one
tff(fact_1111_bits__one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_one_mod_two_eq_one
tff(fact_1112_one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_mod_two_eq_one
tff(fact_1113_one__mod__two__eq__one,axiom,
    aa_int_int(modulo_modulo_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% one_mod_two_eq_one
tff(fact_1114_one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% one_mod_two_eq_one
tff(fact_1115_numeral__plus__one,axiom,
    ! [N: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_1116_numeral__plus__one,axiom,
    ! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(N)),one_one_real) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_1117_numeral__plus__one,axiom,
    ! [N: num] : plus_plus_rat(numeral_numeral_rat(N),one_one_rat) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_1118_numeral__plus__one,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_1119_numeral__plus__one,axiom,
    ! [N: num] : aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,N)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_1120_one__plus__numeral,axiom,
    ! [N: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_1121_one__plus__numeral,axiom,
    ! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),numeral_numeral_real(N)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_1122_one__plus__numeral,axiom,
    ! [N: num] : plus_plus_rat(one_one_rat,numeral_numeral_rat(N)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_1123_one__plus__numeral,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_1124_one__plus__numeral,axiom,
    ! [N: num] : aa_int_int(plus_plus_int(one_one_int),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_1125_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(N)),one_one_real))
    <=> pp(aa_num_bool(ord_less_eq_num(N),one)) ) ).

% numeral_le_one_iff
tff(fact_1126_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(N)),one_one_rat))
    <=> pp(aa_num_bool(ord_less_eq_num(N),one)) ) ).

% numeral_le_one_iff
tff(fact_1127_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(ord_less_eq_num(N),one)) ) ).

% numeral_le_one_iff
tff(fact_1128_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(ord_less_eq_num(N),one)) ) ).

% numeral_le_one_iff
tff(fact_1129_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),numeral_numeral_real(N)))
    <=> pp(aa_num_bool(ord_less_num(one),N)) ) ).

% one_less_numeral_iff
tff(fact_1130_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),numeral_numeral_rat(N)))
    <=> pp(aa_num_bool(ord_less_num(one),N)) ) ).

% one_less_numeral_iff
tff(fact_1131_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(ord_less_num(one),N)) ) ).

% one_less_numeral_iff
tff(fact_1132_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(ord_less_num(one),N)) ) ).

% one_less_numeral_iff
tff(fact_1133_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
    <=> ( X = one_one_complex ) ) ).

% one_reorient
tff(fact_1134_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
    <=> ( X = one_one_real ) ) ).

% one_reorient
tff(fact_1135_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
    <=> ( X = one_one_rat ) ) ).

% one_reorient
tff(fact_1136_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
    <=> ( X = one_one_nat ) ) ).

% one_reorient
tff(fact_1137_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
    <=> ( X = one_one_int ) ) ).

% one_reorient
tff(fact_1138_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_1139_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_1140_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_1141_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_1142_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_1143_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_1144_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_1145_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_1146_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_1147_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_1148_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_1149_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_1150_mult_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(times_times_int(A),one_one_int) = A ).

% mult.comm_neutral
tff(fact_1151_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_1152_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_1153_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_1154_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_1155_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] : aa_int_int(times_times_int(one_one_int),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_1156_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_1157_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_1158_finite__maxlen,axiom,
    ! [M4: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M4)
     => ? [N2: nat] :
        ! [X4: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X4,M4)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(X4)),N2)) ) ) ).

% finite_maxlen
tff(fact_1159_finite__maxlen,axiom,
    ! [M4: set_list_bool] :
      ( finite7866827932528546246t_bool(M4)
     => ? [N2: nat] :
        ! [X4: list_bool] :
          ( member_list_bool(X4,M4)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(X4)),N2)) ) ) ).

% finite_maxlen
tff(fact_1160_finite__maxlen,axiom,
    ! [M4: set_list_nat] :
      ( finite8100373058378681591st_nat(M4)
     => ? [N2: nat] :
        ! [X4: list_nat] :
          ( member_list_nat(X4,M4)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(X4)),N2)) ) ) ).

% finite_maxlen
tff(fact_1161_finite__maxlen,axiom,
    ! [M4: set_list_int] :
      ( finite3922522038869484883st_int(M4)
     => ? [N2: nat] :
        ! [X4: list_int] :
          ( member_list_int(X4,M4)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_int(X4)),N2)) ) ) ).

% finite_maxlen
tff(fact_1162_one__le__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),numeral_numeral_real(N))) ).

% one_le_numeral
tff(fact_1163_one__le__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),numeral_numeral_rat(N))) ).

% one_le_numeral
tff(fact_1164_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_1165_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_1166_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(N)),one_one_real)) ).

% not_numeral_less_one
tff(fact_1167_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(N)),one_one_rat)) ).

% not_numeral_less_one
tff(fact_1168_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_1169_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_1170_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_1171_less__1__mult,axiom,
    ! [M: rat,N: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),M))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(M),N))) ) ) ).

% less_1_mult
tff(fact_1172_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_1173_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(times_times_int(M),N))) ) ) ).

% less_1_mult
tff(fact_1174_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_1175_less__add__one,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),plus_plus_rat(A,one_one_rat))) ).

% less_add_one
tff(fact_1176_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_1177_less__add__one,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(plus_plus_int(A),one_one_int))) ).

% less_add_one
tff(fact_1178_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_1179_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,plus_plus_rat(A,one_one_rat)),plus_plus_rat(B,one_one_rat))) ) ).

% add_mono1
tff(fact_1180_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_1181_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(plus_plus_int(A),one_one_int)),aa_int_int(plus_plus_int(B),one_one_int))) ) ).

% add_mono1
tff(fact_1182_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(X)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(X)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_1183_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),numeral_numeral_real(X)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(X)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_1184_one__plus__numeral__commute,axiom,
    ! [X: num] : plus_plus_rat(one_one_rat,numeral_numeral_rat(X)) = plus_plus_rat(numeral_numeral_rat(X),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_1185_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_1186_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_int_int(plus_plus_int(one_one_int),aa_num_int(numeral_numeral_int,X)) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,X)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_1187_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_1188_numeral__One,axiom,
    numeral_numeral_real(one) = one_one_real ).

% numeral_One
tff(fact_1189_numeral__One,axiom,
    numeral_numeral_rat(one) = one_one_rat ).

% numeral_One
tff(fact_1190_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_1191_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_1192_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_1193_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_1194_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_1195_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_1196_left__right__inverse__power,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( aa_complex_complex(times_times_complex(X),Y) = one_one_complex )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(Y),N)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_1197_left__right__inverse__power,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) = one_one_real )
     => ( 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,Y),N)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_1198_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( aa_rat_rat(times_times_rat(X),Y) = one_one_rat )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),N)),aa_nat_rat(power_power_rat(Y),N)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_1199_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),N)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_1200_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( aa_int_int(times_times_int(X),Y) = one_one_int )
     => ( aa_int_int(times_times_int(aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),N)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_1201_power__one__over,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A)),N) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(A),N)) ).

% power_one_over
tff(fact_1202_power__one__over,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),N) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).

% power_one_over
tff(fact_1203_power__one__over,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(divide_divide_rat(one_one_rat,A)),N) = divide_divide_rat(one_one_rat,aa_nat_rat(power_power_rat(A),N)) ).

% power_one_over
tff(fact_1204_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_1205_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] : vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz),Va) = none_nat ).

% vebt_succ.simps(3)
tff(fact_1206_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_1207_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(plus_plus_int(A),one_one_int)),B)) ) ).

% discrete
tff(fact_1208_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_1209_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,divide_divide_rat(plus_plus_rat(A,B),plus_plus_rat(one_one_rat,one_one_rat))),B)) ) ).

% gt_half_sum
tff(fact_1210_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_1211_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),divide_divide_rat(plus_plus_rat(A,B),plus_plus_rat(one_one_rat,one_one_rat)))) ) ).

% less_half_sum
tff(fact_1212_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_1213_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(times_times_rat(A),aa_nat_rat(power_power_rat(A),N)))) ) ).

% power_gt1_lemma
tff(fact_1214_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_1215_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(times_times_int(A),aa_nat_int(power_power_int(A),N)))) ) ).

% power_gt1_lemma
tff(fact_1216_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_1217_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(times_times_rat(A),aa_nat_rat(power_power_rat(A),N)))) ) ).

% power_less_power_Suc
tff(fact_1218_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_1219_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(times_times_int(A),aa_nat_int(power_power_int(A),N)))) ) ).

% power_less_power_Suc
tff(fact_1220_square__diff__one__factored,axiom,
    ! [X: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),X)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(X),one_one_complex)),aa_complex_complex(minus_minus_complex(X),one_one_complex)) ).

% square_diff_one_factored
tff(fact_1221_square__diff__one__factored,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)) ).

% square_diff_one_factored
tff(fact_1222_square__diff__one__factored,axiom,
    ! [X: rat] : minus_minus_rat(aa_rat_rat(times_times_rat(X),X),one_one_rat) = aa_rat_rat(times_times_rat(plus_plus_rat(X,one_one_rat)),minus_minus_rat(X,one_one_rat)) ).

% square_diff_one_factored
tff(fact_1223_square__diff__one__factored,axiom,
    ! [X: int] : aa_int_int(minus_minus_int(aa_int_int(times_times_int(X),X)),one_one_int) = aa_int_int(times_times_int(aa_int_int(plus_plus_int(X),one_one_int)),aa_int_int(minus_minus_int(X),one_one_int)) ).

% square_diff_one_factored
tff(fact_1224_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_1225_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_1226_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_1227_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_1228_power__strict__increasing,axiom,
    ! [N: nat,N3: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
     => ( 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),N3))) ) ) ).

% power_strict_increasing
tff(fact_1229_power__strict__increasing,axiom,
    ! [N: nat,N3: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
     => ( 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),N3))) ) ) ).

% power_strict_increasing
tff(fact_1230_power__strict__increasing,axiom,
    ! [N: nat,N3: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
     => ( 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),N3))) ) ) ).

% power_strict_increasing
tff(fact_1231_power__strict__increasing,axiom,
    ! [N: nat,N3: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
     => ( 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),N3))) ) ) ).

% power_strict_increasing
tff(fact_1232_power__increasing,axiom,
    ! [N: nat,N3: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
     => ( 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),N3))) ) ) ).

% power_increasing
tff(fact_1233_power__increasing,axiom,
    ! [N: nat,N3: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
     => ( 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),N3))) ) ) ).

% power_increasing
tff(fact_1234_power__increasing,axiom,
    ! [N: nat,N3: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
     => ( 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),N3))) ) ) ).

% power_increasing
tff(fact_1235_power__increasing,axiom,
    ! [N: nat,N3: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
     => ( 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),N3))) ) ) ).

% power_increasing
tff(fact_1236_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_1237_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_1238_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_1239_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_1240_one__power2,axiom,
    aa_nat_rat(power_power_rat(one_one_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat ).

% one_power2
tff(fact_1241_one__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_power2
tff(fact_1242_one__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real ).

% one_power2
tff(fact_1243_one__power2,axiom,
    aa_nat_complex(power_power_complex(one_one_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex ).

% one_power2
tff(fact_1244_one__power2,axiom,
    aa_nat_int(power_power_int(one_one_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int ).

% one_power2
tff(fact_1245_nat__1__add__1,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% nat_1_add_1
tff(fact_1246_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_1247_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_1248__092_060open_062vebt__succ_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_061_ASome_A_I2_A_094_A_Ideg_Adiv_A2_J_A_K_Ahigh_Ax_A_Ideg_Adiv_A2_J_A_L_Asuccy_J_092_060close_062,axiom,
    vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList),summary),xa) = some_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(power_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))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),succy)) ).

% \<open>vebt_succ (Node (Some (mi, ma)) deg treeList summary) x = Some (2 ^ (deg div 2) * high x (deg div 2) + succy)\<close>
tff(fact_1249_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X2),N)) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = N )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_1))
               => ( ! [X2: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
                     => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_1)) )
                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_1250_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,aa_VEB5832953974749077948T_VEBT(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_1251_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),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(aa_VEB4832260192069861672t_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_1252_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),N))
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),Ma))
          & ? [M3: nat] :
              ( ( some_nat(M3) = vEBT_vebt_mint(Summary) )
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),aa_nat_nat(aa_nat_fun_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_1253_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,aa_VEB5832953974749077948T_VEBT(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_1254_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(aa_VEB5832953974749077948T_VEBT(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_1255__C1_C,axiom,
    ( ( ( ( vEBT_vebt_maxt(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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 )
        & pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) )
     => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList),summary),xa) = 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,xa),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,xa),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,xa),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,xa),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 )
          & pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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,xa),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,xa),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 )
         => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList),summary),xa) = none_nat ) )
        & ( ( vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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 )
         => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList),summary),xa) = 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,xa),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_nat(vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) ) ) ) ) ).

% "1"
tff(fact_1256_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(aa_VEB5832953974749077948T_VEBT(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_1257_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))) ) ) ).

% mi_ma_2_deg
tff(fact_1258_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(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary) ) ) ) ).

% insert_simp_mima
tff(fact_1259_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg))
     => ( ( Mi = Ma )
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
             => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_13)) )
          & ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_13)) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_1260_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg))
     => ( ( Mi != Ma )
       => ( the_nat(vEBT_vebt_maxt(Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% summaxma
tff(fact_1261_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( some_nat(the_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_1262_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_1263_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( some_num(the_num(Option)) = Option ) ) ).

% option.collapse
tff(fact_1264_option_Osel,axiom,
    ! [X22: nat] : the_nat(some_nat(X22)) = X22 ).

% option.sel
tff(fact_1265_option_Osel,axiom,
    ! [X22: product_prod_nat_nat] : the_Pr8591224930841456533at_nat(some_P7363390416028606310at_nat(X22)) = X22 ).

% option.sel
tff(fact_1266_option_Osel,axiom,
    ! [X22: num] : the_num(some_num(X22)) = X22 ).

% option.sel
tff(fact_1267_option_Oexpand,axiom,
    ! [Option: option_nat,Option2: option_nat] :
      ( ( ( Option = none_nat )
      <=> ( Option2 = none_nat ) )
     => ( ( ( Option != none_nat )
         => ( ( Option2 != none_nat )
           => ( the_nat(Option) = the_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_1268_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_1269_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_1270_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_1271_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option = some_nat(the_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_1272_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_1273_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option = some_num(the_num(Option)) ) ) ).

% option.exhaust_sel
tff(fact_1274_two__realpow__ge__one,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N))) ).

% two_realpow_ge_one
tff(fact_1275_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X2),N)) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = N )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ! [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_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),X_12))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I3)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X2: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
                       => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),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 )
                         => ! [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),N) = 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),N))) )
                                & ! [X2: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N) = 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),N))) )
                                   => ( 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)) ) ) ) ) )
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_1276_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),N))
     => ( ( N = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),Mi))
         => ( ( Ma != Mi )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ).

% nested_mint
tff(fact_1277_divmod__step__eq,axiom,
    ! [L: num,R: nat,Q2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R))
       => ( unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R)) = 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))),Q2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R),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)),R))
       => ( unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R)) = 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))),Q2)),R) ) ) ) ).

% divmod_step_eq
tff(fact_1278_divmod__step__eq,axiom,
    ! [L: num,R: int,Q2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R))
       => ( unique5024387138958732305ep_int(L,product_Pair_int_int(Q2,R)) = product_Pair_int_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q2)),one_one_int),aa_int_int(minus_minus_int(R),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)),R))
       => ( unique5024387138958732305ep_int(L,product_Pair_int_int(Q2,R)) = product_Pair_int_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q2),R) ) ) ) ).

% divmod_step_eq
tff(fact_1279_divmod__step__eq,axiom,
    ! [L: num,R: code_integer,Q2: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,L)),R))
       => ( unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q2,R)) = produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Q2),one_one_Code_integer),minus_8373710615458151222nteger(R,aa_num_Code_integer(numera6620942414471956472nteger,L))) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,L)),R))
       => ( unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q2,R)) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Q2),R) ) ) ) ).

% divmod_step_eq
tff(fact_1280_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_mint(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)) = some_nat(Mi) ).

% vebt_mint.simps(3)
tff(fact_1281_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_maxt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)) = some_nat(Ma) ).

% vebt_maxt.simps(3)
tff(fact_1282_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X2),N)) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,N) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ! [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_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),X_12))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I3)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X2: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
                       => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),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 )
                         => ! [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),N) = 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),N))) )
                                & ! [X2: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N) = 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),N))) )
                                   => ( 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)) ) ) ) ) )
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_1283_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_mint(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = none_nat ).

% vebt_mint.simps(2)
tff(fact_1284_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_maxt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = none_nat ).

% vebt_maxt.simps(2)
tff(fact_1285_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(aa_VEB5832953974749077948T_VEBT(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_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))))))))))) ) ) ) ) ).

% succ_less_length_list
tff(fact_1286_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(aa_VEB5832953974749077948T_VEBT(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_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) ) ) ) ).

% succ_greatereq_min
tff(fact_1287_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_mint(T) = none_nat )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
tff(fact_1288_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_1289_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_set_vebt(T) = collect_nat(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T)) ).

% set_vebt'_def
tff(fact_1290_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Tree),aa_nat_nat(suc,aa_nat_nat(suc,N))))
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] : Tree = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,N)),TreeList2),S2) ) ).

% deg_SUcn_Node
tff(fact_1291_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( aa_nat_nat(suc,X22) = aa_nat_nat(suc,Y22) )
    <=> ( X22 = Y22 ) ) ).

% nat.inject
tff(fact_1292_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_1293_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_maxt(T) = none_nat )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
tff(fact_1294_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_1295_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_1296_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_1297_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_1298_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_1299_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_1300_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_1301_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_1302_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_1303_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_1304_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_1305_Suc__numeral,axiom,
    ! [N: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% Suc_numeral
tff(fact_1306_Suc__mod__mult__self4,axiom,
    ! [N: nat,K: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)),M))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self4
tff(fact_1307_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)),M))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self3
tff(fact_1308_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self2
tff(fact_1309_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self1
tff(fact_1310_add__2__eq__Suc,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc
tff(fact_1311_add__2__eq__Suc_H,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc'
tff(fact_1312_div2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% div2_Suc_Suc
tff(fact_1313_Suc__1,axiom,
    aa_nat_nat(suc,one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% Suc_1
tff(fact_1314_mod2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% mod2_Suc_Suc
tff(fact_1315_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) != one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),N))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_1316_mult__commute__abs,axiom,
    ! [C: real] : aa_rea7340605660300375871l_real(aTP_Lamm_ag,C) = aa_rea7340605660300375871l_real(times_times_real,C) ).

% mult_commute_abs
tff(fact_1317_mult__commute__abs,axiom,
    ! [C: rat] : aTP_Lamm_ah(C) = times_times_rat(C) ).

% mult_commute_abs
tff(fact_1318_mult__commute__abs,axiom,
    ! [C: nat] : aa_nat_fun_nat_nat(aTP_Lamm_ai,C) = aa_nat_fun_nat_nat(times_times_nat,C) ).

% mult_commute_abs
tff(fact_1319_mult__commute__abs,axiom,
    ! [C: int] : aTP_Lamm_aj(C) = times_times_int(C) ).

% mult_commute_abs
tff(fact_1320_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(suc,X) = aa_nat_nat(suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_1321_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa_nat_nat(suc,N) ).

% n_not_Suc_n
tff(fact_1322_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,A4: nat,B4: nat] : X != produc6699827698519632094on_nat(F2,produc5098337634421038937on_nat(some_nat(A4),some_nat(B4))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1323_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,A4: product_prod_nat_nat,B4: product_prod_nat_nat] : X != produc7814092408141220007at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A4),some_P7363390416028606310at_nat(B4))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1324_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,A4: num,B4: num] : X != produc6360833703471310868on_num(F2,produc8585076106096196333on_num(some_num(A4),some_num(B4))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1325_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,X2: nat,Y2: nat] : X != produc2004789812759622991on_nat(F2,produc5098337634421038937on_nat(some_nat(X2),some_nat(Y2))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1326_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,X2: product_prod_nat_nat,Y2: product_prod_nat_nat] : X != produc5961576194287291153at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(X2),some_P7363390416028606310at_nat(Y2))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1327_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,X2: num,Y2: num] : X != produc8433281988922931063on_num(F2,produc8585076106096196333on_num(some_num(X2),some_num(Y2))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1328_lambda__one,axiom,
    aTP_Lamm_ak = times_times_complex(one_one_complex) ).

% lambda_one
tff(fact_1329_lambda__one,axiom,
    aTP_Lamm_al = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).

% lambda_one
tff(fact_1330_lambda__one,axiom,
    aTP_Lamm_am = times_times_rat(one_one_rat) ).

% lambda_one
tff(fact_1331_lambda__one,axiom,
    aTP_Lamm_an = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_1332_lambda__one,axiom,
    aTP_Lamm_ao = times_times_int(one_one_int) ).

% lambda_one
tff(fact_1333_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_1334_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_1335_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_1336_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_1337_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_1338_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_1339_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_1340_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_1341_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_1342_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_1343_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))
    <=> ? [M5: nat] :
          ( ( M = aa_nat_nat(suc,M5) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M5)) ) ) ).

% Suc_less_eq2
tff(fact_1344_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_1345_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_1346_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_1347_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))
     => ( ! [I3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),aa_nat_nat(suc,I3)))
       => ( ! [I3: nat,J2: nat,K3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),K3))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),J2))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,J2),K3))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),K3)) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I),J)) ) ) ) ).

% less_Suc_induct
tff(fact_1348_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))
     => ( ! [I3: nat] :
            ( ( J = aa_nat_nat(suc,I3) )
           => pp(aa_nat_bool(P,I3)) )
       => ( ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J))
             => ( pp(aa_nat_bool(P,aa_nat_nat(suc,I3)))
               => pp(aa_nat_bool(P,I3)) ) )
         => pp(aa_nat_bool(P,I)) ) ) ) ).

% strict_inc_induct
tff(fact_1349_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_1350_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_1351_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_1352_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_1353_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M6))
     => ? [M3: nat] : M6 = aa_nat_nat(suc,M3) ) ).

% Suc_le_D
tff(fact_1354_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_1355_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_1356_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_1357_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_1358_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_1359_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R2: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( ! [X2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X2),X2))
       => ( ! [X2: nat,Y2: nat,Z4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X2),Y2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,Y2),Z4))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X2),Z4)) ) )
         => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,N2),aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,M),N)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_1360_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_1361_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_1362_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_1363_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_1364_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_1365_numeral__code_I2_J,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit0,N)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N)) ).

% numeral_code(2)
tff(fact_1366_numeral__code_I2_J,axiom,
    ! [N: num] : numeral_numeral_real(aa_num_num(bit0,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(N)),numeral_numeral_real(N)) ).

% numeral_code(2)
tff(fact_1367_numeral__code_I2_J,axiom,
    ! [N: num] : numeral_numeral_rat(aa_num_num(bit0,N)) = plus_plus_rat(numeral_numeral_rat(N),numeral_numeral_rat(N)) ).

% numeral_code(2)
tff(fact_1368_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_code(2)
tff(fact_1369_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N)) ).

% numeral_code(2)
tff(fact_1370_mod__Suc__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% mod_Suc_eq
tff(fact_1371_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),N) ).

% mod_Suc_Suc_eq
tff(fact_1372_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_set_vebt(T) = collect_nat(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T)) ).

% set_vebt_def
tff(fact_1373_power__numeral__even,axiom,
    ! [Z: complex,W: num] : aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_even
tff(fact_1374_power__numeral__even,axiom,
    ! [Z: real,W: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_even
tff(fact_1375_power__numeral__even,axiom,
    ! [Z: rat,W: num] : aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_even
tff(fact_1376_power__numeral__even,axiom,
    ! [Z: nat,W: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_even
tff(fact_1377_power__numeral__even,axiom,
    ! [Z: int,W: num] : aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = aa_int_int(times_times_int(aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_even
tff(fact_1378_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,N: nat,N4: 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),N4))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,N4))) ) ) ).

% lift_Suc_mono_less
tff(fact_1379_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,N: nat,N4: 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),N4))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N4))) ) ) ).

% lift_Suc_mono_less
tff(fact_1380_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,N: nat,N4: nat] :
      ( ! [N2: nat] : pp(aa_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),N4))
       => pp(aa_num_bool(ord_less_num(aa_nat_num(F,N)),aa_nat_num(F,N4))) ) ) ).

% lift_Suc_mono_less
tff(fact_1381_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,N: nat,N4: 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),N4))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N4))) ) ) ).

% lift_Suc_mono_less
tff(fact_1382_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,N: nat,N4: 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),N4))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,N4))) ) ) ).

% lift_Suc_mono_less
tff(fact_1383_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_1384_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_1385_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(ord_less_num(aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_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_1386_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_1387_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_1388_power__Suc2,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),N)),A) ).

% power_Suc2
tff(fact_1389_power__Suc2,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),A) ).

% power_Suc2
tff(fact_1390_power__Suc2,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),N)),A) ).

% power_Suc2
tff(fact_1391_power__Suc2,axiom,
    ! [A: nat,N: 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(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),A) ).

% power_Suc2
tff(fact_1392_power__Suc2,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_int_int(times_times_int(aa_nat_int(power_power_int(A),N)),A) ).

% power_Suc2
tff(fact_1393_power__Suc,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),N)) ).

% power_Suc
tff(fact_1394_power__Suc,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,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_Suc
tff(fact_1395_power__Suc,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N)) ).

% power_Suc
tff(fact_1396_power__Suc,axiom,
    ! [A: nat,N: 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(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_Suc
tff(fact_1397_power__Suc,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_int_int(times_times_int(A),aa_nat_int(power_power_int(A),N)) ).

% power_Suc
tff(fact_1398_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_nat,N: nat,N4: nat] :
      ( ! [N2: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,N2)),aa_nat_set_nat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,N)),aa_nat_set_nat(F,N4))) ) ) ).

% lift_Suc_mono_le
tff(fact_1399_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,N: nat,N4: 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),N4))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N4))) ) ) ).

% lift_Suc_mono_le
tff(fact_1400_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,N: nat,N4: nat] :
      ( ! [N2: nat] : pp(aa_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),N4))
       => pp(aa_num_bool(ord_less_eq_num(aa_nat_num(F,N)),aa_nat_num(F,N4))) ) ) ).

% lift_Suc_mono_le
tff(fact_1401_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N4: 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),N4))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N4))) ) ) ).

% lift_Suc_mono_le
tff(fact_1402_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,N: nat,N4: 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),N4))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,N4))) ) ) ).

% lift_Suc_mono_le
tff(fact_1403_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_nat,N: nat,N4: nat] :
      ( ! [N2: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,aa_nat_nat(suc,N2))),aa_nat_set_nat(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,N4)),aa_nat_set_nat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1404_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,N: nat,N4: 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),N4))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N4)),aa_nat_rat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1405_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,N: nat,N4: nat] :
      ( ! [N2: nat] : pp(aa_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),N4))
       => pp(aa_num_bool(ord_less_eq_num(aa_nat_num(F,N4)),aa_nat_num(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1406_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N4: 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),N4))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N4)),aa_nat_nat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1407_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,N: nat,N4: 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),N4))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N4)),aa_nat_int(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1408_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_1409_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_1410_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_1411_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_1412_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_1413_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_1414_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_1415_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_1416_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_1417_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_1418_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_1419_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_1420_less__iff__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_iff_Suc_add
tff(fact_1421_less__imp__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_imp_Suc_add
tff(fact_1422_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_1423_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_1424_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_1425_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_1426_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_1427_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_1428_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_1429_infinite__growing,axiom,
    ! [X5: set_real] :
      ( ( X5 != bot_bot_set_real )
     => ( ! [X2: real] :
            ( member_real(X2,X5)
           => ? [Xa: real] :
                ( member_real(Xa,X5)
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Xa)) ) )
       => ~ finite_finite_real(X5) ) ) ).

% infinite_growing
tff(fact_1430_infinite__growing,axiom,
    ! [X5: set_rat] :
      ( ( X5 != bot_bot_set_rat )
     => ( ! [X2: rat] :
            ( member_rat(X2,X5)
           => ? [Xa: rat] :
                ( member_rat(Xa,X5)
                & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Xa)) ) )
       => ~ finite_finite_rat(X5) ) ) ).

% infinite_growing
tff(fact_1431_infinite__growing,axiom,
    ! [X5: set_num] :
      ( ( X5 != bot_bot_set_num )
     => ( ! [X2: num] :
            ( member_num(X2,X5)
           => ? [Xa: num] :
                ( member_num(Xa,X5)
                & pp(aa_num_bool(ord_less_num(X2),Xa)) ) )
       => ~ finite_finite_num(X5) ) ) ).

% infinite_growing
tff(fact_1432_infinite__growing,axiom,
    ! [X5: set_nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),X5))
           => ? [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),X5))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Xa)) ) )
       => ~ pp(aa_set_nat_bool(finite_finite_nat,X5)) ) ) ).

% infinite_growing
tff(fact_1433_infinite__growing,axiom,
    ! [X5: set_int] :
      ( ( X5 != bot_bot_set_int )
     => ( ! [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),X5))
           => ? [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),X5))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Xa)) ) )
       => ~ pp(aa_set_int_bool(finite_finite_int,X5)) ) ) ).

% infinite_growing
tff(fact_1434_ex__min__if__finite,axiom,
    ! [S3: set_real] :
      ( finite_finite_real(S3)
     => ( ( S3 != bot_bot_set_real )
       => ? [X2: real] :
            ( member_real(X2,S3)
            & ~ ? [Xa: real] :
                  ( member_real(Xa,S3)
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Xa),X2)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1435_ex__min__if__finite,axiom,
    ! [S3: set_rat] :
      ( finite_finite_rat(S3)
     => ( ( S3 != bot_bot_set_rat )
       => ? [X2: rat] :
            ( member_rat(X2,S3)
            & ~ ? [Xa: rat] :
                  ( member_rat(Xa,S3)
                  & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Xa),X2)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1436_ex__min__if__finite,axiom,
    ! [S3: set_num] :
      ( finite_finite_num(S3)
     => ( ( S3 != bot_bot_set_num )
       => ? [X2: num] :
            ( member_num(X2,S3)
            & ~ ? [Xa: num] :
                  ( member_num(Xa,S3)
                  & pp(aa_num_bool(ord_less_num(Xa),X2)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1437_ex__min__if__finite,axiom,
    ! [S3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( S3 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),S3))
            & ~ ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),S3))
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),X2)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1438_ex__min__if__finite,axiom,
    ! [S3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( S3 != bot_bot_set_int )
       => ? [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),S3))
            & ~ ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),S3))
                  & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Xa),X2)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1439_set__conv__nth,axiom,
    ! [Xs: list_real] : set_real2(Xs) = collect_real(aTP_Lamm_ap(Xs)) ).

% set_conv_nth
tff(fact_1440_set__conv__nth,axiom,
    ! [Xs: list_list_nat] : set_list_nat2(Xs) = collect_list_nat(aTP_Lamm_aq(Xs)) ).

% set_conv_nth
tff(fact_1441_set__conv__nth,axiom,
    ! [Xs: list_set_nat] : set_set_nat2(Xs) = collect_set_nat(aTP_Lamm_ar(Xs)) ).

% set_conv_nth
tff(fact_1442_set__conv__nth,axiom,
    ! [Xs: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs) = collect_VEBT_VEBT(aTP_Lamm_as(Xs)) ).

% set_conv_nth
tff(fact_1443_set__conv__nth,axiom,
    ! [Xs: list_bool] : set_bool2(Xs) = collect_bool(aTP_Lamm_at(Xs)) ).

% set_conv_nth
tff(fact_1444_set__conv__nth,axiom,
    ! [Xs: list_nat] : set_nat2(Xs) = collect_nat(aTP_Lamm_au(Xs)) ).

% set_conv_nth
tff(fact_1445_set__conv__nth,axiom,
    ! [Xs: list_int] : set_int2(Xs) = collect_int(aTP_Lamm_av(Xs)) ).

% set_conv_nth
tff(fact_1446_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_1447_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_1448_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_1449_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_1450_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_1451_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_1452_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_aw(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_1453_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_ax(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_1454_finite__lists__length__eq,axiom,
    ! [A2: set_bool,N: nat] :
      ( finite_finite_bool(A2)
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_ay(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_1455_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_az(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_1456_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_ba(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_1457_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_bb(A2),N))) ) ).

% finite_lists_length_le
tff(fact_1458_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_bc(A2),N))) ) ).

% finite_lists_length_le
tff(fact_1459_finite__lists__length__le,axiom,
    ! [A2: set_bool,N: nat] :
      ( finite_finite_bool(A2)
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_bd(A2),N))) ) ).

% finite_lists_length_le
tff(fact_1460_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_be(A2),N))) ) ).

% finite_lists_length_le
tff(fact_1461_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_bf(A2),N))) ) ).

% finite_lists_length_le
tff(fact_1462_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_1463_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_1464_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_1465_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_1466_div__nat__eqI,axiom,
    ! [N: nat,Q2: 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),Q2)),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,Q2))))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_1467_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),one))),N) ).

% Suc_nat_number_of_add
tff(fact_1468_power__odd__eq,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(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))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_1469_power__odd__eq,axiom,
    ! [A: real,N: nat] : 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))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_1470_power__odd__eq,axiom,
    ! [A: rat,N: nat] : 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))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_1471_power__odd__eq,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,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))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_1472_power__odd__eq,axiom,
    ! [A: int,N: nat] : 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))) = aa_int_int(times_times_int(A),aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_1473_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(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary),X) = some_nat(Mi) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
       => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary),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_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) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_1474_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_1475_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_1476_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_1477_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_1478_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_1479_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_1480_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X2),N)) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,N) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_1))
               => ( ! [X2: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
                     => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_1)) )
                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_1481_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_1482_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_1483_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_1484_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 ) ) )
         => ~ ! [A4: product_prod_nat_nat] :
                ( ( Xa2 = some_P7363390416028606310at_nat(A4) )
               => ! [B4: product_prod_nat_nat] :
                    ( ( Xb2 = some_P7363390416028606310at_nat(B4) )
                   => ( Y != some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(X,A4),B4)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1485_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 ) ) )
         => ~ ! [A4: num] :
                ( ( Xa2 = some_num(A4) )
               => ! [B4: num] :
                    ( ( Xb2 = some_num(B4) )
                   => ( Y != some_num(aa_num_num(aa_num_fun_num_num(X,A4),B4)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1486_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 ) ) )
         => ~ ! [A4: nat] :
                ( ( Xa2 = some_nat(A4) )
               => ! [B4: nat] :
                    ( ( Xb2 = some_nat(B4) )
                   => ( Y != some_nat(aa_nat_nat(aa_nat_fun_nat_nat(X,A4),B4)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1487_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(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)),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(aa_VEB4832260192069861672t_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_1488_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),aa_int_int(plus_plus_int(Z),one_one_int)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).

% zle_add1_eq_le
tff(fact_1489_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(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_1490_finite__Collect__le__nat,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bg,K)))) ).

% finite_Collect_le_nat
tff(fact_1491_finite__Collect__less__nat,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bh,K)))) ).

% finite_Collect_less_nat
tff(fact_1492_buildup__gives__empty,axiom,
    ! [N: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(N)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_1493_finite__Collect__subsets,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => finite6197958912794628473et_int(collect_set_int(aTP_Lamm_bi(A2))) ) ).

% finite_Collect_subsets
tff(fact_1494_finite__Collect__subsets,axiom,
    ! [A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => finite6551019134538273531omplex(collect_set_complex(aTP_Lamm_bj(A2))) ) ).

% finite_Collect_subsets
tff(fact_1495_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => finite1152437895449049373et_nat(collect_set_nat(aTP_Lamm_bk(A2))) ) ).

% finite_Collect_subsets
tff(fact_1496_finite__Collect__bounded__ex,axiom,
    ! [P: fun_real_bool,Q: fun_re1255401033344146673l_bool] :
      ( finite_finite_real(collect_real(P))
     => ( finite_finite_real(collect_real(aa_fun4964148113266127823l_bool(aTP_Lamm_bl(P),Q)))
      <=> ! [Y4: real] :
            ( pp(aa_real_bool(P,Y4))
           => finite_finite_real(collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_bm(Q),Y4))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_1497_finite__Collect__bounded__ex,axiom,
    ! [P: fun_real_bool,Q: fun_na6801867027684862029l_bool] :
      ( finite_finite_real(collect_real(P))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun7391232120346184911t_bool(aTP_Lamm_bn(P),Q))))
      <=> ! [Y4: real] :
            ( pp(aa_real_bool(P,Y4))
           => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_real_fun_nat_bool(aTP_Lamm_bo(Q),Y4)))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_1498_finite__Collect__bounded__ex,axiom,
    ! [P: fun_real_bool,Q: fun_in3970322703178350321l_bool] :
      ( finite_finite_real(collect_real(P))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun69789402421856207t_bool(aTP_Lamm_bp(P),Q))))
      <=> ! [Y4: real] :
            ( pp(aa_real_bool(P,Y4))
           => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_real_fun_int_bool(aTP_Lamm_bq(Q),Y4)))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_1499_finite__Collect__bounded__ex,axiom,
    ! [P: fun_real_bool,Q: fun_co2286642472385812463l_bool] :
      ( finite_finite_real(collect_real(P))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun6039685132559610447x_bool(aTP_Lamm_br(P),Q))))
      <=> ! [Y4: real] :
            ( pp(aa_real_bool(P,Y4))
           => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_rea5887071448072802392x_bool(aTP_Lamm_bs(Q),Y4)))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_1500_finite__Collect__bounded__ex,axiom,
    ! [P: fun_nat_bool,Q: fun_re9121486324101951821t_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => ( finite_finite_real(collect_real(aa_fun4927090880566115059l_bool(aTP_Lamm_bt(P),Q)))
      <=> ! [Y4: nat] :
            ( pp(aa_nat_bool(P,Y4))
           => finite_finite_real(collect_real(aa_nat_fun_real_bool(aTP_Lamm_bu(Q),Y4))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_1501_finite__Collect__bounded__ex,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8399658544460314611t_bool(aTP_Lamm_bv(P),Q))))
      <=> ! [Y4: nat] :
            ( pp(aa_nat_bool(P,Y4))
           => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bw(Q),Y4)))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_1502_finite__Collect__bounded__ex,axiom,
    ! [P: fun_nat_bool,Q: fun_int_fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun6484551404966387955t_bool(aTP_Lamm_bx(P),Q))))
      <=> ! [Y4: nat] :
            ( pp(aa_nat_bool(P,Y4))
           => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_nat_fun_int_bool(aTP_Lamm_by(Q),Y4)))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_1503_finite__Collect__bounded__ex,axiom,
    ! [P: fun_nat_bool,Q: fun_co2841405903858895691t_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun6682930621474858099x_bool(aTP_Lamm_bz(P),Q))))
      <=> ! [Y4: nat] :
            ( pp(aa_nat_bool(P,Y4))
           => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_ca(Q),Y4)))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_1504_finite__Collect__bounded__ex,axiom,
    ! [P: fun_int_bool,Q: fun_re8661203372455305201t_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
     => ( finite_finite_real(collect_real(aa_fun7352647192720469839l_bool(aTP_Lamm_cb(P),Q)))
      <=> ! [Y4: int] :
            ( pp(aa_int_bool(P,Y4))
           => finite_finite_real(collect_real(aa_int_fun_real_bool(aTP_Lamm_cc(Q),Y4))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_1505_finite__Collect__bounded__ex,axiom,
    ! [P: fun_int_bool,Q: fun_nat_fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun7626329022887090767t_bool(aTP_Lamm_cd(P),Q))))
      <=> ! [Y4: int] :
            ( pp(aa_int_bool(P,Y4))
           => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_int_fun_nat_bool(aTP_Lamm_ce(Q),Y4)))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_1506_finite__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => finite_finite_real(collect_real(aa_nat_fun_real_bool(aTP_Lamm_cf,N))) ) ).

% finite_roots_unity
tff(fact_1507_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_cg,N)))) ) ).

% finite_roots_unity
tff(fact_1508_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),aa_int_int(minus_minus_int(Z),one_one_int)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% zle_diff1_eq
tff(fact_1509_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_1510_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_1511_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_1512_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_1513_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_1514_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_1515_finite__Collect__disjI,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( finite_finite_real(collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_ch(P),Q)))
    <=> ( finite_finite_real(collect_real(P))
        & finite_finite_real(collect_real(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_1516_finite__Collect__disjI,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( finite8100373058378681591st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_ci(P),Q)))
    <=> ( finite8100373058378681591st_nat(collect_list_nat(P))
        & finite8100373058378681591st_nat(collect_list_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_1517_finite__Collect__disjI,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( finite1152437895449049373et_nat(collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_cj(P),Q)))
    <=> ( finite1152437895449049373et_nat(collect_set_nat(P))
        & finite1152437895449049373et_nat(collect_set_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_1518_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_ck(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_1519_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_cl(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_1520_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_cm(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_1521_finite__Collect__conjI,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( ( finite_finite_real(collect_real(P))
        | finite_finite_real(collect_real(Q)) )
     => finite_finite_real(collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_cn(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_1522_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_co(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_1523_finite__Collect__conjI,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ( finite1152437895449049373et_nat(collect_set_nat(P))
        | finite1152437895449049373et_nat(collect_set_nat(Q)) )
     => finite1152437895449049373et_nat(collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_cp(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_1524_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_cq(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_1525_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_cr(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_1526_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_cs(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_1527_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_ct(A),B)))) ).

% finite_interval_int4
tff(fact_1528_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_cu(A),B)))) ).

% finite_interval_int3
tff(fact_1529_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_cv(A),B)))) ).

% finite_interval_int2
tff(fact_1530_finite__psubset__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [A5: set_nat] :
            ( pp(aa_set_nat_bool(finite_finite_nat,A5))
           => ( ! [B5: set_nat] :
                  ( pp(aa_set_nat_bool(ord_less_set_nat(B5),A5))
                 => pp(aa_set_nat_bool(P,B5)) )
             => pp(aa_set_nat_bool(P,A5)) ) )
       => pp(aa_set_nat_bool(P,A2)) ) ) ).

% finite_psubset_induct
tff(fact_1531_finite__psubset__induct,axiom,
    ! [A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [A5: set_int] :
            ( pp(aa_set_int_bool(finite_finite_int,A5))
           => ( ! [B5: set_int] :
                  ( pp(aa_set_int_bool(ord_less_set_int(B5),A5))
                 => pp(aa_set_int_bool(P,B5)) )
             => pp(aa_set_int_bool(P,A5)) ) )
       => pp(aa_set_int_bool(P,A2)) ) ) ).

% finite_psubset_induct
tff(fact_1532_finite__psubset__induct,axiom,
    ! [A2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [A5: set_complex] :
            ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A5))
           => ( ! [B5: set_complex] :
                  ( pp(aa_set_complex_bool(ord_less_set_complex(B5),A5))
                 => pp(aa_set_complex_bool(P,B5)) )
             => pp(aa_set_complex_bool(P,A5)) ) )
       => pp(aa_set_complex_bool(P,A2)) ) ) ).

% finite_psubset_induct
tff(fact_1533_Diff__infinite__finite,axiom,
    ! [T2: set_int,S3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( ~ pp(aa_set_int_bool(finite_finite_int,S3))
       => ~ pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(S3,T2))) ) ) ).

% Diff_infinite_finite
tff(fact_1534_Diff__infinite__finite,axiom,
    ! [T2: set_complex,S3: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(S3,T2))) ) ) ).

% Diff_infinite_finite
tff(fact_1535_Diff__infinite__finite,axiom,
    ! [T2: set_nat,S3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( ~ pp(aa_set_nat_bool(finite_finite_nat,S3))
       => ~ pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(S3,T2))) ) ) ).

% Diff_infinite_finite
tff(fact_1536_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(minus_minus_int(K),one_one_int)))
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I3),K))
             => ( pp(aa_int_bool(P,I3))
               => pp(aa_int_bool(P,aa_int_int(minus_minus_int(I3),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_less_induct
tff(fact_1537_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(times_times_int(W),aa_int_int(minus_minus_int(Z1),Z22)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(W),Z1)),aa_int_int(times_times_int(W),Z22)) ).

% int_distrib(4)
tff(fact_1538_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(times_times_int(aa_int_int(minus_minus_int(Z1),Z22)),W) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(Z1),W)),aa_int_int(times_times_int(Z22),W)) ).

% int_distrib(3)
tff(fact_1539_pigeonhole__infinite__rel,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_nat,R2: fun_VE3763956218935105199t_bool] :
      ( ~ finite5795047828879050333T_VEBT(A2)
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ! [X2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B2))
                  & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(R2,X2),Xa)) ) )
         => ? [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),B2))
              & ~ finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_nat8029987864780243636T_bool(aa_fun4018725027000996448T_bool(aTP_Lamm_cw(A2),R2),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_1540_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B2: set_nat,R2: fun_re9121486324101951821t_bool] :
      ( ~ finite_finite_real(A2)
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ! [X2: real] :
              ( member_real(X2,A2)
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B2))
                  & pp(aa_nat_bool(aa_real_fun_nat_bool(R2,X2),Xa)) ) )
         => ? [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),B2))
              & ~ finite_finite_real(collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_cx(A2),R2),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_1541_pigeonhole__infinite__rel,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_int,R2: fun_VE3303673267288458579t_bool] :
      ( ~ finite5795047828879050333T_VEBT(A2)
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( ! [X2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
             => ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),B2))
                  & pp(aa_int_bool(aa_VEB1707969225601237452t_bool(R2,X2),Xa)) ) )
         => ? [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),B2))
              & ~ finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_int3199078826956362512T_bool(aa_fun4417922564241321824T_bool(aTP_Lamm_cy(A2),R2),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_1542_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B2: set_int,R2: fun_re8661203372455305201t_bool] :
      ( ~ finite_finite_real(A2)
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( ! [X2: real] :
              ( member_real(X2,A2)
             => ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),B2))
                  & pp(aa_int_bool(aa_real_fun_int_bool(R2,X2),Xa)) ) )
         => ? [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),B2))
              & ~ finite_finite_real(collect_real(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_cz(A2),R2),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_1543_pigeonhole__infinite__rel,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_complex,R2: fun_VE2128610149277289041x_bool] :
      ( ~ finite5795047828879050333T_VEBT(A2)
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( ! [X2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
             => ? [Xa: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa),B2))
                  & pp(aa_complex_bool(aa_VEB8450443329232412106x_bool(R2,X2),Xa)) ) )
         => ? [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),B2))
              & ~ finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_com4769948546659418770T_bool(aa_fun3341029659329701088T_bool(aTP_Lamm_da(A2),R2),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_1544_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B2: set_complex,R2: fun_re3010880487650781423x_bool] :
      ( ~ finite_finite_real(A2)
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( ! [X2: real] :
              ( member_real(X2,A2)
             => ? [Xa: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa),B2))
                  & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(R2,X2),Xa)) ) )
         => ? [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),B2))
              & ~ finite_finite_real(collect_real(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_db(A2),R2),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_1545_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_nat,R2: fun_nat_fun_nat_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ! [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),A2))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B2))
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X2),Xa)) ) )
         => ? [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),B2))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_dc(A2),R2),X2)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_1546_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_int,R2: fun_nat_fun_int_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( ! [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),A2))
             => ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),B2))
                  & pp(aa_int_bool(aa_nat_fun_int_bool(R2,X2),Xa)) ) )
         => ? [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),B2))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_int_fun_nat_bool(aa_fun8184653438340336124t_bool(aTP_Lamm_dd(A2),R2),X2)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_1547_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_complex,R2: fun_na5941084329111710283x_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( ! [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),A2))
             => ? [Xa: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa),B2))
                  & pp(aa_complex_bool(aa_nat5545486172511849908x_bool(R2,X2),Xa)) ) )
         => ? [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),B2))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_de(A2),R2),X2)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_1548_pigeonhole__infinite__rel,axiom,
    ! [A2: set_int,B2: set_nat,R2: fun_int_fun_nat_bool] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),A2))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B2))
                  & pp(aa_nat_bool(aa_int_fun_nat_bool(R2,X2),Xa)) ) )
         => ? [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),B2))
              & ~ pp(aa_set_int_bool(finite_finite_int,collect_int(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_df(A2),R2),X2)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_1549_not__finite__existsD,axiom,
    ! [P: fun_real_bool] :
      ( ~ finite_finite_real(collect_real(P))
     => ? [X_1: real] : pp(aa_real_bool(P,X_1)) ) ).

% not_finite_existsD
tff(fact_1550_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_1551_not__finite__existsD,axiom,
    ! [P: fun_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_1552_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_1553_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_1554_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_1555_finite__has__maximal2,axiom,
    ! [A2: set_real,A: real] :
      ( finite_finite_real(A2)
     => ( member_real(A,A2)
       => ? [X2: real] :
            ( member_real(X2,A2)
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2))
            & ! [Xa: real] :
                ( member_real(Xa,A2)
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Xa))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1556_finite__has__maximal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( member_set_nat(A,A2)
       => ? [X2: set_nat] :
            ( member_set_nat(X2,A2)
            & pp(aa_set_nat_bool(ord_less_eq_set_nat(A),X2))
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A2)
               => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X2),Xa))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1557_finite__has__maximal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( finite_finite_rat(A2)
     => ( member_rat(A,A2)
       => ? [X2: rat] :
            ( member_rat(X2,A2)
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),X2))
            & ! [Xa: rat] :
                ( member_rat(Xa,A2)
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Xa))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1558_finite__has__maximal2,axiom,
    ! [A2: set_num,A: num] :
      ( finite_finite_num(A2)
     => ( member_num(A,A2)
       => ? [X2: num] :
            ( member_num(X2,A2)
            & pp(aa_num_bool(ord_less_eq_num(A),X2))
            & ! [Xa: num] :
                ( member_num(Xa,A2)
               => ( pp(aa_num_bool(ord_less_eq_num(X2),Xa))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1559_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))
       => ? [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),A2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X2))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Xa))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1560_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))
       => ? [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X2))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Xa))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1561_finite__has__minimal2,axiom,
    ! [A2: set_real,A: real] :
      ( finite_finite_real(A2)
     => ( member_real(A,A2)
       => ? [X2: real] :
            ( member_real(X2,A2)
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),A))
            & ! [Xa: real] :
                ( member_real(Xa,A2)
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa),X2))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1562_finite__has__minimal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( member_set_nat(A,A2)
       => ? [X2: set_nat] :
            ( member_set_nat(X2,A2)
            & pp(aa_set_nat_bool(ord_less_eq_set_nat(X2),A))
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A2)
               => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Xa),X2))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1563_finite__has__minimal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( finite_finite_rat(A2)
     => ( member_rat(A,A2)
       => ? [X2: rat] :
            ( member_rat(X2,A2)
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),A))
            & ! [Xa: rat] :
                ( member_rat(Xa,A2)
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa),X2))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1564_finite__has__minimal2,axiom,
    ! [A2: set_num,A: num] :
      ( finite_finite_num(A2)
     => ( member_num(A,A2)
       => ? [X2: num] :
            ( member_num(X2,A2)
            & pp(aa_num_bool(ord_less_eq_num(X2),A))
            & ! [Xa: num] :
                ( member_num(Xa,A2)
               => ( pp(aa_num_bool(ord_less_eq_num(Xa),X2))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1565_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))
       => ? [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),A2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),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),X2))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1566_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))
       => ? [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),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),X2))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1567_finite_OemptyI,axiom,
    pp(aa_set_complex_bool(finite3207457112153483333omplex,bot_bot_set_complex)) ).

% finite.emptyI
tff(fact_1568_finite_OemptyI,axiom,
    pp(aa_set_nat_bool(finite_finite_nat,bot_bot_set_nat)) ).

% finite.emptyI
tff(fact_1569_finite_OemptyI,axiom,
    pp(aa_set_int_bool(finite_finite_int,bot_bot_set_int)) ).

% finite.emptyI
tff(fact_1570_finite_OemptyI,axiom,
    finite_finite_real(bot_bot_set_real) ).

% finite.emptyI
tff(fact_1571_infinite__imp__nonempty,axiom,
    ! [S3: set_complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( S3 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
tff(fact_1572_infinite__imp__nonempty,axiom,
    ! [S3: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( S3 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
tff(fact_1573_infinite__imp__nonempty,axiom,
    ! [S3: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S3))
     => ( S3 != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
tff(fact_1574_infinite__imp__nonempty,axiom,
    ! [S3: set_real] :
      ( ~ finite_finite_real(S3)
     => ( S3 != bot_bot_set_real ) ) ).

% infinite_imp_nonempty
tff(fact_1575_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_1576_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_1577_finite__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_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_1578_infinite__super,axiom,
    ! [S3: set_int,T2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T2))
     => ( ~ pp(aa_set_int_bool(finite_finite_int,S3))
       => ~ pp(aa_set_int_bool(finite_finite_int,T2)) ) ) ).

% infinite_super
tff(fact_1579_infinite__super,axiom,
    ! [S3: set_complex,T2: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T2))
     => ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,T2)) ) ) ).

% infinite_super
tff(fact_1580_infinite__super,axiom,
    ! [S3: set_nat,T2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(S3),T2))
     => ( ~ pp(aa_set_nat_bool(finite_finite_nat,S3))
       => ~ pp(aa_set_nat_bool(finite_finite_nat,T2)) ) ) ).

% infinite_super
tff(fact_1581_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_1582_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_1583_rev__finite__subset,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
       => pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ) ).

% rev_finite_subset
tff(fact_1584_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),K))
     => ( pp(aa_int_bool(P,K))
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I3),K))
             => ( pp(aa_int_bool(P,I3))
               => pp(aa_int_bool(P,aa_int_int(minus_minus_int(I3),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_le_induct
tff(fact_1585_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(plus_plus_int(K),one_one_int)))
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I3))
             => ( pp(aa_int_bool(P,I3))
               => pp(aa_int_bool(P,aa_int_int(plus_plus_int(I3),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_gr_induct
tff(fact_1586_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),aa_int_int(plus_plus_int(Z),one_one_int)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z))
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_1587_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(Z1),Z22)),W) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(Z1),W)),aa_int_int(times_times_int(Z22),W)) ).

% int_distrib(1)
tff(fact_1588_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(times_times_int(W),aa_int_int(plus_plus_int(Z1),Z22)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(W),Z1)),aa_int_int(times_times_int(W),Z22)) ).

% int_distrib(2)
tff(fact_1589_finite__image__set2,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool,F: fun_re7681423839302404990l_real] :
      ( finite_finite_real(collect_real(P))
     => ( finite_finite_real(collect_real(Q))
       => finite_finite_real(collect_real(aa_fun955784922888763656l_bool(aa_fun2902298374912565143l_bool(aTP_Lamm_dg(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_1590_finite__image__set2,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool,F: fun_re7428432522468557474al_nat] :
      ( finite_finite_real(collect_real(P))
     => ( finite_finite_real(collect_real(Q))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun528418799539591104t_bool(aa_fun8259732966411813455t_bool(aTP_Lamm_dh(P),Q),F)))) ) ) ).

% finite_image_set2
tff(fact_1591_finite__image__set2,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool,F: fun_re4938349377536830718al_int] :
      ( finite_finite_real(collect_real(P))
     => ( finite_finite_real(collect_real(Q))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1238366933954993416t_bool(aa_fun2410317078794417559t_bool(aTP_Lamm_di(P),Q),F)))) ) ) ).

% finite_image_set2
tff(fact_1592_finite__image__set2,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool,F: fun_re7812779276277946752omplex] :
      ( finite_finite_real(collect_real(P))
     => ( finite_finite_real(collect_real(Q))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun8363404835654190852x_bool(aa_fun7464652376466389139x_bool(aTP_Lamm_dj(P),Q),F)))) ) ) ).

% finite_image_set2
tff(fact_1593_finite__image__set2,axiom,
    ! [P: fun_real_bool,Q: fun_nat_bool,F: fun_re4238438383274954274t_real] :
      ( finite_finite_real(collect_real(P))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(Q)))
       => finite_finite_real(collect_real(aa_fun5008594764412271332l_bool(aa_fun7835077198776561559l_bool(aTP_Lamm_dk(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_1594_finite__image__set2,axiom,
    ! [P: fun_real_bool,Q: fun_nat_bool,F: fun_real_fun_nat_nat] :
      ( finite_finite_real(collect_real(P))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(Q)))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun3685766209695167900t_bool(aa_fun2965812958138651215t_bool(aTP_Lamm_dl(P),Q),F)))) ) ) ).

% finite_image_set2
tff(fact_1595_finite__image__set2,axiom,
    ! [P: fun_real_bool,Q: fun_nat_bool,F: fun_real_fun_nat_int] :
      ( finite_finite_real(collect_real(P))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(Q)))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun4395714344110570212t_bool(aa_fun6339769107376031127t_bool(aTP_Lamm_dm(P),Q),F)))) ) ) ).

% finite_image_set2
tff(fact_1596_finite__image__set2,axiom,
    ! [P: fun_real_bool,Q: fun_nat_bool,F: fun_re2084383175013312036omplex] :
      ( finite_finite_real(collect_real(P))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(Q)))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9170867047750497504x_bool(aa_fun2493215948360222867x_bool(aTP_Lamm_dn(P),Q),F)))) ) ) ).

% finite_image_set2
tff(fact_1597_finite__image__set2,axiom,
    ! [P: fun_real_bool,Q: fun_int_bool,F: fun_re7457141977439324286t_real] :
      ( finite_finite_real(collect_real(P))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(Q)))
       => finite_finite_real(collect_real(aa_fun6686107996194020744l_bool(aa_fun7698435294645330071l_bool(aTP_Lamm_do(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_1598_finite__image__set2,axiom,
    ! [P: fun_real_bool,Q: fun_int_bool,F: fun_real_fun_int_nat] :
      ( finite_finite_real(collect_real(P))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(Q)))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun6723608294159301696t_bool(aa_fun543251173574902607t_bool(aTP_Lamm_dp(P),Q),F)))) ) ) ).

% finite_image_set2
tff(fact_1599_finite__image__set,axiom,
    ! [P: fun_real_bool,F: fun_real_real] :
      ( finite_finite_real(collect_real(P))
     => finite_finite_real(collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_dq(P),F))) ) ).

% finite_image_set
tff(fact_1600_finite__image__set,axiom,
    ! [P: fun_real_bool,F: fun_real_nat] :
      ( finite_finite_real(collect_real(P))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun3670751239728173737t_bool(aTP_Lamm_dr(P),F)))) ) ).

% finite_image_set
tff(fact_1601_finite__image__set,axiom,
    ! [P: fun_real_bool,F: fun_real_int] :
      ( finite_finite_real(collect_real(P))
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun6938287985607813617t_bool(aTP_Lamm_ds(P),F)))) ) ).

% finite_image_set
tff(fact_1602_finite__image__set,axiom,
    ! [P: fun_real_bool,F: fun_real_complex] :
      ( finite_finite_real(collect_real(P))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun6713666175893136621x_bool(aTP_Lamm_dt(P),F)))) ) ).

% finite_image_set
tff(fact_1603_finite__image__set,axiom,
    ! [P: fun_nat_bool,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => finite_finite_real(collect_real(aa_fun2369234836118212941l_bool(aTP_Lamm_du(P),F))) ) ).

% finite_image_set
tff(fact_1604_finite__image__set,axiom,
    ! [P: fun_nat_bool,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1783405840085286277t_bool(aTP_Lamm_dv(P),F)))) ) ).

% finite_image_set
tff(fact_1605_finite__image__set,axiom,
    ! [P: fun_nat_bool,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5050942585964926157t_bool(aTP_Lamm_dw(P),F)))) ) ).

% finite_image_set
tff(fact_1606_finite__image__set,axiom,
    ! [P: fun_nat_bool,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun678042768005816265x_bool(aTP_Lamm_dx(P),F)))) ) ).

% finite_image_set
tff(fact_1607_finite__image__set,axiom,
    ! [P: fun_int_bool,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
     => finite_finite_real(collect_real(aa_fun1272151855474214385l_bool(aTP_Lamm_dy(P),F))) ) ).

% finite_image_set
tff(fact_1608_finite__image__set,axiom,
    ! [P: fun_int_bool,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8751864718750488105t_bool(aTP_Lamm_dz(P),F)))) ) ).

% finite_image_set
tff(fact_1609_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( finite_finite_real(A2)
     => ( ( A2 != bot_bot_set_real )
       => ? [X2: real] :
            ( member_real(X2,A2)
            & ! [Xa: real] :
                ( member_real(Xa,A2)
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Xa))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1610_finite__has__maximal,axiom,
    ! [A2: set_set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X2: set_nat] :
            ( member_set_nat(X2,A2)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A2)
               => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X2),Xa))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1611_finite__has__maximal,axiom,
    ! [A2: set_rat] :
      ( finite_finite_rat(A2)
     => ( ( A2 != bot_bot_set_rat )
       => ? [X2: rat] :
            ( member_rat(X2,A2)
            & ! [Xa: rat] :
                ( member_rat(Xa,A2)
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Xa))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1612_finite__has__maximal,axiom,
    ! [A2: set_num] :
      ( finite_finite_num(A2)
     => ( ( A2 != bot_bot_set_num )
       => ? [X2: num] :
            ( member_num(X2,A2)
            & ! [Xa: num] :
                ( member_num(Xa,A2)
               => ( pp(aa_num_bool(ord_less_eq_num(X2),Xa))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1613_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),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,X2),Xa))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1614_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ? [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),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,X2),Xa))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1615_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( finite_finite_real(A2)
     => ( ( A2 != bot_bot_set_real )
       => ? [X2: real] :
            ( member_real(X2,A2)
            & ! [Xa: real] :
                ( member_real(Xa,A2)
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa),X2))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1616_finite__has__minimal,axiom,
    ! [A2: set_set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X2: set_nat] :
            ( member_set_nat(X2,A2)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A2)
               => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Xa),X2))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1617_finite__has__minimal,axiom,
    ! [A2: set_rat] :
      ( finite_finite_rat(A2)
     => ( ( A2 != bot_bot_set_rat )
       => ? [X2: rat] :
            ( member_rat(X2,A2)
            & ! [Xa: rat] :
                ( member_rat(Xa,A2)
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa),X2))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1618_finite__has__minimal,axiom,
    ! [A2: set_num] :
      ( finite_finite_num(A2)
     => ( ( A2 != bot_bot_set_num )
       => ? [X2: num] :
            ( member_num(X2,A2)
            & ! [Xa: num] :
                ( member_num(Xa,A2)
               => ( pp(aa_num_bool(ord_less_eq_num(Xa),X2))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1619_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),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),X2))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1620_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ? [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),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),X2))
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1621_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(W),one_one_int)),Z)) ) ).

% zless_imp_add1_zle
tff(fact_1622_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I))
     => ( pp(aa_int_bool(P,K))
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I3))
             => ( pp(aa_int_bool(P,I3))
               => pp(aa_int_bool(P,aa_int_int(plus_plus_int(I3),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_ge_induct
tff(fact_1623_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(W),one_one_int)),Z))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% add1_zle_eq
tff(fact_1624_int__induct,axiom,
    ! [P: fun_int_bool,K: int,I: int] :
      ( pp(aa_int_bool(P,K))
     => ( ! [I3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I3))
           => ( pp(aa_int_bool(P,I3))
             => pp(aa_int_bool(P,aa_int_int(plus_plus_int(I3),one_one_int))) ) )
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I3),K))
             => ( pp(aa_int_bool(P,I3))
               => pp(aa_int_bool(P,aa_int_int(minus_minus_int(I3),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_induct
tff(fact_1625_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)),X)) ).

% vebt_member.simps(2)
tff(fact_1626_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc))) ).

% VEBT_internal.minNull.simps(5)
tff(fact_1627_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy))) ).

% VEBT_internal.minNull.simps(4)
tff(fact_1628_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_1629_Diff__eq__empty__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( minus_minus_set_real(A2,B2) = bot_bot_set_real )
    <=> ord_less_eq_set_real(A2,B2) ) ).

% Diff_eq_empty_iff
tff(fact_1630_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(ord_less_eq_set_nat(A2),B2)) ) ).

% Diff_eq_empty_iff
tff(fact_1631_double__not__eq__Suc__double,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% double_not_eq_Suc_double
tff(fact_1632_Suc__double__not__eq__double,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% Suc_double_not_eq_double
tff(fact_1633_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(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList),Vc)),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(aa_VEB4832260192069861672t_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_1634_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(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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_1635_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList),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(aa_VEB4832260192069861672t_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_1636_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,vEBT_vebt_buildup(N)),X)) ).

% buildup_nothing_in_leaf
tff(fact_1637_unset__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se8260200283734997820nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se8260200283734997820nteger(N,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_1638_unset__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,aa_nat_nat(suc,N)),A) = aa_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(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_1639_unset__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_1640_flip__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se1345352211410354436nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se1345352211410354436nteger(N,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_1641_flip__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,aa_nat_nat(suc,N)),A) = aa_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(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_1642_flip__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_1643_set__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se2793503036327961859nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se2793503036327961859nteger(N,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_1644_set__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,aa_nat_nat(suc,N)),A) = aa_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(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_1645_set__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_1646_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,vEBT_vebt_buildup(N)),X)) ).

% buildup_nothing_in_min_max
tff(fact_1647_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(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,T),X))
        | pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,T),X)) ) ) ).

% both_member_options_def
tff(fact_1648_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_1649_Diff__iff,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( member_real(C,minus_minus_set_real(A2,B2))
    <=> ( member_real(C,A2)
        & ~ member_real(C,B2) ) ) ).

% Diff_iff
tff(fact_1650_Diff__iff,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),minus_5127226145743854075T_VEBT(A2,B2)))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),A2))
        & ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),B2)) ) ) ).

% Diff_iff
tff(fact_1651_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_1652_Diff__iff,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(member_complex(C),minus_811609699411566653omplex(A2,B2)))
    <=> ( pp(aa_set_complex_bool(member_complex(C),A2))
        & ~ pp(aa_set_complex_bool(member_complex(C),B2)) ) ) ).

% Diff_iff
tff(fact_1653_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_1654_DiffI,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( member_real(C,A2)
     => ( ~ member_real(C,B2)
       => member_real(C,minus_minus_set_real(A2,B2)) ) ) ).

% DiffI
tff(fact_1655_DiffI,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),A2))
     => ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),B2))
       => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),minus_5127226145743854075T_VEBT(A2,B2))) ) ) ).

% DiffI
tff(fact_1656_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_1657_DiffI,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(member_complex(C),A2))
     => ( ~ pp(aa_set_complex_bool(member_complex(C),B2))
       => pp(aa_set_complex_bool(member_complex(C),minus_811609699411566653omplex(A2,B2))) ) ) ).

% DiffI
tff(fact_1658_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_1659_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Tree),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,Tree),X))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,Tree),X))
          | pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,Tree),X)) ) ) ) ).

% member_valid_both_member_options
tff(fact_1660_Diff__empty,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,bot_bot_set_int) = A2 ).

% Diff_empty
tff(fact_1661_Diff__empty,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,bot_bot_set_real) = A2 ).

% Diff_empty
tff(fact_1662_Diff__empty,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(A2,bot_bot_set_nat) = A2 ).

% Diff_empty
tff(fact_1663_empty__Diff,axiom,
    ! [A2: set_int] : minus_minus_set_int(bot_bot_set_int,A2) = bot_bot_set_int ).

% empty_Diff
tff(fact_1664_empty__Diff,axiom,
    ! [A2: set_real] : minus_minus_set_real(bot_bot_set_real,A2) = bot_bot_set_real ).

% empty_Diff
tff(fact_1665_empty__Diff,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(bot_bot_set_nat,A2) = bot_bot_set_nat ).

% empty_Diff
tff(fact_1666_Diff__cancel,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,A2) = bot_bot_set_int ).

% Diff_cancel
tff(fact_1667_Diff__cancel,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,A2) = bot_bot_set_real ).

% Diff_cancel
tff(fact_1668_Diff__cancel,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(A2,A2) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_1669_psubsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
     => ( ( A2 != B2 )
       => pp(aa_set_nat_bool(ord_less_set_nat(A2),B2)) ) ) ).

% psubsetI
tff(fact_1670_set__diff__eq,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A2,B2) = collect_VEBT_VEBT(aa_set1728193350978918526T_bool(aTP_Lamm_ea(A2),B2)) ).

% set_diff_eq
tff(fact_1671_set__diff__eq,axiom,
    ! [A2: set_complex,B2: set_complex] : minus_811609699411566653omplex(A2,B2) = collect_complex(aa_set3741564773079358752x_bool(aTP_Lamm_eb(A2),B2)) ).

% set_diff_eq
tff(fact_1672_set__diff__eq,axiom,
    ! [A2: set_real,B2: set_real] : minus_minus_set_real(A2,B2) = collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_ec(A2),B2)) ).

% set_diff_eq
tff(fact_1673_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_ed(A2),B2)) ).

% set_diff_eq
tff(fact_1674_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_ee(A2),B2)) ).

% set_diff_eq
tff(fact_1675_set__diff__eq,axiom,
    ! [A2: set_int,B2: set_int] : minus_minus_set_int(A2,B2) = collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_ef(A2),B2)) ).

% set_diff_eq
tff(fact_1676_set__diff__eq,axiom,
    ! [A2: set_nat,B2: set_nat] : minus_minus_set_nat(A2,B2) = collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_eg(A2),B2)) ).

% set_diff_eq
tff(fact_1677_minus__set__def,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A2,B2) = collect_VEBT_VEBT(minus_1873103217924207103T_bool(aTP_Lamm_a(A2),aTP_Lamm_a(B2))) ).

% minus_set_def
tff(fact_1678_minus__set__def,axiom,
    ! [A2: set_complex,B2: set_complex] : minus_811609699411566653omplex(A2,B2) = collect_complex(minus_5423061390740260265x_bool(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2))) ).

% minus_set_def
tff(fact_1679_minus__set__def,axiom,
    ! [A2: set_real,B2: set_real] : minus_minus_set_real(A2,B2) = collect_real(minus_6582619742941512235l_bool(aTP_Lamm_ab(A2),aTP_Lamm_ab(B2))) ).

% minus_set_def
tff(fact_1680_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_ac(A2),aTP_Lamm_ac(B2))) ).

% minus_set_def
tff(fact_1681_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_ad(A2),aTP_Lamm_ad(B2))) ).

% minus_set_def
tff(fact_1682_minus__set__def,axiom,
    ! [A2: set_int,B2: set_int] : minus_minus_set_int(A2,B2) = collect_int(minus_7727386867957376939t_bool(aTP_Lamm_af(A2),aTP_Lamm_af(B2))) ).

% minus_set_def
tff(fact_1683_minus__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] : minus_minus_set_nat(A2,B2) = collect_nat(minus_1628305797571225351t_bool(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2))) ).

% minus_set_def
tff(fact_1684_less__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
    <=> ord_le6841235830588351564t_bool(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2)) ) ).

% less_set_def
tff(fact_1685_less__set__def,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ord_less_set_real(A2,B2)
    <=> ord_le7519368973566061808l_bool(aTP_Lamm_ab(A2),aTP_Lamm_ab(B2)) ) ).

% less_set_def
tff(fact_1686_less__set__def,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ord_le3480810397992357184T_VEBT(A2,B2)
    <=> ord_le763421453198481914T_bool(aTP_Lamm_a(A2),aTP_Lamm_a(B2)) ) ).

% less_set_def
tff(fact_1687_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_af(A2),aTP_Lamm_af(B2)) ) ).

% less_set_def
tff(fact_1688_less__set__def,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(ord_less_set_complex(A2),B2))
    <=> ord_le3776645642988763374x_bool(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2)) ) ).

% less_set_def
tff(fact_1689_psubsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( pp(aa_set_nat_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_1690_psubsetD,axiom,
    ! [A2: set_real,B2: set_real,C: real] :
      ( ord_less_set_real(A2,B2)
     => ( member_real(C,A2)
       => member_real(C,B2) ) ) ).

% psubsetD
tff(fact_1691_psubsetD,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,C: vEBT_VEBT] :
      ( ord_le3480810397992357184T_VEBT(A2,B2)
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),A2))
       => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),B2)) ) ) ).

% psubsetD
tff(fact_1692_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_1693_psubsetD,axiom,
    ! [A2: set_complex,B2: set_complex,C: complex] :
      ( pp(aa_set_complex_bool(ord_less_set_complex(A2),B2))
     => ( pp(aa_set_complex_bool(member_complex(C),A2))
       => pp(aa_set_complex_bool(member_complex(C),B2)) ) ) ).

% psubsetD
tff(fact_1694_DiffD2,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( member_real(C,minus_minus_set_real(A2,B2))
     => ~ member_real(C,B2) ) ).

% DiffD2
tff(fact_1695_DiffD2,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),minus_5127226145743854075T_VEBT(A2,B2)))
     => ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),B2)) ) ).

% DiffD2
tff(fact_1696_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_1697_DiffD2,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(member_complex(C),minus_811609699411566653omplex(A2,B2)))
     => ~ pp(aa_set_complex_bool(member_complex(C),B2)) ) ).

% DiffD2
tff(fact_1698_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_1699_DiffD1,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( member_real(C,minus_minus_set_real(A2,B2))
     => member_real(C,A2) ) ).

% DiffD1
tff(fact_1700_DiffD1,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),minus_5127226145743854075T_VEBT(A2,B2)))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),A2)) ) ).

% DiffD1
tff(fact_1701_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_1702_DiffD1,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(member_complex(C),minus_811609699411566653omplex(A2,B2)))
     => pp(aa_set_complex_bool(member_complex(C),A2)) ) ).

% DiffD1
tff(fact_1703_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_1704_DiffE,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( member_real(C,minus_minus_set_real(A2,B2))
     => ~ ( member_real(C,A2)
         => member_real(C,B2) ) ) ).

% DiffE
tff(fact_1705_DiffE,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),minus_5127226145743854075T_VEBT(A2,B2)))
     => ~ ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),A2))
         => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),B2)) ) ) ).

% DiffE
tff(fact_1706_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_1707_DiffE,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(member_complex(C),minus_811609699411566653omplex(A2,B2)))
     => ~ ( pp(aa_set_complex_bool(member_complex(C),A2))
         => pp(aa_set_complex_bool(member_complex(C),B2)) ) ) ).

% DiffE
tff(fact_1708_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_1709_psubset__imp__ex__mem,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ord_less_set_real(A2,B2)
     => ? [B4: real] : member_real(B4,minus_minus_set_real(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_1710_psubset__imp__ex__mem,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ord_le3480810397992357184T_VEBT(A2,B2)
     => ? [B4: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(B2,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_1711_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
     => ? [B4: int] : pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(B2,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_1712_psubset__imp__ex__mem,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(ord_less_set_complex(A2),B2))
     => ? [B4: complex] : pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(B2,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_1713_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
     => ? [B4: nat] : pp(aa_set_nat_bool(member_nat(B4),minus_minus_set_nat(B2,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_1714_unset__bit__less__eq,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N),K)),K)) ).

% unset_bit_less_eq
tff(fact_1715_set__bit__greater__eq,axiom,
    ! [K: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),K))) ).

% set_bit_greater_eq
tff(fact_1716_not__psubset__empty,axiom,
    ! [A2: set_nat] : ~ pp(aa_set_nat_bool(ord_less_set_nat(A2),bot_bot_set_nat)) ).

% not_psubset_empty
tff(fact_1717_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_1718_not__psubset__empty,axiom,
    ! [A2: set_real] : ~ ord_less_set_real(A2,bot_bot_set_real) ).

% not_psubset_empty
tff(fact_1719_psubsetE,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
     => ~ ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
         => pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2)) ) ) ).

% psubsetE
tff(fact_1720_psubset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
        & ( A2 != B2 ) ) ) ).

% psubset_eq
tff(fact_1721_psubset__imp__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2)) ) ).

% psubset_imp_subset
tff(fact_1722_psubset__subset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C4: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),C4))
       => pp(aa_set_nat_bool(ord_less_set_nat(A2),C4)) ) ) ).

% psubset_subset_trans
tff(fact_1723_subset__not__subset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
        & ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2)) ) ) ).

% subset_not_subset_eq
tff(fact_1724_subset__psubset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C4: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
     => ( pp(aa_set_nat_bool(ord_less_set_nat(B2),C4))
       => pp(aa_set_nat_bool(ord_less_set_nat(A2),C4)) ) ) ).

% subset_psubset_trans
tff(fact_1725_subset__iff__psubset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
    <=> ( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
        | ( A2 = B2 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1726_double__diff,axiom,
    ! [A2: set_nat,B2: set_nat,C4: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),C4))
       => ( minus_minus_set_nat(B2,minus_minus_set_nat(C4,A2)) = A2 ) ) ) ).

% double_diff
tff(fact_1727_Diff__subset,axiom,
    ! [A2: set_nat,B2: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(minus_minus_set_nat(A2,B2)),A2)) ).

% Diff_subset
tff(fact_1728_Diff__mono,axiom,
    ! [A2: set_nat,C4: set_nat,D3: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),C4))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(D3),B2))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(minus_minus_set_nat(A2,B2)),minus_minus_set_nat(C4,D3))) ) ) ).

% Diff_mono
tff(fact_1729_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X),Xa2))
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2),Vb2)
           => ~ ( ( Xa2 = Mi2 )
                | ( Xa2 = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)
             => ~ ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 )
                  | ( ( 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(aa_VEB4832260192069861672t_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 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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(aa_VEB4832260192069861672t_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_1730_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(X,Xa2) = Y )
     => ( ! [Uu2: bool,B4: bool] :
            ( ( X = vEBT_Leaf(Uu2,B4) )
           => ( ( Xa2 = zero_zero_nat )
             => ~ ( ( pp(B4)
                   => ( Y = some_nat(one_one_nat) ) )
                  & ( ~ pp(B4)
                   => ( 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 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)
             => ( Y != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2)
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg),Vh)
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                     => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                           => ( Y = some_nat(Mi2) ) )
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                           => ( 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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(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,Va3))),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_succ(Summary2,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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_1731_product__nth,axiom,
    ! [N: nat,Xs: list_num,Ys: list_num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_num(Xs)),size_size_list_num(Ys))))
     => ( nth_Pr6456567536196504476um_num(product_num_num(Xs,Ys),N) = product_Pair_num_num(aa_nat_num(nth_num(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_num(Ys))),aa_nat_num(nth_num(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_num(Ys)))) ) ) ).

% product_nth
tff(fact_1732_product__nth,axiom,
    ! [N: nat,Xs: list_Code_integer,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_s3445333598471063425nteger(Xs)),size_size_list_bool(Ys))))
     => ( nth_Pr136253548056570280r_bool(produc2469108709554126195r_bool(Xs,Ys),N) = produc5135875973608410785r_bool(aa_nat_Code_integer(nth_Code_integer(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_1733_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_1734_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_1735_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(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_1736_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_1737_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_1738_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_1739_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(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_1740_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_1741_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,vEBT_vebt_buildup(N)),N)) ) ).

% buildup_gives_valid
tff(fact_1742_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X: fun_VE5569070535936029429omplex,Y: fun_VE5569070535936029429omplex] :
      ( finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aTP_Lamm_eh(I5),X)))
     => ( finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aTP_Lamm_eh(I5),Y)))
       => finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aa_fun4024526296178165685T_bool(aTP_Lamm_ei(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_1743_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_complex,Y: fun_real_complex] :
      ( finite_finite_real(collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_ej(I5),X)))
     => ( finite_finite_real(collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_ej(I5),Y)))
       => finite_finite_real(collect_real(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_ek(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_1744_prod_Ofinite__Collect__op,axiom,
    ! [I5: 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_el(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_el(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_em(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_1745_prod_Ofinite__Collect__op,axiom,
    ! [I5: 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_en(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_en(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_eo(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_1746_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_complex_complex,Y: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_ep(I5),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_ep(I5),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_eq(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_1747_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X: fun_VEBT_VEBT_real,Y: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_er(I5),X)))
     => ( finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_er(I5),Y)))
       => finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aa_fun4082248197810269365T_bool(aTP_Lamm_es(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_1748_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_real,Y: fun_real_real] :
      ( finite_finite_real(collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_et(I5),X)))
     => ( finite_finite_real(collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_et(I5),Y)))
       => finite_finite_real(collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_eu(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_1749_prod_Ofinite__Collect__op,axiom,
    ! [I5: 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_ev(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_ev(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_ew(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_1750_prod_Ofinite__Collect__op,axiom,
    ! [I5: 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_ex(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_ex(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_ey(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_1751_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_complex_real,Y: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_ez(I5),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_ez(I5),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_fa(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_1752_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(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(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))))))),aa_VEB5832953974749077948T_VEBT(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_1753_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(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,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))))))),aa_VEB5832953974749077948T_VEBT(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_1754_set__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,zero_zero_nat),A) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_1755_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_1756_valid__0__not,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),zero_zero_nat)) ).

% valid_0_not
tff(fact_1757_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),zero_zero_nat)) ).

% valid_tree_deg_neq_0
tff(fact_1758_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% deg_not_0
tff(fact_1759_Leaf__0__not,axiom,
    ! [A: bool,B: bool] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,vEBT_Leaf(A,B)),zero_zero_nat)) ).

% Leaf_0_not
tff(fact_1760_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),one_one_nat))
    <=> ? [A6: bool,B6: bool] : T = vEBT_Leaf(A6,B6) ) ).

% deg1Leaf
tff(fact_1761_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),one_one_nat))
     => ? [A4: bool,B4: bool] : T = vEBT_Leaf(A4,B4) ) ).

% deg_1_Leaf
tff(fact_1762_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( N = one_one_nat )
       => ? [A4: bool,B4: bool] : T = vEBT_Leaf(A4,B4) ) ) ).

% deg_1_Leafy
tff(fact_1763_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_1764_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_1765_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_1766_mult__zero__left,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A) = zero_zero_complex ).

% mult_zero_left
tff(fact_1767_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_1768_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_1769_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_1770_mult__zero__left,axiom,
    ! [A: int] : aa_int_int(times_times_int(zero_zero_int),A) = zero_zero_int ).

% mult_zero_left
tff(fact_1771_mult__zero__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_1772_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_1773_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_1774_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_1775_mult__zero__right,axiom,
    ! [A: int] : aa_int_int(times_times_int(A),zero_zero_int) = zero_zero_int ).

% mult_zero_right
tff(fact_1776_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_1777_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_1778_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_1779_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_1780_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(times_times_int(A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_1781_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_1782_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_1783_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1784_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_1785_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( aa_int_int(times_times_int(C),A) = aa_int_int(times_times_int(C),B) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1786_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_1787_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_1788_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1789_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_1790_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(times_times_int(A),C) = aa_int_int(times_times_int(B),C) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1791_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_1792_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat(A,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_1793_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(plus_plus_int(A),A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% double_eq_0_iff
tff(fact_1794_add__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add_0
tff(fact_1795_add__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add_0
tff(fact_1796_add__0,axiom,
    ! [A: rat] : plus_plus_rat(zero_zero_rat,A) = A ).

% add_0
tff(fact_1797_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_1798_add__0,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(zero_zero_int),A) = A ).

% add_0
tff(fact_1799_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_1800_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_1801_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_1802_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_1803_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = plus_plus_rat(A,B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_1804_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_1805_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(plus_plus_int(A),B) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_1806_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_1807_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_1808_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = plus_plus_rat(B,A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_1809_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_1810_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(plus_plus_int(B),A) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_1811_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_1812_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_1813_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat(A,B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_1814_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_1815_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_1816_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_1817_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_1818_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( plus_plus_rat(B,A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_1819_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_1820_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(plus_plus_int(B),A) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_1821_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_1822_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = plus_plus_rat(A,A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_1823_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(plus_plus_int(A),A) )
    <=> ( A = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_1824_add_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.right_neutral
tff(fact_1825_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_1826_add_Oright__neutral,axiom,
    ! [A: rat] : plus_plus_rat(A,zero_zero_rat) = A ).

% add.right_neutral
tff(fact_1827_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_1828_add_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(A),zero_zero_int) = A ).

% add.right_neutral
tff(fact_1829_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_1830_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_1831_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : minus_minus_rat(A,A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1832_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_1833_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] : aa_int_int(minus_minus_int(A),A) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1834_diff__zero,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_zero
tff(fact_1835_diff__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_zero
tff(fact_1836_diff__zero,axiom,
    ! [A: rat] : minus_minus_rat(A,zero_zero_rat) = A ).

% diff_zero
tff(fact_1837_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_1838_diff__zero,axiom,
    ! [A: int] : aa_int_int(minus_minus_int(A),zero_zero_int) = A ).

% diff_zero
tff(fact_1839_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_1840_diff__0__right,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_0_right
tff(fact_1841_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_1842_diff__0__right,axiom,
    ! [A: rat] : minus_minus_rat(A,zero_zero_rat) = A ).

% diff_0_right
tff(fact_1843_diff__0__right,axiom,
    ! [A: int] : aa_int_int(minus_minus_int(A),zero_zero_int) = A ).

% diff_0_right
tff(fact_1844_diff__self,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% diff_self
tff(fact_1845_diff__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% diff_self
tff(fact_1846_diff__self,axiom,
    ! [A: rat] : minus_minus_rat(A,A) = zero_zero_rat ).

% diff_self
tff(fact_1847_diff__self,axiom,
    ! [A: int] : aa_int_int(minus_minus_int(A),A) = zero_zero_int ).

% diff_self
tff(fact_1848_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_1849_bits__div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_1850_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_1851_bits__div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_1852_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_1853_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_1854_division__ring__divide__zero,axiom,
    ! [A: rat] : divide_divide_rat(A,zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_1855_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_1856_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_1857_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_1858_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_1859_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_1860_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( divide_divide_rat(C,A) = divide_divide_rat(C,B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_1861_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_1862_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_1863_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat(A,B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_1864_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_1865_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_1866_div__by__0,axiom,
    ! [A: rat] : divide_divide_rat(A,zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_1867_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_1868_div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_1869_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_1870_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_1871_div__0,axiom,
    ! [A: rat] : divide_divide_rat(zero_zero_rat,A) = zero_zero_rat ).

% div_0
tff(fact_1872_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_1873_div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_1874_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_1875_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_1876_bits__mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% bits_mod_0
tff(fact_1877_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_1878_mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_1879_mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% mod_0
tff(fact_1880_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_1881_mod__by__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),zero_zero_int) = A ).

% mod_by_0
tff(fact_1882_mod__by__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,zero_z3403309356797280102nteger) = A ).

% mod_by_0
tff(fact_1883_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_1884_mod__self,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),A) = zero_zero_int ).

% mod_self
tff(fact_1885_mod__self,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,A) = zero_z3403309356797280102nteger ).

% mod_self
tff(fact_1886_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_1887_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_1888_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_1889_le0,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% le0
tff(fact_1890_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_1891_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_1892_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_1893_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_1894_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_1895_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_1896_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_1897_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_1898_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_1899_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_1900_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_1901_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_1902_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_1903_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_1904_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_1905_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_1906_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_1907_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_1908_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_1909_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_1910_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] : list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),I)) = Xs ).

% list_update_id
tff(fact_1911_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_1912_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_1913_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_nat,X: nat] :
      ( ( I != J )
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_1914_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_1915_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_1916_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_1917_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),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_1918_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(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_1919_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_1920_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,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_1921_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(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_1922_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_1923_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),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_1924_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_1925_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(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_1926_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_1927_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),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_1928_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_1929_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(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_1930_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_1931_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_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_1932_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_1933_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(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_1934_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_1935_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_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_1936_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_1937_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(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_1938_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_1939_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),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_1940_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(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_1941_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_1942_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),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_1943_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(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_1944_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_1945_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_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_1946_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(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_1947_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_1948_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),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_1949_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_1950_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(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_1951_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_1952_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),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_1953_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_1954_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(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_1955_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_1956_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_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_1957_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_1958_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_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_1959_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_1960_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_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_1961_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_1962_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_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_1963_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_1964_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),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_1965_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(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_1966_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_1967_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_1968_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_1969_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(times_times_int(C),B) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_1970_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_1971_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_1972_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_1973_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( aa_int_int(times_times_int(C),A) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_1974_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_1975_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_1976_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_1977_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(times_times_int(B),C) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_1978_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_1979_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_1980_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_1981_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( aa_int_int(times_times_int(A),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_1982_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_1983_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( plus_plus_rat(aa_rat_rat(times_times_rat(X),X),aa_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_1984_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(X),X)),aa_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_1985_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_1986_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult1
tff(fact_1987_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_1988_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult2
tff(fact_1989_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_1990_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( C = zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% div_mult_mult1_if
tff(fact_1991_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_1992_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_1993_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( C = zero_zero_rat )
       => ( divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(C),B)) = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(C),B)) = divide_divide_rat(A,B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_1994_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_1995_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_1996_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(C),B)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1997_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_1998_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_1999_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(B),C)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2000_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_2001_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_2002_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2003_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_2004_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_2005_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(C),B)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2006_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_2007_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_2008_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A),B),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2009_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_2010_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2011_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_2012_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_2013_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A),B),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2014_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_2015_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2016_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_2017_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_2018_diff__numeral__special_I9_J,axiom,
    minus_minus_rat(one_one_rat,one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_2019_diff__numeral__special_I9_J,axiom,
    aa_int_int(minus_minus_int(one_one_int),one_one_int) = zero_zero_int ).

% diff_numeral_special(9)
tff(fact_2020_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_2021_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_2022_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = divide_divide_rat(one_one_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_2023_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_2024_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat(one_one_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_2025_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_2026_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat = divide_divide_rat(B,A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_2027_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_2028_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( divide_divide_rat(B,A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_2029_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_2030_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_2031_divide__self__if,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( divide_divide_rat(A,A) = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( divide_divide_rat(A,A) = one_one_rat ) ) ) ).

% divide_self_if
tff(fact_2032_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_2033_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_2034_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,A) = one_one_rat ) ) ).

% divide_self
tff(fact_2035_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_2036_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_2037_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat = divide_divide_rat(A,B) )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2038_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_2039_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_2040_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat(A,B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2041_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_2042_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_2043_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,A) = one_one_rat ) ) ).

% div_self
tff(fact_2044_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_2045_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),A) = one_one_int ) ) ).

% div_self
tff(fact_2046_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_2047_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_2048_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_2049_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_2050_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_2051_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_2052_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_2053_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_2054_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_2055_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_2056_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_2057_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(B),A)),B) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_2058_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_2059_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_2060_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),B)),B) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_2061_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_2062_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_2063_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_2064_bits__mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_2065_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_2066_mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_2067_mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_2068_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_2069_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(modulo_modulo_int(A),B)),B) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_2070_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_2071_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_2072_mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(modulo_modulo_int(A),B)),B) = zero_zero_int ).

% mod_div_trivial
tff(fact_2073_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_2074_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_2075_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_2076_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_2077_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_2078_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_2079_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_2080_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)))
       => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(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)))
       => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)) = numera1916890842035813515d_enat(U) ) ) ) ).

% max_number_of(1)
tff(fact_2081_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(U)),numeral_numeral_real(V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(U)),numeral_numeral_real(V)) = numeral_numeral_real(V) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(U)),numeral_numeral_real(V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(U)),numeral_numeral_real(V)) = numeral_numeral_real(U) ) ) ) ).

% max_number_of(1)
tff(fact_2082_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(U)),numeral_numeral_rat(V)))
       => ( ord_max_rat(numeral_numeral_rat(U),numeral_numeral_rat(V)) = numeral_numeral_rat(V) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(U)),numeral_numeral_rat(V)))
       => ( ord_max_rat(numeral_numeral_rat(U),numeral_numeral_rat(V)) = numeral_numeral_rat(U) ) ) ) ).

% max_number_of(1)
tff(fact_2083_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_2084_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_2085_max__0__1_I4_J,axiom,
    ! [X: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(X)),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(X) ).

% max_0_1(4)
tff(fact_2086_max__0__1_I4_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(X)),zero_zero_real) = numeral_numeral_real(X) ).

% max_0_1(4)
tff(fact_2087_max__0__1_I4_J,axiom,
    ! [X: num] : ord_max_rat(numeral_numeral_rat(X),zero_zero_rat) = numeral_numeral_rat(X) ).

% max_0_1(4)
tff(fact_2088_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_2089_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_2090_max__0__1_I3_J,axiom,
    ! [X: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(X)) = numera1916890842035813515d_enat(X) ).

% max_0_1(3)
tff(fact_2091_max__0__1_I3_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),numeral_numeral_real(X)) = numeral_numeral_real(X) ).

% max_0_1(3)
tff(fact_2092_max__0__1_I3_J,axiom,
    ! [X: num] : ord_max_rat(zero_zero_rat,numeral_numeral_rat(X)) = numeral_numeral_rat(X) ).

% max_0_1(3)
tff(fact_2093_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_2094_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_2095_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_2096_max__0__1_I1_J,axiom,
    ord_max_rat(zero_zero_rat,one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_2097_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_2098_max__0__1_I1_J,axiom,
    aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(1)
tff(fact_2099_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_2100_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_2101_max__0__1_I2_J,axiom,
    ord_max_rat(one_one_rat,zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_2102_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_2103_max__0__1_I2_J,axiom,
    aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,one_on7984719198319812577d_enat),zero_z5237406670263579293d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(2)
tff(fact_2104_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_2105_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_2106_max__0__1_I6_J,axiom,
    ! [X: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(X)),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(X) ).

% max_0_1(6)
tff(fact_2107_max__0__1_I6_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(X)),one_one_real) = numeral_numeral_real(X) ).

% max_0_1(6)
tff(fact_2108_max__0__1_I6_J,axiom,
    ! [X: num] : ord_max_rat(numeral_numeral_rat(X),one_one_rat) = numeral_numeral_rat(X) ).

% max_0_1(6)
tff(fact_2109_max__0__1_I6_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) = aa_num_nat(numeral_numeral_nat,X) ).

% max_0_1(6)
tff(fact_2110_max__0__1_I6_J,axiom,
    ! [X: num] : aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,X)),one_one_int) = aa_num_int(numeral_numeral_int,X) ).

% max_0_1(6)
tff(fact_2111_max__0__1_I5_J,axiom,
    ! [X: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,one_on7984719198319812577d_enat),numera1916890842035813515d_enat(X)) = numera1916890842035813515d_enat(X) ).

% max_0_1(5)
tff(fact_2112_max__0__1_I5_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),numeral_numeral_real(X)) = numeral_numeral_real(X) ).

% max_0_1(5)
tff(fact_2113_max__0__1_I5_J,axiom,
    ! [X: num] : ord_max_rat(one_one_rat,numeral_numeral_rat(X)) = numeral_numeral_rat(X) ).

% max_0_1(5)
tff(fact_2114_max__0__1_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_num_nat(numeral_numeral_nat,X) ).

% max_0_1(5)
tff(fact_2115_max__0__1_I5_J,axiom,
    ! [X: num] : aa_int_int(ord_max_int(one_one_int),aa_num_int(numeral_numeral_int,X)) = aa_num_int(numeral_numeral_int,X) ).

% max_0_1(5)
tff(fact_2116_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_2117_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_2118_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_2119_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_2120_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_2121_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_2122_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_2123_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_2124_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_2125_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_2126_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_2127_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_2128_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_2129_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_2130_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_2131_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_2132_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_2133_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_2134_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_2135_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_2136_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_2137_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_2138_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_2139_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_2140_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_2141_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_2142_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_2143_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_2144_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_2145_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_2146_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_2147_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_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_2148_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_2149_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_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_2150_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_2151_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_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_2152_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_2153_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),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_2154_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_2155_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),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_2156_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_2157_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,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_2158_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_2159_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,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_2160_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_2161_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_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_2162_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) )
    <=> ( ( ( numera6690914467698888265omplex(W) != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)) = B ) )
        & ( ( numera6690914467698888265omplex(W) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2163_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)) )
    <=> ( ( ( numeral_numeral_real(W) != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W)) = B ) )
        & ( ( numeral_numeral_real(W) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2164_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = divide_divide_rat(B,numeral_numeral_rat(W)) )
    <=> ( ( ( numeral_numeral_rat(W) != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W)) = B ) )
        & ( ( numeral_numeral_rat(W) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2165_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) = A )
    <=> ( ( ( numera6690914467698888265omplex(W) != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)) ) )
        & ( ( numera6690914467698888265omplex(W) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2166_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)) = A )
    <=> ( ( ( numeral_numeral_real(W) != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W)) ) )
        & ( ( numeral_numeral_real(W) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2167_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( divide_divide_rat(B,numeral_numeral_rat(W)) = A )
    <=> ( ( ( numeral_numeral_rat(W) != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W)) ) )
        & ( ( numeral_numeral_rat(W) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2168_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_2169_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_2170_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,aa_rat_rat(times_times_rat(A),B)) = divide_divide_rat(one_one_rat,B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2171_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_2172_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_2173_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( divide_divide_rat(B,aa_rat_rat(times_times_rat(A),B)) = divide_divide_rat(one_one_rat,A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2174_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_2175_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),C)),A)),B) = aa_int_int(plus_plus_int(C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self4
tff(fact_2176_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_2177_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(C),B)),A)),B) = aa_int_int(plus_plus_int(C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self3
tff(fact_2178_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_2179_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(B),C))),B) = aa_int_int(plus_plus_int(C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self2
tff(fact_2180_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_2181_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(C),B))),B) = aa_int_int(plus_plus_int(C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self1
tff(fact_2182_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_2183_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_2184_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_2185_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_2186_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_2187_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_2188_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_2189_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_2190_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_2191_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_2192_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_2193_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_2194_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_2195_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(nth_nat(list_update_nat(Xs,I,X)),I) = X ) ) ).

% nth_list_update_eq
tff(fact_2196_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_2197_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_2198_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),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_2199_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_2200_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),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_2201_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_2202_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,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_2203_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_2204_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,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_2205_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_2206_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_2207_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_2208_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_2209_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_2210_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_2211_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_2212_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_2213_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_2214_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_2215_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_2216_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_2217_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_2218_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_2219_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_2220_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_2221_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_nat(Xs)))
       => ( set_nat2(list_update_nat(list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),I))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_2222_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_2223_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_2224_one__div__two__eq__zero,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_2225_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_2226_bits__1__div__2,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_2227_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_2228_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_2229_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_2230_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_2231_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_2232_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_2233_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_2234_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_2235_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_2236_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_2237_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_2238_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_2239_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_2240_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_2241_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: 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_2242_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_2243_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_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_2244_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_2245_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_2246_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_2247_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_2248_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_2249_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_2250_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_2251_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_2252_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_2253_unset__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,zero_zero_nat),A) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_2254_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_2255_zero__reorient,axiom,
    ! [X: complex] :
      ( ( zero_zero_complex = X )
    <=> ( X = zero_zero_complex ) ) ).

% zero_reorient
tff(fact_2256_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
    <=> ( X = zero_zero_real ) ) ).

% zero_reorient
tff(fact_2257_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
    <=> ( X = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_2258_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
    <=> ( X = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_2259_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
    <=> ( X = zero_zero_int ) ) ).

% zero_reorient
tff(fact_2260_list__update__swap,axiom,
    ! [I: nat,I6: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT,X6: vEBT_VEBT] :
      ( ( I != I6 )
     => ( list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),I6,X6) = list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I6,X6),I,X) ) ) ).

% list_update_swap
tff(fact_2261_invar__vebt_Ointros_I1_J,axiom,
    ! [A: bool,B: bool] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,vEBT_Leaf(A,B)),aa_nat_nat(suc,zero_zero_nat))) ).

% invar_vebt.intros(1)
tff(fact_2262_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_2263_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z: rat] : plus_plus_rat(X,ord_max_rat(Y,Z)) = ord_max_rat(plus_plus_rat(X,Y),plus_plus_rat(X,Z)) ).

% max_add_distrib_right
tff(fact_2264_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_2265_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(plus_plus_int(X),aa_int_int(ord_max_int(Y),Z)) = aa_int_int(ord_max_int(aa_int_int(plus_plus_int(X),Y)),aa_int_int(plus_plus_int(X),Z)) ).

% max_add_distrib_right
tff(fact_2266_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_2267_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] : plus_plus_rat(ord_max_rat(X,Y),Z) = ord_max_rat(plus_plus_rat(X,Z),plus_plus_rat(Y,Z)) ).

% max_add_distrib_left
tff(fact_2268_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_2269_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(plus_plus_int(aa_int_int(ord_max_int(X),Y)),Z) = aa_int_int(ord_max_int(aa_int_int(plus_plus_int(X),Z)),aa_int_int(plus_plus_int(Y),Z)) ).

% max_add_distrib_left
tff(fact_2270_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_2271_max__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] : minus_minus_rat(ord_max_rat(X,Y),Z) = ord_max_rat(minus_minus_rat(X,Z),minus_minus_rat(Y,Z)) ).

% max_diff_distrib_left
tff(fact_2272_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(minus_minus_int(aa_int_int(ord_max_int(X),Y)),Z) = aa_int_int(ord_max_int(aa_int_int(minus_minus_int(X),Z)),aa_int_int(minus_minus_int(Y),Z)) ).

% max_diff_distrib_left
tff(fact_2273_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_2274_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_2275_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_2276_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_2277_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_2278_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_2279_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_2280_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_2281_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_2282_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_2283_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q2: 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),Q2)) = 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),Q2)) ).

% nat_add_max_right
tff(fact_2284_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q2: 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)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),Q2)) ).

% nat_add_max_left
tff(fact_2285_vebt__member_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_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_2286_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q2: 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)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ).

% nat_mult_max_left
tff(fact_2287_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q2: 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),Q2)) = 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),Q2)) ).

% nat_mult_max_right
tff(fact_2288_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_2289_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_2290_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_2291_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_2292_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_2293_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D22))
       => ? [E2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E2),D1))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E2),D22)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_2294_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),D1))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),D22))
       => ? [E2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E2),D1))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E2),D22)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_2295_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_2296_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_2297_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_2298_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_2299_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_2300_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_2301_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_2302_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_2303_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_complex != numera6690914467698888265omplex(N) ).

% zero_neq_numeral
tff(fact_2304_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_real != numeral_numeral_real(N) ).

% zero_neq_numeral
tff(fact_2305_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_rat != numeral_numeral_rat(N) ).

% zero_neq_numeral
tff(fact_2306_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,N) ).

% zero_neq_numeral
tff(fact_2307_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_num_int(numeral_numeral_int,N) ).

% zero_neq_numeral
tff(fact_2308_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_2309_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_2310_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_2311_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_2312_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(times_times_int(A),B) != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_2313_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_2314_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_2315_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_2316_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_2317_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(times_times_int(A),B) = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_2318_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_2319_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_2320_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_2321_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_2322_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( aa_int_int(times_times_int(A),B) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_2323_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_2324_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_2325_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2326_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_2327_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(times_times_int(C),A) = aa_int_int(times_times_int(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2328_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_2329_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_2330_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2331_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_2332_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(times_times_int(A),C) = aa_int_int(times_times_int(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2333_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_2334_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_2335_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_2336_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_2337_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_2338_add_Ogroup__left__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add.group_left_neutral
tff(fact_2339_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_2340_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : plus_plus_rat(zero_zero_rat,A) = A ).

% add.group_left_neutral
tff(fact_2341_add_Ogroup__left__neutral,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(zero_zero_int),A) = A ).

% add.group_left_neutral
tff(fact_2342_add_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.comm_neutral
tff(fact_2343_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_2344_add_Ocomm__neutral,axiom,
    ! [A: rat] : plus_plus_rat(A,zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_2345_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_2346_add_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(A),zero_zero_int) = A ).

% add.comm_neutral
tff(fact_2347_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_2348_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_2349_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : plus_plus_rat(zero_zero_rat,A) = A ).

% comm_monoid_add_class.add_0
tff(fact_2350_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_2351_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(zero_zero_int),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_2352_verit__sum__simplify,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% verit_sum_simplify
tff(fact_2353_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_2354_verit__sum__simplify,axiom,
    ! [A: rat] : plus_plus_rat(A,zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_2355_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_2356_verit__sum__simplify,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(A),zero_zero_int) = A ).

% verit_sum_simplify
tff(fact_2357_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_2358_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_2359_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( minus_minus_rat(A,B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_2360_eq__iff__diff__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_int(minus_minus_int(A),B) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_2361_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_2362_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_2363_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_2364_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_2365_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_2366_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_2367_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M3: nat] : N = aa_nat_nat(suc,M3) ) ).

% not0_implies_Suc
tff(fact_2368_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_2369_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_2370_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_2371_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_2372_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
      ( ! [X2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X2),zero_zero_nat))
     => ( ! [Y2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,zero_zero_nat),aa_nat_nat(suc,Y2)))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(suc,X2)),aa_nat_nat(suc,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ) ).

% diff_induct
tff(fact_2373_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_2374_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_2375_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa_nat_nat(suc,X22) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_2376_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_2377_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_2378_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero_nat != aa_nat_nat(suc,X22) ).

% nat.distinct(1)
tff(fact_2379_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_2380_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_2381_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_2382_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_2383_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_2384_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_2385_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_2386_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_2387_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_2388_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_2389_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_2390_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_2391_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_2392_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_2393_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_2394_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_2395_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_2396_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_2397_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_2398_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_2399_max__def__raw,axiom,
    ! [X4: extended_enat,Xa: extended_enat] :
      ( ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(X4),Xa))
       => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X4),Xa) = Xa ) )
      & ( ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(X4),Xa))
       => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X4),Xa) = X4 ) ) ) ).

% max_def_raw
tff(fact_2400_max__def__raw,axiom,
    ! [X4: set_nat,Xa: set_nat] :
      ( ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X4),Xa))
       => ( ord_max_set_nat(X4,Xa) = Xa ) )
      & ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(X4),Xa))
       => ( ord_max_set_nat(X4,Xa) = X4 ) ) ) ).

% max_def_raw
tff(fact_2401_max__def__raw,axiom,
    ! [X4: rat,Xa: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Xa))
       => ( ord_max_rat(X4,Xa) = Xa ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Xa))
       => ( ord_max_rat(X4,Xa) = X4 ) ) ) ).

% max_def_raw
tff(fact_2402_max__def__raw,axiom,
    ! [X4: num,Xa: num] :
      ( ( pp(aa_num_bool(ord_less_eq_num(X4),Xa))
       => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X4),Xa) = Xa ) )
      & ( ~ pp(aa_num_bool(ord_less_eq_num(X4),Xa))
       => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X4),Xa) = X4 ) ) ) ).

% max_def_raw
tff(fact_2403_max__def__raw,axiom,
    ! [X4: nat,Xa: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X4),Xa) = Xa ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X4),Xa) = X4 ) ) ) ).

% max_def_raw
tff(fact_2404_max__def__raw,axiom,
    ! [X4: int,Xa: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Xa))
       => ( aa_int_int(ord_max_int(X4),Xa) = Xa ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Xa))
       => ( aa_int_int(ord_max_int(X4),Xa) = X4 ) ) ) ).

% max_def_raw
tff(fact_2405_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_2406_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_2407_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_2408_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_2409_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_2410_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_2411_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_2412_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_2413_lambda__zero,axiom,
    aTP_Lamm_fb = times_times_complex(zero_zero_complex) ).

% lambda_zero
tff(fact_2414_lambda__zero,axiom,
    aTP_Lamm_fc = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_2415_lambda__zero,axiom,
    aTP_Lamm_fd = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_2416_lambda__zero,axiom,
    aTP_Lamm_fe = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_2417_lambda__zero,axiom,
    aTP_Lamm_ff = times_times_int(zero_zero_int) ).

% lambda_zero
tff(fact_2418_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X: fun_VE5569070535936029429omplex,Y: fun_VE5569070535936029429omplex] :
      ( finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aTP_Lamm_fg(I5),X)))
     => ( finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aTP_Lamm_fg(I5),Y)))
       => finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aa_fun4024526296178165685T_bool(aTP_Lamm_fh(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_2419_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_complex,Y: fun_real_complex] :
      ( finite_finite_real(collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_fi(I5),X)))
     => ( finite_finite_real(collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_fi(I5),Y)))
       => finite_finite_real(collect_real(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_fj(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_2420_sum_Ofinite__Collect__op,axiom,
    ! [I5: 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_fk(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_fk(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_fl(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_2421_sum_Ofinite__Collect__op,axiom,
    ! [I5: 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_fm(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_fm(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_fn(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_2422_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_complex_complex,Y: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_fo(I5),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_fo(I5),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_fp(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_2423_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X: fun_VEBT_VEBT_real,Y: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_fq(I5),X)))
     => ( finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_fq(I5),Y)))
       => finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aa_fun4082248197810269365T_bool(aTP_Lamm_fr(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_2424_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_real,Y: fun_real_real] :
      ( finite_finite_real(collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_fs(I5),X)))
     => ( finite_finite_real(collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_fs(I5),Y)))
       => finite_finite_real(collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_ft(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_2425_sum_Ofinite__Collect__op,axiom,
    ! [I5: 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_fu(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_fu(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_fv(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_2426_sum_Ofinite__Collect__op,axiom,
    ! [I5: 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_fw(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_fw(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_fx(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_2427_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_complex_real,Y: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_fy(I5),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_fy(I5),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_fz(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_2428_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_2429_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_2430_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_2431_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_2432_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_2433_set__update__subsetI,axiom,
    ! [Xs: list_real,A2: set_real,X: real,I: nat] :
      ( ord_less_eq_set_real(set_real2(Xs),A2)
     => ( member_real(X,A2)
       => ord_less_eq_set_real(set_real2(list_update_real(Xs,I,X)),A2) ) ) ).

% set_update_subsetI
tff(fact_2434_set__update__subsetI,axiom,
    ! [Xs: list_complex,A2: set_complex,X: complex,I: nat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Xs)),A2))
     => ( pp(aa_set_complex_bool(member_complex(X),A2))
       => pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(list_update_complex(Xs,I,X))),A2)) ) ) ).

% set_update_subsetI
tff(fact_2435_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_2436_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_2437_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,X: nat,I: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Xs)),A2))
     => ( pp(aa_set_nat_bool(member_nat(X),A2))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,X))),A2)) ) ) ).

% set_update_subsetI
tff(fact_2438_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_2439_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_2440_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),numeral_numeral_real(N))) ).

% zero_le_numeral
tff(fact_2441_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),numeral_numeral_rat(N))) ).

% zero_le_numeral
tff(fact_2442_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_2443_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_2444_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(N)),zero_zero_real)) ).

% not_numeral_le_zero
tff(fact_2445_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(N)),zero_zero_rat)) ).

% not_numeral_le_zero
tff(fact_2446_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_2447_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_2448_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_2449_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(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2450_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_2451_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(times_times_int(C),A)),aa_int_int(times_times_int(C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2452_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_2453_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(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_2454_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(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_2455_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_2456_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(times_times_rat(B),A)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2457_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_2458_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(times_times_int(B),A)),zero_zero_int)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2459_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_2460_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(times_times_rat(A),B)),zero_zero_rat)) ) ) ).

% mult_nonpos_nonneg
tff(fact_2461_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_2462_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(times_times_int(A),B)),zero_zero_int)) ) ) ).

% mult_nonpos_nonneg
tff(fact_2463_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_2464_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(times_times_rat(A),B)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos
tff(fact_2465_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_2466_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(times_times_int(A),B)),zero_zero_int)) ) ) ).

% mult_nonneg_nonpos
tff(fact_2467_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_2468_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(times_times_rat(A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_2469_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_2470_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(times_times_int(A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_2471_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_2472_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(times_times_rat(A),B)),zero_zero_rat)) ) ).

% split_mult_neg_le
tff(fact_2473_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_2474_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(times_times_int(A),B)),zero_zero_int)) ) ).

% split_mult_neg_le
tff(fact_2475_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_2476_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(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_2477_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(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_2478_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_2479_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).

% mult_right_mono
tff(fact_2480_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_2481_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(times_times_int(A),C)),aa_int_int(times_times_int(B),C))) ) ) ).

% mult_right_mono
tff(fact_2482_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_2483_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_2484_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(times_times_int(A),C)),aa_int_int(times_times_int(B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_2485_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_2486_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(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% mult_left_mono
tff(fact_2487_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_2488_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(times_times_int(C),A)),aa_int_int(times_times_int(C),B))) ) ) ).

% mult_left_mono
tff(fact_2489_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_2490_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(times_times_rat(A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_2491_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(times_times_int(A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_2492_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_2493_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(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_2494_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(times_times_int(C),A)),aa_int_int(times_times_int(C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_2495_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_2496_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(times_times_rat(A),B))) ) ).

% split_mult_pos_le
tff(fact_2497_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(times_times_int(A),B))) ) ).

% split_mult_pos_le
tff(fact_2498_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_2499_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(times_times_rat(A),A))) ).

% zero_le_square
tff(fact_2500_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(times_times_int(A),A))) ).

% zero_le_square
tff(fact_2501_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_2502_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2503_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_2504_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(times_times_int(A),C)),aa_int_int(times_times_int(B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2505_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_2506_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_mono
tff(fact_2507_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_2508_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(times_times_int(A),C)),aa_int_int(times_times_int(B),D))) ) ) ) ) ).

% mult_mono
tff(fact_2509_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_2510_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(N)),zero_zero_real)) ).

% not_numeral_less_zero
tff(fact_2511_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(N)),zero_zero_rat)) ).

% not_numeral_less_zero
tff(fact_2512_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_2513_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_2514_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),numeral_numeral_real(N))) ).

% zero_less_numeral
tff(fact_2515_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),numeral_numeral_rat(N))) ).

% zero_less_numeral
tff(fact_2516_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_2517_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_2518_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_2519_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_2520_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_2521_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_2522_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_2523_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_2524_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_2525_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_2526_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_2527_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_2528_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_2529_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_2530_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_2531_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))
       => ( ( plus_plus_rat(X,Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2532_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_2533_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(plus_plus_int(X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2534_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_2535_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))
       => ( ( plus_plus_rat(X,Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2536_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_2537_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(plus_plus_int(X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2538_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_2539_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,plus_plus_rat(A,B)),zero_zero_rat)) ) ) ).

% add_nonpos_nonpos
tff(fact_2540_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_2541_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(plus_plus_int(A),B)),zero_zero_int)) ) ) ).

% add_nonpos_nonpos
tff(fact_2542_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_2543_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),plus_plus_rat(A,B))) ) ) ).

% add_nonneg_nonneg
tff(fact_2544_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_2545_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(plus_plus_int(A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_2546_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_2547_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),plus_plus_rat(A,C))) ) ) ).

% add_increasing2
tff(fact_2548_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_2549_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(plus_plus_int(A),C))) ) ) ).

% add_increasing2
tff(fact_2550_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_2551_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,plus_plus_rat(A,C)),B)) ) ) ).

% add_decreasing2
tff(fact_2552_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_2553_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(plus_plus_int(A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_2554_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_2555_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),plus_plus_rat(A,C))) ) ) ).

% add_increasing
tff(fact_2556_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_2557_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(plus_plus_int(A),C))) ) ) ).

% add_increasing
tff(fact_2558_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_2559_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,plus_plus_rat(A,C)),B)) ) ) ).

% add_decreasing
tff(fact_2560_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_2561_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(plus_plus_int(A),C)),B)) ) ) ).

% add_decreasing
tff(fact_2562_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_2563_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(times_times_rat(A),B))) ) ) ).

% mult_neg_neg
tff(fact_2564_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(times_times_int(A),B))) ) ) ).

% mult_neg_neg
tff(fact_2565_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_2566_not__square__less__zero,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),A)),zero_zero_rat)) ).

% not_square_less_zero
tff(fact_2567_not__square__less__zero,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(A),A)),zero_zero_int)) ).

% not_square_less_zero
tff(fact_2568_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_2569_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_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_2570_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_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_2571_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_2572_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(times_times_rat(A),B)),zero_zero_rat)) ) ) ).

% mult_neg_pos
tff(fact_2573_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_2574_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(times_times_int(A),B)),zero_zero_int)) ) ) ).

% mult_neg_pos
tff(fact_2575_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_2576_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(times_times_rat(A),B)),zero_zero_rat)) ) ) ).

% mult_pos_neg
tff(fact_2577_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_2578_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(times_times_int(A),B)),zero_zero_int)) ) ) ).

% mult_pos_neg
tff(fact_2579_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_2580_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(times_times_rat(A),B))) ) ) ).

% mult_pos_pos
tff(fact_2581_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_2582_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(times_times_int(A),B))) ) ) ).

% mult_pos_pos
tff(fact_2583_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_2584_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(times_times_rat(B),A)),zero_zero_rat)) ) ) ).

% mult_pos_neg2
tff(fact_2585_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_2586_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(times_times_int(B),A)),zero_zero_int)) ) ) ).

% mult_pos_neg2
tff(fact_2587_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_2588_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(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_2589_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(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_2590_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_2591_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(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_2592_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_2593_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(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_2594_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_2595_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(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_2596_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_2597_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(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_2598_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_2599_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(times_times_rat(C),A)),aa_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_2600_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(times_times_int(C),A)),aa_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_2601_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_2602_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(times_times_rat(C),A)),aa_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_2603_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(times_times_int(C),A)),aa_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_2604_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_2605_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(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2606_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(times_times_int(C),A)),aa_int_int(times_times_int(C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2607_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_2608_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(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_2609_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_2610_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(times_times_int(C),A)),aa_int_int(times_times_int(C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_2611_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_2612_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(times_times_rat(C),A)),aa_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_2613_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(times_times_int(C),A)),aa_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_2614_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_2615_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2616_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(times_times_int(A),C)),aa_int_int(times_times_int(B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2617_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_2618_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_2619_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_2620_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(times_times_int(A),C)),aa_int_int(times_times_int(B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_2621_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_2622_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(times_times_rat(A),C)),aa_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_2623_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(times_times_int(A),C)),aa_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_2624_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_2625_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(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2626_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_2627_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(times_times_int(C),A)),aa_int_int(times_times_int(C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2628_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_2629_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,minus_minus_rat(A,B)),zero_zero_rat)) ) ).

% le_iff_diff_le_0
tff(fact_2630_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(minus_minus_int(A),B)),zero_zero_int)) ) ).

% le_iff_diff_le_0
tff(fact_2631_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_2632_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_2633_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_2634_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_2635_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_2636_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_2637_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_2638_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_2639_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_2640_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_2641_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_2642_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_2643_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_2644_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),plus_plus_rat(A,C))) ) ) ).

% pos_add_strict
tff(fact_2645_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_2646_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(plus_plus_int(A),C))) ) ) ).

% pos_add_strict
tff(fact_2647_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))
     => ~ ! [C2: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_2648_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_2649_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),plus_plus_rat(A,B))) ) ) ).

% add_pos_pos
tff(fact_2650_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_2651_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(plus_plus_int(A),B))) ) ) ).

% add_pos_pos
tff(fact_2652_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_2653_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,plus_plus_rat(A,B)),zero_zero_rat)) ) ) ).

% add_neg_neg
tff(fact_2654_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_2655_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(plus_plus_int(A),B)),zero_zero_int)) ) ) ).

% add_neg_neg
tff(fact_2656_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_2657_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_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_2658_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_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_2659_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_2660_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,divide_divide_rat(B,C)),divide_divide_rat(A,C))) ) ) ).

% divide_right_mono_neg
tff(fact_2661_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_2662_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),divide_divide_rat(X,Y))) ) ) ).

% divide_nonpos_nonpos
tff(fact_2663_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_2664_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,divide_divide_rat(X,Y)),zero_zero_rat)) ) ) ).

% divide_nonpos_nonneg
tff(fact_2665_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_2666_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,divide_divide_rat(X,Y)),zero_zero_rat)) ) ) ).

% divide_nonneg_nonpos
tff(fact_2667_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_2668_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),divide_divide_rat(X,Y))) ) ) ).

% divide_nonneg_nonneg
tff(fact_2669_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_2670_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),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_2671_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_2672_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,divide_divide_rat(A,C)),divide_divide_rat(B,C))) ) ) ).

% divide_right_mono
tff(fact_2673_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_2674_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_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_2675_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_2676_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,minus_minus_rat(A,B)),zero_zero_rat)) ) ).

% less_iff_diff_less_0
tff(fact_2677_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(minus_minus_int(A),B)),zero_zero_int)) ) ).

% less_iff_diff_less_0
tff(fact_2678_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_2679_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,divide_divide_rat(A,C)),divide_divide_rat(B,C))) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2680_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_2681_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,divide_divide_rat(A,C)),divide_divide_rat(B,C))) ) ) ).

% divide_strict_right_mono
tff(fact_2682_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_2683_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_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_2684_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_2685_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(A,C)),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_2686_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_2687_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_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_2688_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_2689_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),divide_divide_rat(X,Y))) ) ) ).

% divide_pos_pos
tff(fact_2690_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_2691_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,divide_divide_rat(X,Y)),zero_zero_rat)) ) ) ).

% divide_pos_neg
tff(fact_2692_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_2693_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,divide_divide_rat(X,Y)),zero_zero_rat)) ) ) ).

% divide_neg_pos
tff(fact_2694_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_2695_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),divide_divide_rat(X,Y))) ) ) ).

% divide_neg_neg
tff(fact_2696_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_2697_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_2698_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_2699_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_2700_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_2701_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_2702_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_2703_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_2704_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_2705_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_2706_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_2707_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_2708_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( aa_complex_complex(divide1717551699836669952omplex(X),Y) = aa_complex_complex(divide1717551699836669952omplex(W),Z) )
        <=> ( aa_complex_complex(times_times_complex(X),Z) = aa_complex_complex(times_times_complex(W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2709_frac__eq__eq,axiom,
    ! [Y: real,Z: real,X: real,W: 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,W),Z) )
        <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2710_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( divide_divide_rat(X,Y) = divide_divide_rat(W,Z) )
        <=> ( aa_rat_rat(times_times_rat(X),Z) = aa_rat_rat(times_times_rat(W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2711_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_2712_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_2713_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( divide_divide_rat(B,C) = A )
    <=> ( ( ( C != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(A),C) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
tff(fact_2714_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_2715_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_2716_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = divide_divide_rat(B,C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),C) = B ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
tff(fact_2717_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_2718_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_2719_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B = aa_rat_rat(times_times_rat(A),C) )
       => ( divide_divide_rat(B,C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2720_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_2721_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_2722_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = B )
       => ( A = divide_divide_rat(B,C) ) ) ) ).

% eq_divide_imp
tff(fact_2723_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_2724_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_2725_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat(B,C) = A )
      <=> ( B = aa_rat_rat(times_times_rat(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2726_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_2727_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_2728_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A = divide_divide_rat(B,C) )
      <=> ( aa_rat_rat(times_times_rat(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2729_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_2730_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_2731_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat(A,B) = one_one_rat )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2732_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_2733_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_2734_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_2735_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_2736_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_2737_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_2738_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_2739_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_2740_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_2741_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_2742_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_2743_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_2744_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = A )
    <=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_2745_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_2746_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_2747_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ? [M7: nat] : N = aa_nat_nat(suc,M7) ) ).

% gr0_conv_Suc
tff(fact_2748_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_2749_gr0__implies__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_implies_Suc
tff(fact_2750_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_2751_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_2752_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_2753_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))
       => ? [K3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N))
            & ! [I2: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K3))
               => ~ pp(aa_nat_bool(P,I2)) )
            & pp(aa_nat_bool(P,K3)) ) ) ) ).

% ex_least_nat_le
tff(fact_2754_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_2755_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_2756_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_2757_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3) = J ) ) ) ).

% less_imp_add_positive
tff(fact_2758_option_Osize_I3_J,axiom,
    size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_2759_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_2760_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_2761_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_2762_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_2763_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_2764_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_2765_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_2766_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_2767_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_2768_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_2769_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_2770_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_2771_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_2772_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_2773_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_2774_vebt__insert_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A4: bool,B4: bool,X2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),X2)
     => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S2),X2)
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S2),X2)
         => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2),X2)
           => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),X2) ) ) ) ) ).

% vebt_insert.cases
tff(fact_2775_vebt__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A4: bool,B4: bool,X2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),X2)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2),X2)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2),X2)
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2),X2)
           => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),X2) ) ) ) ) ).

% vebt_member.cases
tff(fact_2776_vebt__succ_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,B4: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,B4),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,Va2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2),Va2)
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2),Ve)
           => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg),Vh),Vi)
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),X2) ) ) ) ) ) ).

% vebt_succ.cases
tff(fact_2777_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(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2),Uz2)
       => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2),Vb2),X2)
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2),X2)
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2),X2) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_2778_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 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_2779_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)))
     => member_real(X,set_real2(list_update_real(Xs,N,X))) ) ).

% set_update_memI
tff(fact_2780_set__update__memI,axiom,
    ! [N: nat,Xs: list_complex,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3451745648224563538omplex(Xs)))
     => pp(aa_set_complex_bool(member_complex(X),set_complex2(list_update_complex(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_2781_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_2782_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_2783_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_2784_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_2785_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_2786_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_2787_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(nth_nat(list_update_nat(Xs,I,X)),J) = X ) )
        & ( ( I != J )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_2788_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_2789_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_2790_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_2791_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(nth_nat(Xs),I) = X ) ) ) ).

% list_update_same_conv
tff(fact_2792_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_2793_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_2794_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,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_2795_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_2796_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(times_times_rat(C),A)),aa_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_2797_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(times_times_int(C),A)),aa_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_2798_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_2799_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(times_times_rat(A),C)),aa_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_2800_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(times_times_int(A),C)),aa_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_2801_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_2802_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(times_times_rat(C),A)),aa_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_2803_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_2804_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(times_times_int(C),A)),aa_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_2805_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_2806_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2807_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_2808_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(times_times_int(A),C)),aa_int_int(times_times_int(B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2809_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_2810_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(times_times_rat(C),A)),aa_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_2811_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(times_times_int(C),A)),aa_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_2812_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_2813_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(times_times_rat(A),C)),aa_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_2814_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_2815_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(times_times_int(A),C)),aa_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_2816_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_2817_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2818_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_2819_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(times_times_int(A),C)),aa_int_int(times_times_int(B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2820_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_2821_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(times_times_rat(A),C)),aa_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_2822_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(times_times_int(A),C)),aa_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_2823_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_2824_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(times_times_rat(C),A)),aa_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_2825_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(times_times_int(C),A)),aa_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_2826_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_2827_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(times_times_rat(C),A)),aa_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_2828_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(times_times_int(C),A)),aa_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_2829_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_2830_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(times_times_rat(C),A)),aa_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_2831_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_2832_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(times_times_int(C),A)),aa_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_2833_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_2834_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(times_times_rat(A),C)),aa_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_2835_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_2836_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(times_times_int(A),C)),aa_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_2837_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_2838_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2839_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_2840_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(times_times_int(A),C)),aa_int_int(times_times_int(B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2841_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_2842_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2843_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_2844_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(times_times_int(A),C)),aa_int_int(times_times_int(B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2845_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_2846_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),plus_plus_rat(A,C))) ) ) ).

% add_strict_increasing2
tff(fact_2847_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_2848_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(plus_plus_int(A),C))) ) ) ).

% add_strict_increasing2
tff(fact_2849_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_2850_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),plus_plus_rat(A,C))) ) ) ).

% add_strict_increasing
tff(fact_2851_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_2852_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(plus_plus_int(A),C))) ) ) ).

% add_strict_increasing
tff(fact_2853_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_2854_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),plus_plus_rat(A,B))) ) ) ).

% add_pos_nonneg
tff(fact_2855_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_2856_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(plus_plus_int(A),B))) ) ) ).

% add_pos_nonneg
tff(fact_2857_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_2858_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,plus_plus_rat(A,B)),zero_zero_rat)) ) ) ).

% add_nonpos_neg
tff(fact_2859_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_2860_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(plus_plus_int(A),B)),zero_zero_int)) ) ) ).

% add_nonpos_neg
tff(fact_2861_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_2862_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),plus_plus_rat(A,B))) ) ) ).

% add_nonneg_pos
tff(fact_2863_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_2864_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(plus_plus_int(A),B))) ) ) ).

% add_nonneg_pos
tff(fact_2865_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_2866_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,plus_plus_rat(A,B)),zero_zero_rat)) ) ) ).

% add_neg_nonpos
tff(fact_2867_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_2868_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(plus_plus_int(A),B)),zero_zero_int)) ) ) ).

% add_neg_nonpos
tff(fact_2869_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% field_le_epsilon
tff(fact_2870_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),plus_plus_rat(Y,E2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% field_le_epsilon
tff(fact_2871_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_2872_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,divide_divide_rat(X,Y)),zero_zero_rat)) ) ) ).

% divide_nonpos_pos
tff(fact_2873_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_2874_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),divide_divide_rat(X,Y))) ) ) ).

% divide_nonpos_neg
tff(fact_2875_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_2876_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),divide_divide_rat(X,Y))) ) ) ).

% divide_nonneg_pos
tff(fact_2877_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_2878_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,divide_divide_rat(X,Y)),zero_zero_rat)) ) ) ).

% divide_nonneg_neg
tff(fact_2879_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_2880_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(A,C)),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_2881_frac__less2,axiom,
    ! [X: real,Y: real,W: 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),W))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W))) ) ) ) ) ).

% frac_less2
tff(fact_2882_frac__less2,axiom,
    ! [X: rat,Y: rat,W: 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),W))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W),Z))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(X,Z)),divide_divide_rat(Y,W))) ) ) ) ) ).

% frac_less2
tff(fact_2883_frac__less,axiom,
    ! [X: real,Y: real,W: 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),W))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W))) ) ) ) ) ).

% frac_less
tff(fact_2884_frac__less,axiom,
    ! [X: rat,Y: rat,W: 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),W))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W),Z))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(X,Z)),divide_divide_rat(Y,W))) ) ) ) ) ).

% frac_less
tff(fact_2885_frac__le,axiom,
    ! [Y: real,X: real,W: 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),W))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W))) ) ) ) ) ).

% frac_le
tff(fact_2886_frac__le,axiom,
    ! [Y: rat,X: rat,W: 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),W))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W),Z))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(X,Z)),divide_divide_rat(Y,W))) ) ) ) ) ).

% frac_le
tff(fact_2887_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_2888_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(aa_int_fun_int_int(divide_divide_int,A),B))) ) ) ).

% div_positive
tff(fact_2889_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_2890_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(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2891_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_2892_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(times_times_rat(A),C)),A)) ) ) ).

% mult_left_le
tff(fact_2893_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_2894_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(times_times_int(A),C)),A)) ) ) ).

% mult_left_le
tff(fact_2895_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_2896_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(times_times_rat(A),B)),one_one_rat)) ) ) ) ).

% mult_le_one
tff(fact_2897_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_2898_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(times_times_int(A),B)),one_one_int)) ) ) ) ).

% mult_le_one
tff(fact_2899_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_2900_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(times_times_rat(X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_2901_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(times_times_int(X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_2902_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_2903_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(times_times_rat(Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_2904_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(times_times_int(Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_2905_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_2906_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(X),X),aa_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_2907_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(plus_plus_int(aa_int_int(times_times_int(X),X)),aa_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_2908_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_2909_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),plus_plus_rat(aa_rat_rat(times_times_rat(X),X),aa_rat_rat(times_times_rat(Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_2910_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(plus_plus_int(aa_int_int(times_times_int(X),X)),aa_int_int(times_times_int(Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_2911_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_2912_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_2913_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_2914_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_2915_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_2916_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),plus_plus_rat(aa_rat_rat(times_times_rat(X),X),aa_rat_rat(times_times_rat(Y),Y))))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2917_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(plus_plus_int(aa_int_int(times_times_int(X),X)),aa_int_int(times_times_int(Y),Y))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2918_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_2919_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(X),X),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat)) ).

% not_sum_squares_lt_zero
tff(fact_2920_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(plus_plus_int(aa_int_int(times_times_int(X),X)),aa_int_int(times_times_int(Y),Y))),zero_zero_int)) ).

% not_sum_squares_lt_zero
tff(fact_2921_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_2922_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(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(times_times_int(B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2923_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_2924_zero__less__two,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),plus_plus_rat(one_one_rat,one_one_rat))) ).

% zero_less_two
tff(fact_2925_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_2926_zero__less__two,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(one_one_int),one_one_int))) ).

% zero_less_two
tff(fact_2927_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_2928_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(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B))) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2929_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_2930_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(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B))) ) ) ) ).

% divide_strict_left_mono
tff(fact_2931_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_2932_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(times_times_rat(Z),Y)),X))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),divide_divide_rat(X,Y))) ) ) ).

% mult_imp_less_div_pos
tff(fact_2933_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_2934_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(times_times_rat(Z),Y)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(X,Y)),Z)) ) ) ).

% mult_imp_div_pos_less
tff(fact_2935_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_2936_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),divide_divide_rat(B,C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).

% pos_less_divide_eq
tff(fact_2937_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_2938_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,divide_divide_rat(B,C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% pos_divide_less_eq
tff(fact_2939_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_2940_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),divide_divide_rat(B,C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% neg_less_divide_eq
tff(fact_2941_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_2942_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,divide_divide_rat(B,C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).

% neg_divide_less_eq
tff(fact_2943_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_2944_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),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(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(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_2945_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_2946_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_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(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(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_2947_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_2948_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_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_2949_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_2950_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_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_2951_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 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_2952_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_2953_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_2954_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_2955_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_2956_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( numera6690914467698888265omplex(W) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C) = B ) )
        & ( ( C = zero_zero_complex )
         => ( numera6690914467698888265omplex(W) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2957_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( numeral_numeral_real(W) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C) = B ) )
        & ( ( C = zero_zero_real )
         => ( numeral_numeral_real(W) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2958_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( numeral_numeral_rat(W) = divide_divide_rat(B,C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C) = B ) )
        & ( ( C = zero_zero_rat )
         => ( numeral_numeral_rat(W) = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2959_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = numera6690914467698888265omplex(W) )
    <=> ( ( ( C != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C) ) )
        & ( ( C = zero_zero_complex )
         => ( numera6690914467698888265omplex(W) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2960_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = numeral_numeral_real(W) )
    <=> ( ( ( C != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C) ) )
        & ( ( C = zero_zero_real )
         => ( numeral_numeral_real(W) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2961_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( divide_divide_rat(B,C) = numeral_numeral_rat(W) )
    <=> ( ( ( C != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C) ) )
        & ( ( C = zero_zero_rat )
         => ( numeral_numeral_rat(W) = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2962_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_2963_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_2964_divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( plus_plus_rat(divide_divide_rat(X,Z),Y) = divide_divide_rat(plus_plus_rat(X,aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2965_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_2966_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_2967_add__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( plus_plus_rat(X,divide_divide_rat(Y,Z)) = divide_divide_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),Z),Y),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2968_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_2969_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_2970_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( plus_plus_rat(Z,divide_divide_rat(X,Y)) = divide_divide_rat(plus_plus_rat(X,aa_rat_rat(times_times_rat(Z),Y)),Y) ) ) ).

% add_num_frac
tff(fact_2971_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_2972_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_2973_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( plus_plus_rat(divide_divide_rat(X,Y),Z) = divide_divide_rat(plus_plus_rat(X,aa_rat_rat(times_times_rat(Z),Y)),Y) ) ) ).

% add_frac_num
tff(fact_2974_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: 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(W),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(W),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2975_add__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W: 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,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2976_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( plus_plus_rat(divide_divide_rat(X,Y),divide_divide_rat(W,Z)) = divide_divide_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2977_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_2978_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_2979_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( plus_plus_rat(A,divide_divide_rat(B,Z)) = A ) )
      & ( ( Z != zero_zero_rat )
       => ( plus_plus_rat(A,divide_divide_rat(B,Z)) = divide_divide_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),Z),B),Z) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_2980_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_2981_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_2982_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( plus_plus_rat(divide_divide_rat(A,Z),B) = B ) )
      & ( ( Z != zero_zero_rat )
       => ( plus_plus_rat(divide_divide_rat(A,Z),B) = divide_divide_rat(plus_plus_rat(A,aa_rat_rat(times_times_rat(B),Z)),Z) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_2983_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_2984_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_2985_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_2986_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_2987_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_2988_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_2989_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_2990_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_2991_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_2992_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A),B)),B) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self2
tff(fact_2993_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_2994_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(B),A)),B) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self1
tff(fact_2995_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_2996_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_2997_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( minus_minus_rat(A,divide_divide_rat(B,Z)) = A ) )
      & ( ( Z != zero_zero_rat )
       => ( minus_minus_rat(A,divide_divide_rat(B,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),Z),B),Z) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_2998_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: 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(W),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(W),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_2999_diff__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W: 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,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3000_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( minus_minus_rat(divide_divide_rat(X,Y),divide_divide_rat(W,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3001_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_3002_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_3003_diff__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(X,divide_divide_rat(Y,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z),Y),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3004_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_3005_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_3006_divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(divide_divide_rat(X,Z),Y) = divide_divide_rat(minus_minus_rat(X,aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3007_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_3008_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_3009_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_3010_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_3011_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_3012_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_3013_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: 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,Q2))) = 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,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_3014_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: 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,Q2))) = zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_3015_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,N)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,N),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_3016_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_3017_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_3018_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,N),aa_num_Code_integer(numera6620942414471956472nteger,one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_3019_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_3020_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_3021_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))
       => ? [K3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
            & ! [I2: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),K3))
               => ~ pp(aa_nat_bool(P,I2)) )
            & pp(aa_nat_bool(P,aa_nat_nat(suc,K3))) ) ) ) ).

% ex_least_nat_less
tff(fact_3022_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_3023_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_3024_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_3025_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_3026_length__pos__if__in__set,axiom,
    ! [X: real,Xs: list_real] :
      ( 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_3027_length__pos__if__in__set,axiom,
    ! [X: complex,Xs: list_complex] :
      ( pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s3451745648224563538omplex(Xs))) ) ).

% length_pos_if_in_set
tff(fact_3028_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_3029_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_3030_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_3031_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_3032_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_3033_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_3034_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)) )
          | ? [D4: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D4) )
              & ~ pp(aa_nat_bool(P,D4)) ) ) ) ).

% nat_diff_split_asm
tff(fact_3035_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)) )
        & ! [D4: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D4) )
           => pp(aa_nat_bool(P,D4)) ) ) ) ).

% nat_diff_split
tff(fact_3036_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_3037_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_3038_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_3039_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_3040_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q2)),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),Q2))) ) ) ).

% div_less_iff_less_mult
tff(fact_3041_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_3042_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_3043_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_3044_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_3045_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_3046_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz)),X)) ).

% vebt_member.simps(3)
tff(fact_3047_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy)),Uz)) ).

% VEBT_internal.membermima.simps(2)
tff(fact_3048_vebt__mint_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A4: bool,B4: bool] : X != vEBT_Leaf(A4,B4)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2) ) ) ).

% vebt_mint.cases
tff(fact_3049_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 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)
               => ~ pp(Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2)
                 => pp(Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_3050_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z4))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z4),X)),Y)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% field_le_mult_one_interval
tff(fact_3051_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z4))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),one_one_rat))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z4),X)),Y)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% field_le_mult_one_interval
tff(fact_3052_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_3053_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(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_3054_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(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_3055_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_3056_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(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_3057_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(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_3058_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_3059_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(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_3060_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(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_3061_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_3062_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(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_3063_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(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_3064_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_3065_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(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_3066_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(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_3067_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_3068_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_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_3069_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_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_3070_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_3071_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(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_3072_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(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_3073_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_3074_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_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_3075_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_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_3076_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_3077_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(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B))) ) ) ) ).

% divide_left_mono_neg
tff(fact_3078_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_3079_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(times_times_rat(Z),Y)),X))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),divide_divide_rat(X,Y))) ) ) ).

% mult_imp_le_div_pos
tff(fact_3080_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_3081_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(times_times_rat(Z),Y)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(X,Y)),Z)) ) ) ).

% mult_imp_div_pos_le
tff(fact_3082_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_3083_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),divide_divide_rat(B,C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).

% pos_le_divide_eq
tff(fact_3084_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_3085_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,divide_divide_rat(B,C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% pos_divide_le_eq
tff(fact_3086_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_3087_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),divide_divide_rat(B,C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% neg_le_divide_eq
tff(fact_3088_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_3089_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,divide_divide_rat(B,C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).

% neg_divide_le_eq
tff(fact_3090_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_3091_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(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B))) ) ) ) ).

% divide_left_mono
tff(fact_3092_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_3093_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),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(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(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_3094_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_3095_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_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(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(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_3096_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_3097_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),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_3098_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_3099_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_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_3100_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_3101_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))
           => ( ( plus_plus_rat(U,V) = one_one_rat )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(U),X),aa_rat_rat(times_times_rat(V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3102_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(plus_plus_int(U),V) = one_one_int )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(U),X)),aa_int_int(times_times_int(V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3103_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(W)),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,numeral_numeral_real(W)),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,numeral_numeral_real(W)),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,numeral_numeral_real(W)),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_3104_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(W)),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(times_times_rat(numeral_numeral_rat(W)),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(times_times_rat(numeral_numeral_rat(W)),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,numeral_numeral_rat(W)),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_3105_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),numeral_numeral_real(W)))
    <=> ( ( 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,numeral_numeral_real(W)),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,numeral_numeral_real(W)),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),numeral_numeral_real(W))) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_3106_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,C)),numeral_numeral_rat(W)))
    <=> ( ( 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(times_times_rat(numeral_numeral_rat(W)),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(times_times_rat(numeral_numeral_rat(W)),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),numeral_numeral_rat(W))) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_3107_frac__le__eq,axiom,
    ! [Y: real,Z: real,X: real,W: 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,W),Z)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real)) ) ) ) ).

% frac_le_eq
tff(fact_3108_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(X,Y)),divide_divide_rat(W,Z)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat)) ) ) ) ).

% frac_le_eq
tff(fact_3109_frac__less__eq,axiom,
    ! [Y: real,Z: real,X: real,W: 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,W),Z)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real)) ) ) ) ).

% frac_less_eq
tff(fact_3110_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(X,Y)),divide_divide_rat(W,Z)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat)) ) ) ) ).

% frac_less_eq
tff(fact_3111_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_3112_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(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_3113_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_3114_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(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_3115_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_3116_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_3117_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_3118_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_3119_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_3120_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_3121_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_3122_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_3123_power__strict__decreasing,axiom,
    ! [N: nat,N3: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
     => ( 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),N3)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_3124_power__strict__decreasing,axiom,
    ! [N: nat,N3: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
     => ( 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),N3)),aa_nat_rat(power_power_rat(A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_3125_power__strict__decreasing,axiom,
    ! [N: nat,N3: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
     => ( 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),N3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_3126_power__strict__decreasing,axiom,
    ! [N: nat,N3: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
     => ( 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),N3)),aa_nat_int(power_power_int(A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_3127_power__decreasing,axiom,
    ! [N: nat,N3: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
     => ( 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),N3)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ) ).

% power_decreasing
tff(fact_3128_power__decreasing,axiom,
    ! [N: nat,N3: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
     => ( 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),N3)),aa_nat_rat(power_power_rat(A),N))) ) ) ) ).

% power_decreasing
tff(fact_3129_power__decreasing,axiom,
    ! [N: nat,N3: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
     => ( 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),N3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ) ).

% power_decreasing
tff(fact_3130_power__decreasing,axiom,
    ! [N: nat,N3: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
     => ( 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),N3)),aa_nat_int(power_power_int(A),N))) ) ) ) ).

% power_decreasing
tff(fact_3131_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_3132_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_3133_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_3134_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_3135_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_3136_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(X) = Y )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ~ ( ( pp(B4)
                 => ( Y = some_nat(one_one_nat) ) )
                & ( ~ pp(B4)
                 => ( ( pp(A4)
                     => ( Y = some_nat(zero_zero_nat) ) )
                    & ( ~ pp(A4)
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2)
               => ( Y != some_nat(Ma2) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_3137_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(X) = Y )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ~ ( ( pp(A4)
                 => ( Y = some_nat(zero_zero_nat) ) )
                & ( ~ pp(A4)
                 => ( ( pp(B4)
                     => ( Y = some_nat(one_one_nat) ) )
                    & ( ~ pp(B4)
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2)
               => ( Y != some_nat(Mi2) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_3138_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa2) = Y )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => ( Y = vEBT_Leaf(fTrue,B4) ) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => ( Y = vEBT_Leaf(A4,fTrue) ) )
                    & ( ( Xa2 != one_one_nat )
                     => ( Y = vEBT_Leaf(A4,B4) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
              ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S2) )
             => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S2) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S2) )
               => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S2) ) )
           => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) )
                 => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),Xa2)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                   => ( Y != aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2)))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_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),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,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),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary2))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2)) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_3139_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_3140_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_3141_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_3142_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_3143_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_3144_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_3145_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_3146_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_3147_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_3148_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_3149_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_3150_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_3151_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)) = divide_divide_rat(aa_nat_rat(power_power_rat(A),M),aa_nat_rat(power_power_rat(A),N)) ) ) ) ).

% power_diff
tff(fact_3152_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_3153_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(aa_int_fun_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_3154_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_3155_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_3156_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_3157_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_3158_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_3159_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q2))
     => ( 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),Q2)))
      <=> 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),Q2)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_3160_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_3161_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_3162_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_3163_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_3164_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_3165_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb),Vc)),X)) ).

% vebt_member.simps(4)
tff(fact_3166_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va),Vb)),X))
    <=> ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_3167_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve2: nat] : vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc),Vd),Ve2) = none_nat ).

% vebt_succ.simps(4)
tff(fact_3168_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_3169_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))
           => ( ( plus_plus_rat(U,V) = one_one_rat )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(U),X),aa_rat_rat(times_times_rat(V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3170_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(plus_plus_int(U),V) = one_one_int )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(U),X)),aa_int_int(times_times_int(V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3171_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(W)),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,numeral_numeral_real(W)),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,numeral_numeral_real(W)),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,numeral_numeral_real(W)),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3172_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(W)),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(times_times_rat(numeral_numeral_rat(W)),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(times_times_rat(numeral_numeral_rat(W)),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,numeral_numeral_rat(W)),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3173_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),numeral_numeral_real(W)))
    <=> ( ( 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,numeral_numeral_real(W)),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,numeral_numeral_real(W)),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),numeral_numeral_real(W))) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3174_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,C)),numeral_numeral_rat(W)))
    <=> ( ( 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(times_times_rat(numeral_numeral_rat(W)),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(times_times_rat(numeral_numeral_rat(W)),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),numeral_numeral_rat(W))) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3175_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),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_3176_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),divide_divide_rat(A,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_3177_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),numeral_numeral_real(aa_num_num(bit0,one))))) ) ).

% half_gt_zero
tff(fact_3178_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),divide_divide_rat(A,numeral_numeral_rat(aa_num_num(bit0,one))))) ) ).

% half_gt_zero
tff(fact_3179_scaling__mono,axiom,
    ! [U: real,V: real,R: 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),R))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R),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,R),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,V),U))),S))),V)) ) ) ) ).

% scaling_mono
tff(fact_3180_scaling__mono,axiom,
    ! [U: rat,V: rat,R: 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),R))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,R),S))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(U,divide_divide_rat(aa_rat_rat(times_times_rat(R),minus_minus_rat(V,U)),S))),V)) ) ) ) ).

% scaling_mono
tff(fact_3181_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_3182_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_3183_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_3184_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_3185_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_3186_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_3187_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_3188_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_3189_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_3190_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_3191_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_3192_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_3193_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_3194_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_3195_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_3196_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_3197_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(times_times_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_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_3198_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_3199_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_3200_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_3201_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_3202_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_3203_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_3204_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_3205_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(aa_int_fun_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(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) = aa_int_int(aa_int_fun_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_3206_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_3207_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_3208_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_3209_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_3210_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_3211_power__eq__if,axiom,
    ! [M: nat,P2: rat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(P2),M) = one_one_rat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(P2),M) = aa_rat_rat(times_times_rat(P2),aa_nat_rat(power_power_rat(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_3212_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_3213_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(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_3214_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_3215_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_3216_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(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_3217_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_3218_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(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_3219_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_3220_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)) )
        | ? [Q4: 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))))
            & pp(aa_nat_bool(P,Q4)) ) ) ) ).

% split_div'
tff(fact_3221_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_3222_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2),Vi2) = none_nat ).

% vebt_succ.simps(5)
tff(fact_3223_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_3224_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_3225_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_3226_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_3227_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_3228_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_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_3229_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(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_3230_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_3231_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),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_3232_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(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_3233_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_3234_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),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_3235_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(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_3236_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_3237_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_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_3238_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(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_3239_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_3240_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(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(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_3241_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,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),B))
       => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)) = modulo364778990260209775nteger(A,B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_3242_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_3243_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_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_3244_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A )
     => ( plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_3245_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_3246_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_3247_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_3248_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_3249_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_3250_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_3251_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_3252_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X),Xa2))
      <=> pp(Y) )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa2 = zero_zero_nat )
                   => pp(A4) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B4) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2)
           => pp(Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S2)
               => ( 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(aa_VEB4832260192069861672t_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_3253_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X),Xa2))
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => pp(A4) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => pp(B4) )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S2)
             => ~ ( ( 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(aa_VEB4832260192069861672t_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_3254_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X),Xa2))
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ( ( ( Xa2 = zero_zero_nat )
               => pp(A4) )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => pp(B4) )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S2)
               => ( ( 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(aa_VEB4832260192069861672t_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_3255_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_3256_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(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B))
       => ( aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_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(divide_divide_int,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3257_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,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),B))
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))) = divide6298287555418463151nteger(A,B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3258_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_3259_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_3260_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_3261_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_3262_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_3263_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_3264_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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)))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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))))))),aa_VEB5832953974749077948T_VEBT(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))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)) ).

% vebt_insert.simps(5)
tff(fact_3265_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_3266_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_3267_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X),Xa2))
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => pp(A4) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => pp(B4) )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2)
             => ~ ( ( Xa2 != Mi2 )
                 => ( ( Xa2 != Ma2 )
                   => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,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,Va3))),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_3268_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X),Xa2))
     => ( ! [Uu2: bool,Uv2: bool] : X != vEBT_Leaf(Uu2,Uv2)
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2),Vb2)
               => ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)
                 => ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 )
                    | ( ( 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(aa_VEB4832260192069861672t_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 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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(aa_VEB4832260192069861672t_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_3269_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_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 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)
           => pp(Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2),Vb2)
               => ( pp(Y)
                <=> ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)
                 => ( pp(Y)
                  <=> ~ ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( 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(aa_VEB4832260192069861672t_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 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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_3270_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,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) = modulo364778990260209775nteger(X,M) )
          | ( modulo364778990260209775nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(X,M),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_3271_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_3272_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(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(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_3273_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,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))))
         => ( minus_8373710615458151222nteger(modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)),B) = modulo364778990260209775nteger(A,B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_3274_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_3275_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(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))))
         => ( aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(A),aa_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_3276_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X),Xa2))
      <=> pp(Y) )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa2 = zero_zero_nat )
                   => pp(A4) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B4) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => pp(Y) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(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 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)
               => pp(Y) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2)
                   => ( pp(Y)
                    <=> ~ ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,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,Va3))),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_3277_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X),Xa2))
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ( ( ( Xa2 = zero_zero_nat )
               => pp(A4) )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => pp(B4) )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2)
                   => ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,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,Va3))),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_3278_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)),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ).

% arith_geo_mean
tff(fact_3279_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(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),divide_divide_rat(plus_plus_rat(X,Y),numeral_numeral_rat(aa_num_num(bit0,one))))) ) ) ) ).

% arith_geo_mean
tff(fact_3280_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,A1),A22))
     => ( ( ? [A4: bool,B4: bool] : A1 = vEBT_Leaf(A4,B4)
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
              ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
             => ( ( A22 = Deg2 )
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X4),N2)) )
                 => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary2),M3))
                   => ( ( 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))),M3) )
                     => ( ( M3 = N2 )
                       => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
                         => ( ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_13))
                           => ~ ! [X4: vEBT_VEBT] :
                                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                                 => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_13)) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
                ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
               => ( ( A22 = Deg2 )
                 => ( ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X4),N2)) )
                   => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary2),M3))
                     => ( ( 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))),M3) )
                       => ( ( M3 = aa_nat_nat(suc,N2) )
                         => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
                           => ( ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_13))
                             => ~ ! [X4: vEBT_VEBT] :
                                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                                   => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_13)) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2),Summary2) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X4: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X4),N2)) )
                     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary2),M3))
                       => ( ( 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))),M3) )
                         => ( ( M3 = N2 )
                           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
                             => ( ! [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))),M3)))
                                   => ( ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),X_12))
                                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I2)) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X4: vEBT_VEBT] :
                                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                                       => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_13)) ) )
                                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma2))
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [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))),M3)))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N2) = I2 )
                                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N2))) )
                                                & ! [X4: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N2) = I2 )
                                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N2))) )
                                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X4))
                                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2),Summary2) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X4: vEBT_VEBT] :
                            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X4),N2)) )
                       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary2),M3))
                         => ( ( 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))),M3) )
                           => ( ( M3 = aa_nat_nat(suc,N2) )
                             => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
                               => ( ! [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))),M3)))
                                     => ( ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),X_12))
                                      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I2)) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X4: vEBT_VEBT] :
                                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                                         => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_13)) ) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma2))
                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [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))),M3)))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N2) = I2 )
                                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N2))) )
                                                  & ! [X4: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N2) = I2 )
                                                        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N2))) )
                                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X4))
                                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_3281_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_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,N5: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3),Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X3),N5)) )
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary3),N5))
            & ( 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))),N5) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N5),N5) )
            & ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_12))
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_12)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N5: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3),Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X3),N5)) )
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary3),aa_nat_nat(suc,N5)))
            & ( 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,N5)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N5),aa_nat_nat(suc,N5)) )
            & ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_12))
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_12)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N5: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3),Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X3),N5)) )
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary3),N5))
            & ( 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))),N5) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N5),N5) )
            & ! [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))),N5)))
               => ( ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),X_12))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I4)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
                 => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),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))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N5) = 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,Ma3),N5))) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N5) = 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),N5))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X3))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma3)) ) ) ) ) ) )
        | ? [TreeList3: list_VEBT_VEBT,N5: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3),Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X3),N5)) )
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary3),aa_nat_nat(suc,N5)))
            & ( 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,N5)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N5),aa_nat_nat(suc,N5)) )
            & ! [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,N5))))
               => ( ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),X_12))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I4)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
                 => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),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))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N5))))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N5) = 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,Ma3),N5))) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N5) = 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),N5))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X3))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma3)) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_3282_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,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))))
         => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),one_one_Code_integer) = divide6298287555418463151nteger(A,B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3283_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_3284_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(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))))
         => ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))),one_one_int) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3285_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(vEBT_VEBT_set_vebt(T)),aa_nat_set_nat(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_3286_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_ga(N),C)))) ) ).

% finite_nth_roots
tff(fact_3287_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,B4: bool] :
              ( ( X = vEBT_Leaf(Uu2,B4) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( ( pp(B4)
                     => ( Y = some_nat(one_one_nat) ) )
                    & ( ~ pp(B4)
                     => ( Y = none_nat ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,B4),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 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2) )
                 => ( ( Y = none_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2),Xa2)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2),Xa2)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg),Vh) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg),Vh),Xa2)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                       => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                             => ( Y = some_nat(Mi2) ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                             => ( 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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(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,Va3))),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_succ(Summary2,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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa2)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_3288_max__less__iff__conj,axiom,
    ! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X),Y)),Z))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(X),Z))
        & pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_3289_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_3290_max__less__iff__conj,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_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_3291_max__less__iff__conj,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(ord_less_num(aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y)),Z))
    <=> ( pp(aa_num_bool(ord_less_num(X),Z))
        & pp(aa_num_bool(ord_less_num(Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_3292_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_3293_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_3294_max_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(A),B))
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = B ) ) ).

% max.absorb4
tff(fact_3295_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_3296_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ord_max_rat(A,B) = B ) ) ).

% max.absorb4
tff(fact_3297_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( pp(aa_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_3298_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_3299_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_3300_max_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B),A))
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = A ) ) ).

% max.absorb3
tff(fact_3301_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_3302_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( ord_max_rat(A,B) = A ) ) ).

% max.absorb3
tff(fact_3303_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( pp(aa_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_3304_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_3305_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_3306_max_Oabsorb1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = A ) ) ).

% max.absorb1
tff(fact_3307_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( ord_max_rat(A,B) = A ) ) ).

% max.absorb1
tff(fact_3308_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( pp(aa_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_3309_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_3310_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_3311_max_Oabsorb2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),B))
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = B ) ) ).

% max.absorb2
tff(fact_3312_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ord_max_rat(A,B) = B ) ) ).

% max.absorb2
tff(fact_3313_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( pp(aa_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_3314_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_3315_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_3316_max_Obounded__iff,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)),A))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
        & pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),A)) ) ) ).

% max.bounded_iff
tff(fact_3317_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_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_3318_max_Obounded__iff,axiom,
    ! [B: num,C: num,A: num] :
      ( pp(aa_num_bool(ord_less_eq_num(aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A))
    <=> ( pp(aa_num_bool(ord_less_eq_num(B),A))
        & pp(aa_num_bool(ord_less_eq_num(C),A)) ) ) ).

% max.bounded_iff
tff(fact_3319_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_3320_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_3321_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_3322_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(aa_nat_fun_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_3323_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_3324_idiff__0,axiom,
    ! [N: extended_enat] : minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat,N) = zero_z5237406670263579293d_enat ).

% idiff_0
tff(fact_3325_idiff__0__right,axiom,
    ! [N: extended_enat] : minus_3235023915231533773d_enat(N,zero_z5237406670263579293d_enat) = N ).

% idiff_0_right
tff(fact_3326_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_3327_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_3328_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(aa_nat_fun_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_3329_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% set_bit_nonnegative_int_iff
tff(fact_3330_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(aa_nat_fun_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_3331_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_3332_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),K))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_3333_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_3334_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_3335_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% half_nonnegative_int_iff
tff(fact_3336_half__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% half_negative_int_iff
tff(fact_3337_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_3338_times__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(times_times_int(zero_zero_int),L) = zero_zero_int ).

% times_int_code(2)
tff(fact_3339_times__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(times_times_int(K),zero_zero_int) = zero_zero_int ).

% times_int_code(1)
tff(fact_3340_plus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(plus_plus_int(zero_zero_int),L) = L ).

% plus_int_code(2)
tff(fact_3341_plus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(plus_plus_int(K),zero_zero_int) = K ).

% plus_int_code(1)
tff(fact_3342_minus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(minus_minus_int(K),zero_zero_int) = K ).

% minus_int_code(1)
tff(fact_3343_not__iless0,axiom,
    ! [N: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(N),zero_z5237406670263579293d_enat)) ).

% not_iless0
tff(fact_3344_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_3345_iadd__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( plus_p3455044024723400733d_enat(M,N) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        & ( N = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
tff(fact_3346_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(N),zero_z5237406670263579293d_enat))
    <=> ( N = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_3347_i0__lb,axiom,
    ! [N: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),N)) ).

% i0_lb
tff(fact_3348_all__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M7: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N))
         => pp(aa_nat_bool(P,M7)) )
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
         => pp(aa_nat_bool(P,X3)) ) ) ).

% all_nat_less
tff(fact_3349_ex__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M7: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N))
          & pp(aa_nat_bool(P,M7)) )
    <=> ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
          & pp(aa_nat_bool(P,X3)) ) ) ).

% ex_nat_less
tff(fact_3350_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(times_times_int(K),I)),aa_int_int(times_times_int(K),J))) ) ) ).

% zmult_zless_mono2
tff(fact_3351_odd__nonzero,axiom,
    ! [Z: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(one_one_int),Z)),Z) != zero_zero_int ).

% odd_nonzero
tff(fact_3352_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(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)) ) ) ).

% div_neg_pos_less0
tff(fact_3353_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(aa_int_fun_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_3354_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(aa_int_fun_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_3355_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),M))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(M),K)),M)) ) ).

% zmod_le_nonneg_dividend
tff(fact_3356_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_3357_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_3358_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(modulo_modulo_int(M),D) = zero_zero_int )
    <=> ? [Q4: int] : M = aa_int_int(times_times_int(D),Q4) ) ).

% zmod_eq_0_iff
tff(fact_3359_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(modulo_modulo_int(M),D) = zero_zero_int )
     => ? [Q3: int] : M = aa_int_int(times_times_int(D),Q3) ) ).

% zmod_eq_0D
tff(fact_3360_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_3361_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_3362_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_3363_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(times_times_int(M),N) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_3364_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(aa_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_3365_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(aa_int_fun_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_3366_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(aa_int_fun_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_3367_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(aa_int_fun_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_3368_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I)) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_3369_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(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_3370_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(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_3371_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L))) ) ) ).

% div_positive_int
tff(fact_3372_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)))
    <=> ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ) ).

% div_int_pos_iff
tff(fact_3373_zdiv__mono2__neg,axiom,
    ! [A: int,B3: 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),B3))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B3)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) ) ) ) ).

% zdiv_mono2_neg
tff(fact_3374_zdiv__mono1__neg,axiom,
    ! [A: int,A3: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A3))
     => ( 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(aa_int_fun_int_int(divide_divide_int,A3),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) ) ) ).

% zdiv_mono1_neg
tff(fact_3375_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I)) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_3376_zdiv__mono2,axiom,
    ! [A: int,B3: 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),B3))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B3))) ) ) ) ).

% zdiv_mono2
tff(fact_3377_zdiv__mono1,axiom,
    ! [A: int,A3: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A3))
     => ( 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(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),B))) ) ) ).

% zdiv_mono1
tff(fact_3378_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(aa_int_fun_int_int(divide_divide_int,X),K)),X)) ) ) ).

% int_div_less_self
tff(fact_3379_zdiv__zmult2__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(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(times_times_int(B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_3380_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_3381_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_3382_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_3383_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_3384_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_3385_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(aa_int_fun_int_int(divide_divide_int,A2),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),N))) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_3386_not__exp__less__eq__0__int,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),zero_zero_int)) ).

% not_exp_less_eq_0_int
tff(fact_3387_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))
       => ? [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),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 = X2 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_3388_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_3389_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(plus_plus_int(one_one_int),Z))) ) ).

% le_imp_0_less
tff(fact_3390_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q5: int,R4: int,Q2: int,R: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q5)),R4)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R))
         => ( 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,Q2),Q5)) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_3391_unique__quotient__lemma,axiom,
    ! [B: int,Q5: int,R4: int,Q2: int,R: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q5)),R4)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R)))
     => ( 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,R),B))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q5),Q2)) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_3392_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q2: int,R: int,B3: int,Q5: int,R4: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B3),Q5)),R4) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(B3),Q5)),R4)),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R),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),B3))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q5),Q2)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_3393_zdiv__mono2__lemma,axiom,
    ! [B: int,Q2: int,R: int,B3: int,Q5: int,R4: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B3),Q5)),R4) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B3),Q5)),R4)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B3))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R))
           => ( 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(ord_less_eq_int,B3),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q2),Q5)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_3394_q__pos__lemma,axiom,
    ! [B3: int,Q5: int,R4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B3),Q5)),R4)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),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(ord_less_eq_int,zero_zero_int),Q5)) ) ) ) ).

% q_pos_lemma
tff(fact_3395_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(plus_plus_int(K),L)),zero_zero_int))
       => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(plus_plus_int(K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_3396_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(minus_minus_int(K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_3397_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R: int] :
      ( ( A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R),B))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_3398_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R: int] :
      ( ( A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_3399_split__zdiv,axiom,
    ! [P: fun_int_bool,N: int,K: int] :
      ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)))
    <=> ( ( ( K = zero_zero_int )
         => pp(aa_int_bool(P,zero_zero_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
         => ! [I4: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
                & ( N = aa_int_int(plus_plus_int(aa_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(plus_plus_int(aa_int_int(times_times_int(K),I4)),J3) ) )
             => pp(aa_int_bool(P,I4)) ) ) ) ) ).

% split_zdiv
tff(fact_3400_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R: int] :
      ( ( A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R),B))
         => ( aa_int_int(modulo_modulo_int(A),B) = R ) ) ) ) ).

% int_mod_pos_eq
tff(fact_3401_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R: int] :
      ( ( A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R))
         => ( aa_int_int(modulo_modulo_int(A),B) = R ) ) ) ) ).

% int_mod_neg_eq
tff(fact_3402_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(plus_plus_int(aa_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(plus_plus_int(aa_int_int(times_times_int(K),I4)),J3) ) )
             => pp(aa_int_bool(P,J3)) ) ) ) ) ).

% split_zmod
tff(fact_3403_zmod__zmult2__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(times_times_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_3404_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(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_3405_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(minus_minus_int(K),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_3406_max_Omono,axiom,
    ! [C: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),A))
     => ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(D),B))
       => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,C),D)),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ) ) ).

% max.mono
tff(fact_3407_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,ord_max_rat(C,D)),ord_max_rat(A,B))) ) ) ).

% max.mono
tff(fact_3408_max_Omono,axiom,
    ! [C: num,A: num,D: num,B: num] :
      ( pp(aa_num_bool(ord_less_eq_num(C),A))
     => ( pp(aa_num_bool(ord_less_eq_num(D),B))
       => pp(aa_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_3409_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_3410_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_3411_max_OorderE,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
     => ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) ) ) ).

% max.orderE
tff(fact_3412_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( A = ord_max_rat(A,B) ) ) ).

% max.orderE
tff(fact_3413_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( pp(aa_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_3414_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_3415_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_3416_max_OorderI,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) )
     => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A)) ) ).

% max.orderI
tff(fact_3417_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = ord_max_rat(A,B) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% max.orderI
tff(fact_3418_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(ord_less_eq_num(B),A)) ) ).

% max.orderI
tff(fact_3419_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_3420_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_3421_max_OboundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)),A))
     => ~ ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
         => ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),A)) ) ) ).

% max.boundedE
tff(fact_3422_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_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_3423_max_OboundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( pp(aa_num_bool(ord_less_eq_num(aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A))
     => ~ ( pp(aa_num_bool(ord_less_eq_num(B),A))
         => ~ pp(aa_num_bool(ord_less_eq_num(C),A)) ) ) ).

% max.boundedE
tff(fact_3424_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_3425_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_3426_max_OboundedI,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
     => ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),A))
       => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)),A)) ) ) ).

% max.boundedI
tff(fact_3427_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,ord_max_rat(B,C)),A)) ) ) ).

% max.boundedI
tff(fact_3428_max_OboundedI,axiom,
    ! [B: num,A: num,C: num] :
      ( pp(aa_num_bool(ord_less_eq_num(B),A))
     => ( pp(aa_num_bool(ord_less_eq_num(C),A))
       => pp(aa_num_bool(ord_less_eq_num(aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)) ) ) ).

% max.boundedI
tff(fact_3429_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_3430_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_3431_max_Oorder__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
    <=> ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) ) ) ).

% max.order_iff
tff(fact_3432_max_Oorder__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
    <=> ( A = ord_max_rat(A,B) ) ) ).

% max.order_iff
tff(fact_3433_max_Oorder__iff,axiom,
    ! [B: num,A: num] :
      ( pp(aa_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_3434_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_3435_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_3436_max_Ocobounded1,axiom,
    ! [A: extended_enat,B: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ).

% max.cobounded1
tff(fact_3437_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),ord_max_rat(A,B))) ).

% max.cobounded1
tff(fact_3438_max_Ocobounded1,axiom,
    ! [A: num,B: num] : pp(aa_num_bool(ord_less_eq_num(A),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ).

% max.cobounded1
tff(fact_3439_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_3440_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_3441_max_Ocobounded2,axiom,
    ! [B: extended_enat,A: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ).

% max.cobounded2
tff(fact_3442_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),ord_max_rat(A,B))) ).

% max.cobounded2
tff(fact_3443_max_Ocobounded2,axiom,
    ! [B: num,A: num] : pp(aa_num_bool(ord_less_eq_num(B),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ).

% max.cobounded2
tff(fact_3444_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_3445_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_3446_le__max__iff__disj,axiom,
    ! [Z: extended_enat,X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Z),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X),Y)))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Z),X))
        | pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_3447_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),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_3448_le__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( pp(aa_num_bool(ord_less_eq_num(Z),aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y)))
    <=> ( pp(aa_num_bool(ord_less_eq_num(Z),X))
        | pp(aa_num_bool(ord_less_eq_num(Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_3449_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_3450_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_3451_max_Oabsorb__iff1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
    <=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_3452_max_Oabsorb__iff1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
    <=> ( ord_max_rat(A,B) = A ) ) ).

% max.absorb_iff1
tff(fact_3453_max_Oabsorb__iff1,axiom,
    ! [B: num,A: num] :
      ( pp(aa_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_3454_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_3455_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_3456_max_Oabsorb__iff2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),B))
    <=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_3457_max_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> ( ord_max_rat(A,B) = B ) ) ).

% max.absorb_iff2
tff(fact_3458_max_Oabsorb__iff2,axiom,
    ! [A: num,B: num] :
      ( pp(aa_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_3459_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_3460_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_3461_max_OcoboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),A))
     => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ) ).

% max.coboundedI1
tff(fact_3462_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),ord_max_rat(A,B))) ) ).

% max.coboundedI1
tff(fact_3463_max_OcoboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( pp(aa_num_bool(ord_less_eq_num(C),A))
     => pp(aa_num_bool(ord_less_eq_num(C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).

% max.coboundedI1
tff(fact_3464_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_3465_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_3466_max_OcoboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),B))
     => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ) ).

% max.coboundedI2
tff(fact_3467_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),ord_max_rat(A,B))) ) ).

% max.coboundedI2
tff(fact_3468_max_OcoboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( pp(aa_num_bool(ord_less_eq_num(C),B))
     => pp(aa_num_bool(ord_less_eq_num(C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).

% max.coboundedI2
tff(fact_3469_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_3470_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_3471_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(plus_plus_int(aa_int_int(aa_int_fun_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(aa_int_fun_int_int(divide_divide_int,B2),N))) ) ) ).

% verit_le_mono_div_int
tff(fact_3472_less__max__iff__disj,axiom,
    ! [Z: extended_enat,X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Z),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X),Y)))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Z),X))
        | pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_3473_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_3474_less__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),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_3475_less__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( pp(aa_num_bool(ord_less_num(Z),aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y)))
    <=> ( pp(aa_num_bool(ord_less_num(Z),X))
        | pp(aa_num_bool(ord_less_num(Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_3476_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_3477_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_3478_max_Ostrict__boundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)),A))
     => ~ ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B),A))
         => ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C),A)) ) ) ).

% max.strict_boundedE
tff(fact_3479_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_3480_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_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_3481_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( pp(aa_num_bool(ord_less_num(aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A))
     => ~ ( pp(aa_num_bool(ord_less_num(B),A))
         => ~ pp(aa_num_bool(ord_less_num(C),A)) ) ) ).

% max.strict_boundedE
tff(fact_3482_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_3483_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_3484_max_Ostrict__order__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B),A))
    <=> ( ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_3485_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_3486_max_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
    <=> ( ( A = ord_max_rat(A,B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_3487_max_Ostrict__order__iff,axiom,
    ! [B: num,A: num] :
      ( pp(aa_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_3488_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_3489_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_3490_max_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C),A))
     => pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_3491_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_3492_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),ord_max_rat(A,B))) ) ).

% max.strict_coboundedI1
tff(fact_3493_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( pp(aa_num_bool(ord_less_num(C),A))
     => pp(aa_num_bool(ord_less_num(C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_3494_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_3495_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_3496_max_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C),B))
     => pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_3497_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_3498_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),ord_max_rat(A,B))) ) ).

% max.strict_coboundedI2
tff(fact_3499_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( pp(aa_num_bool(ord_less_num(C),B))
     => pp(aa_num_bool(ord_less_num(C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_3500_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_3501_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_3502_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_bool,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)),aa_int_int(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(plus_plus_int(aa_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_3503_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_bool,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)),aa_int_int(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(plus_plus_int(aa_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_3504_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_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(divide_divide_int,aa_int_int(plus_plus_int(B),one_one_int)),A) ) ) ).

% neg_zdiv_mult_2
tff(fact_3505_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_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(divide_divide_int,B),A) ) ) ).

% pos_zdiv_mult_2
tff(fact_3506_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(B),A))) ) ) ).

% pos_zmod_mult_2
tff(fact_3507_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(B),one_one_int)),A))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_3508_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))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( ( ( Xa2 = zero_zero_nat )
                   => ( Y = vEBT_Leaf(fTrue,B4) ) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => ( Y = vEBT_Leaf(A4,fTrue) ) )
                      & ( ( Xa2 != one_one_nat )
                       => ( Y = vEBT_Leaf(A4,B4) ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa2)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S2) )
               => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S2),Xa2)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S2) )
                 => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S2),Xa2)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) )
                   => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),Xa2)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2),Xa2)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                     => ( ( Y = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2)))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_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),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,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),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary2))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2)) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_3509_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa2))
               => ( ( ( Xa2 = zero_zero_nat )
                   => pp(A4) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B4) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2),Xa2)) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2),Xa2)) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2),Xa2)) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa2))
                       => ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,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,Va3))),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_3510_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( pp(Y)
                <=> ( ( ( Xa2 = zero_zero_nat )
                     => pp(A4) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B4) )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa2)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2),Xa2)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2) )
                 => ( ~ pp(Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(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 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2),Xa2)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                     => ( ( pp(Y)
                        <=> ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                                 => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,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,Va3))),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_3511_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa2))
               => ( ( ( Xa2 = zero_zero_nat )
                   => pp(A4) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B4) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2),Xa2)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S2),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(aa_VEB4832260192069861672t_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_3512_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa2))
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => pp(A4) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B4) )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S2),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(aa_VEB4832260192069861672t_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_3513_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( pp(Y)
                <=> ( ( ( Xa2 = zero_zero_nat )
                     => pp(A4) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B4) )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa2)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2),Xa2)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S2) )
                 => ( ( 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(aa_VEB4832260192069861672t_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(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S2),Xa2)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_3514_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa2))
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => pp(A4) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B4) )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa2))
                 => ~ ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,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,Va3))),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_3515_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_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 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2),Xa2)) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2),Vb2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2),Vb2),Xa2))
                   => ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2),Xa2))
                     => ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( 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(aa_VEB4832260192069861672t_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 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2),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(aa_VEB4832260192069861672t_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_3516_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Q2),zero_z5237406670263579293d_enat) = Q2 ).

% max_enat_simps(2)
tff(fact_3517_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),Q2) = Q2 ).

% max_enat_simps(3)
tff(fact_3518_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_3519_imult__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( times_7803423173614009249d_enat(M,N) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        | ( N = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_3520_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2),Vb2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2),Vb2),Xa2))
               => ~ ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2),Xa2))
                 => ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 )
                      | ( ( 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(aa_VEB4832260192069861672t_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 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2),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(aa_VEB4832260192069861672t_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_3521_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_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 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2),Xa2)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2),Vb2) )
                 => ( ( pp(Y)
                    <=> ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2),Vb2),Xa2)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2) )
                   => ( ( pp(Y)
                      <=> ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( 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(aa_VEB4832260192069861672t_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(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2),Xa2)) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2),Xa2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_3522_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ 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_3523_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ~ 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_3524_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_3525_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(ord_less_num(B),A))
     => ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_3526_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_3527_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_3528_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_3529_atLeastatMost__subset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(A,B),set_or4548717258645045905et_nat(C,D))
    <=> ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
        | ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C),A))
          & pp(aa_set_nat_bool(ord_less_eq_set_nat(B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3530_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( 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_3531_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ord_less_eq_set_num(set_or7049704709247886629st_num(A,B),set_or7049704709247886629st_num(C,D))
    <=> ( ~ pp(aa_num_bool(ord_less_eq_num(A),B))
        | ( pp(aa_num_bool(ord_less_eq_num(C),A))
          & pp(aa_num_bool(ord_less_eq_num(B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3532_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_nat_set_nat(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_3533_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_or1266510415728281911st_int(A,B)),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_3534_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( 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_3535_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( set_or4548717258645045905et_nat(A,B) = bot_bot_set_set_nat )
    <=> ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_3536_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_3537_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num )
    <=> ~ pp(aa_num_bool(ord_less_eq_num(A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_3538_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_set_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_3539_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: 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_3540_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_3541_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_set_nat = set_or4548717258645045905et_nat(A,B) )
    <=> ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_3542_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_3543_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A,B) )
    <=> ~ pp(aa_num_bool(ord_less_eq_num(A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_3544_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = aa_nat_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_3545_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_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_3546_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_3547_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(L),U))) ).

% finite_atLeastAtMost
tff(fact_3548_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))
     => ( ! [X2: int] :
            ( pp(aa_int_bool(P,X2))
           => pp(aa_int_bool(P,aa_int_int(minus_minus_int(X2),D))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X4: int] :
              ( pp(aa_int_bool(P,X4))
             => pp(aa_int_bool(P,aa_int_int(minus_minus_int(X4),aa_int_int(times_times_int(K),D)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_3549_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))
     => ( ! [X2: int] :
            ( pp(aa_int_bool(P,X2))
           => pp(aa_int_bool(P,aa_int_int(plus_plus_int(X2),D))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X4: int] :
              ( pp(aa_int_bool(P,X4))
             => pp(aa_int_bool(P,aa_int_int(plus_plus_int(X4),aa_int_int(times_times_int(K),D)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_3550_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( member_set_nat(I,set_or4548717258645045905et_nat(L,U))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(L),I))
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_3551_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( 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_3552_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( member_num(I,set_or7049704709247886629st_num(L,U))
    <=> ( pp(aa_num_bool(ord_less_eq_num(L),I))
        & pp(aa_num_bool(ord_less_eq_num(I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_3553_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(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_3554_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( pp(aa_set_int_bool(member_int(I),set_or1266510415728281911st_int(L,U)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),I))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_3555_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( 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_3556_Icc__eq__Icc,axiom,
    ! [L: set_nat,H: set_nat,L2: set_nat,H2: set_nat] :
      ( ( set_or4548717258645045905et_nat(L,H) = set_or4548717258645045905et_nat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(L),H))
          & ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_3557_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_3558_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(ord_less_eq_num(L),H))
          & ~ pp(aa_num_bool(ord_less_eq_num(L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_3559_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L2: nat,H2: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(L),H) = aa_nat_set_nat(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_3560_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L2: int,H2: int] :
      ( ( set_or1266510415728281911st_int(L,H) = 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_3561_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_3562_aset_I2_J,axiom,
    ! [D3: int,A2: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
             => ! [Xb: int] :
                  ( pp(aa_set_int_bool(member_int(Xb),A2))
                 => ( X2 != aa_int_int(minus_minus_int(Xb),Xa) ) ) )
         => ( pp(aa_int_bool(P,X2))
           => pp(aa_int_bool(P,aa_int_int(plus_plus_int(X2),D3))) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
               => ! [Xb: int] :
                    ( pp(aa_set_int_bool(member_int(Xb),A2))
                   => ( X2 != aa_int_int(minus_minus_int(Xb),Xa) ) ) )
           => ( pp(aa_int_bool(Q,X2))
             => pp(aa_int_bool(Q,aa_int_int(plus_plus_int(X2),D3))) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A2))
                   => ( X4 != aa_int_int(minus_minus_int(Xb3),Xa3) ) ) )
           => ( ( pp(aa_int_bool(P,X4))
                | pp(aa_int_bool(Q,X4)) )
             => ( pp(aa_int_bool(P,aa_int_int(plus_plus_int(X4),D3)))
                | pp(aa_int_bool(Q,aa_int_int(plus_plus_int(X4),D3))) ) ) ) ) ) ).

% aset(2)
tff(fact_3563_aset_I1_J,axiom,
    ! [D3: int,A2: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
             => ! [Xb: int] :
                  ( pp(aa_set_int_bool(member_int(Xb),A2))
                 => ( X2 != aa_int_int(minus_minus_int(Xb),Xa) ) ) )
         => ( pp(aa_int_bool(P,X2))
           => pp(aa_int_bool(P,aa_int_int(plus_plus_int(X2),D3))) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
               => ! [Xb: int] :
                    ( pp(aa_set_int_bool(member_int(Xb),A2))
                   => ( X2 != aa_int_int(minus_minus_int(Xb),Xa) ) ) )
           => ( pp(aa_int_bool(Q,X2))
             => pp(aa_int_bool(Q,aa_int_int(plus_plus_int(X2),D3))) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A2))
                   => ( X4 != aa_int_int(minus_minus_int(Xb3),Xa3) ) ) )
           => ( ( pp(aa_int_bool(P,X4))
                & pp(aa_int_bool(Q,X4)) )
             => ( pp(aa_int_bool(P,aa_int_int(plus_plus_int(X4),D3)))
                & pp(aa_int_bool(Q,aa_int_int(plus_plus_int(X4),D3))) ) ) ) ) ) ).

% aset(1)
tff(fact_3564_bset_I2_J,axiom,
    ! [D3: int,B2: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
             => ! [Xb: int] :
                  ( pp(aa_set_int_bool(member_int(Xb),B2))
                 => ( X2 != aa_int_int(plus_plus_int(Xb),Xa) ) ) )
         => ( pp(aa_int_bool(P,X2))
           => pp(aa_int_bool(P,aa_int_int(minus_minus_int(X2),D3))) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
               => ! [Xb: int] :
                    ( pp(aa_set_int_bool(member_int(Xb),B2))
                   => ( X2 != aa_int_int(plus_plus_int(Xb),Xa) ) ) )
           => ( pp(aa_int_bool(Q,X2))
             => pp(aa_int_bool(Q,aa_int_int(minus_minus_int(X2),D3))) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B2))
                   => ( X4 != aa_int_int(plus_plus_int(Xb3),Xa3) ) ) )
           => ( ( pp(aa_int_bool(P,X4))
                | pp(aa_int_bool(Q,X4)) )
             => ( pp(aa_int_bool(P,aa_int_int(minus_minus_int(X4),D3)))
                | pp(aa_int_bool(Q,aa_int_int(minus_minus_int(X4),D3))) ) ) ) ) ) ).

% bset(2)
tff(fact_3565_bset_I1_J,axiom,
    ! [D3: int,B2: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
             => ! [Xb: int] :
                  ( pp(aa_set_int_bool(member_int(Xb),B2))
                 => ( X2 != aa_int_int(plus_plus_int(Xb),Xa) ) ) )
         => ( pp(aa_int_bool(P,X2))
           => pp(aa_int_bool(P,aa_int_int(minus_minus_int(X2),D3))) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
               => ! [Xb: int] :
                    ( pp(aa_set_int_bool(member_int(Xb),B2))
                   => ( X2 != aa_int_int(plus_plus_int(Xb),Xa) ) ) )
           => ( pp(aa_int_bool(Q,X2))
             => pp(aa_int_bool(Q,aa_int_int(minus_minus_int(X2),D3))) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B2))
                   => ( X4 != aa_int_int(plus_plus_int(Xb3),Xa3) ) ) )
           => ( ( pp(aa_int_bool(P,X4))
                & pp(aa_int_bool(Q,X4)) )
             => ( pp(aa_int_bool(P,aa_int_int(minus_minus_int(X4),D3)))
                & pp(aa_int_bool(Q,aa_int_int(minus_minus_int(X4),D3))) ) ) ) ) ) ).

% bset(1)
tff(fact_3566_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z4))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X4)) ) ).

% minf(7)
tff(fact_3567_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z4))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X4)) ) ).

% minf(7)
tff(fact_3568_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(ord_less_num(X4),Z4))
     => ~ pp(aa_num_bool(ord_less_num(T),X4)) ) ).

% minf(7)
tff(fact_3569_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z4))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X4)) ) ).

% minf(7)
tff(fact_3570_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z4))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X4)) ) ).

% minf(7)
tff(fact_3571_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z4))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),T)) ) ).

% minf(5)
tff(fact_3572_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z4))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),T)) ) ).

% minf(5)
tff(fact_3573_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(ord_less_num(X4),Z4))
     => pp(aa_num_bool(ord_less_num(X4),T)) ) ).

% minf(5)
tff(fact_3574_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z4))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),T)) ) ).

% minf(5)
tff(fact_3575_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z4))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),T)) ) ).

% minf(5)
tff(fact_3576_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z4))
     => ( X4 != T ) ) ).

% minf(4)
tff(fact_3577_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z4))
     => ( X4 != T ) ) ).

% minf(4)
tff(fact_3578_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(ord_less_num(X4),Z4))
     => ( X4 != T ) ) ).

% minf(4)
tff(fact_3579_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z4))
     => ( X4 != T ) ) ).

% minf(4)
tff(fact_3580_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z4))
     => ( X4 != T ) ) ).

% minf(4)
tff(fact_3581_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z4))
     => ( X4 != T ) ) ).

% minf(3)
tff(fact_3582_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z4))
     => ( X4 != T ) ) ).

% minf(3)
tff(fact_3583_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(ord_less_num(X4),Z4))
     => ( X4 != T ) ) ).

% minf(3)
tff(fact_3584_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z4))
     => ( X4 != T ) ) ).

% minf(3)
tff(fact_3585_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z4))
     => ( X4 != T ) ) ).

% minf(3)
tff(fact_3586_minf_I2_J,axiom,
    ! [P: fun_real_bool,P3: fun_real_bool,Q: fun_real_bool,Q6: fun_real_bool] :
      ( ? [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(P3,X2)) ) )
     => ( ? [Z3: real] :
          ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z3))
           => ( pp(aa_real_bool(Q,X2))
            <=> pp(aa_real_bool(Q6,X2)) ) )
       => ? [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(Q,X4)) )
            <=> ( pp(aa_real_bool(P3,X4))
                | pp(aa_real_bool(Q6,X4)) ) ) ) ) ) ).

% minf(2)
tff(fact_3587_minf_I2_J,axiom,
    ! [P: fun_rat_bool,P3: fun_rat_bool,Q: fun_rat_bool,Q6: fun_rat_bool] :
      ( ? [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(P3,X2)) ) )
     => ( ? [Z3: rat] :
          ! [X2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z3))
           => ( pp(aa_rat_bool(Q,X2))
            <=> pp(aa_rat_bool(Q6,X2)) ) )
       => ? [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(Q,X4)) )
            <=> ( pp(aa_rat_bool(P3,X4))
                | pp(aa_rat_bool(Q6,X4)) ) ) ) ) ) ).

% minf(2)
tff(fact_3588_minf_I2_J,axiom,
    ! [P: fun_num_bool,P3: fun_num_bool,Q: fun_num_bool,Q6: fun_num_bool] :
      ( ? [Z3: num] :
        ! [X2: num] :
          ( pp(aa_num_bool(ord_less_num(X2),Z3))
         => ( pp(aa_num_bool(P,X2))
          <=> pp(aa_num_bool(P3,X2)) ) )
     => ( ? [Z3: num] :
          ! [X2: num] :
            ( pp(aa_num_bool(ord_less_num(X2),Z3))
           => ( pp(aa_num_bool(Q,X2))
            <=> pp(aa_num_bool(Q6,X2)) ) )
       => ? [Z4: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(ord_less_num(X4),Z4))
           => ( ( pp(aa_num_bool(P,X4))
                | pp(aa_num_bool(Q,X4)) )
            <=> ( pp(aa_num_bool(P3,X4))
                | pp(aa_num_bool(Q6,X4)) ) ) ) ) ) ).

% minf(2)
tff(fact_3589_minf_I2_J,axiom,
    ! [P: fun_nat_bool,P3: fun_nat_bool,Q: fun_nat_bool,Q6: fun_nat_bool] :
      ( ? [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(P3,X2)) ) )
     => ( ? [Z3: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z3))
           => ( pp(aa_nat_bool(Q,X2))
            <=> pp(aa_nat_bool(Q6,X2)) ) )
       => ? [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(Q,X4)) )
            <=> ( pp(aa_nat_bool(P3,X4))
                | pp(aa_nat_bool(Q6,X4)) ) ) ) ) ) ).

% minf(2)
tff(fact_3590_minf_I2_J,axiom,
    ! [P: fun_int_bool,P3: fun_int_bool,Q: fun_int_bool,Q6: fun_int_bool] :
      ( ? [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(P3,X2)) ) )
     => ( ? [Z3: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z3))
           => ( pp(aa_int_bool(Q,X2))
            <=> pp(aa_int_bool(Q6,X2)) ) )
       => ? [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(Q,X4)) )
            <=> ( pp(aa_int_bool(P3,X4))
                | pp(aa_int_bool(Q6,X4)) ) ) ) ) ) ).

% minf(2)
tff(fact_3591_minf_I1_J,axiom,
    ! [P: fun_real_bool,P3: fun_real_bool,Q: fun_real_bool,Q6: fun_real_bool] :
      ( ? [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(P3,X2)) ) )
     => ( ? [Z3: real] :
          ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z3))
           => ( pp(aa_real_bool(Q,X2))
            <=> pp(aa_real_bool(Q6,X2)) ) )
       => ? [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(Q,X4)) )
            <=> ( pp(aa_real_bool(P3,X4))
                & pp(aa_real_bool(Q6,X4)) ) ) ) ) ) ).

% minf(1)
tff(fact_3592_minf_I1_J,axiom,
    ! [P: fun_rat_bool,P3: fun_rat_bool,Q: fun_rat_bool,Q6: fun_rat_bool] :
      ( ? [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(P3,X2)) ) )
     => ( ? [Z3: rat] :
          ! [X2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z3))
           => ( pp(aa_rat_bool(Q,X2))
            <=> pp(aa_rat_bool(Q6,X2)) ) )
       => ? [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(Q,X4)) )
            <=> ( pp(aa_rat_bool(P3,X4))
                & pp(aa_rat_bool(Q6,X4)) ) ) ) ) ) ).

% minf(1)
tff(fact_3593_minf_I1_J,axiom,
    ! [P: fun_num_bool,P3: fun_num_bool,Q: fun_num_bool,Q6: fun_num_bool] :
      ( ? [Z3: num] :
        ! [X2: num] :
          ( pp(aa_num_bool(ord_less_num(X2),Z3))
         => ( pp(aa_num_bool(P,X2))
          <=> pp(aa_num_bool(P3,X2)) ) )
     => ( ? [Z3: num] :
          ! [X2: num] :
            ( pp(aa_num_bool(ord_less_num(X2),Z3))
           => ( pp(aa_num_bool(Q,X2))
            <=> pp(aa_num_bool(Q6,X2)) ) )
       => ? [Z4: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(ord_less_num(X4),Z4))
           => ( ( pp(aa_num_bool(P,X4))
                & pp(aa_num_bool(Q,X4)) )
            <=> ( pp(aa_num_bool(P3,X4))
                & pp(aa_num_bool(Q6,X4)) ) ) ) ) ) ).

% minf(1)
tff(fact_3594_minf_I1_J,axiom,
    ! [P: fun_nat_bool,P3: fun_nat_bool,Q: fun_nat_bool,Q6: fun_nat_bool] :
      ( ? [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(P3,X2)) ) )
     => ( ? [Z3: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z3))
           => ( pp(aa_nat_bool(Q,X2))
            <=> pp(aa_nat_bool(Q6,X2)) ) )
       => ? [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(Q,X4)) )
            <=> ( pp(aa_nat_bool(P3,X4))
                & pp(aa_nat_bool(Q6,X4)) ) ) ) ) ) ).

% minf(1)
tff(fact_3595_minf_I1_J,axiom,
    ! [P: fun_int_bool,P3: fun_int_bool,Q: fun_int_bool,Q6: fun_int_bool] :
      ( ? [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(P3,X2)) ) )
     => ( ? [Z3: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z3))
           => ( pp(aa_int_bool(Q,X2))
            <=> pp(aa_int_bool(Q6,X2)) ) )
       => ? [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(Q,X4)) )
            <=> ( pp(aa_int_bool(P3,X4))
                & pp(aa_int_bool(Q6,X4)) ) ) ) ) ) ).

% minf(1)
tff(fact_3596_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X4))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X4)) ) ).

% pinf(7)
tff(fact_3597_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X4))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X4)) ) ).

% pinf(7)
tff(fact_3598_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(ord_less_num(Z4),X4))
     => pp(aa_num_bool(ord_less_num(T),X4)) ) ).

% pinf(7)
tff(fact_3599_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X4))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X4)) ) ).

% pinf(7)
tff(fact_3600_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X4))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X4)) ) ).

% pinf(7)
tff(fact_3601_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X4))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),T)) ) ).

% pinf(5)
tff(fact_3602_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X4))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),T)) ) ).

% pinf(5)
tff(fact_3603_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(ord_less_num(Z4),X4))
     => ~ pp(aa_num_bool(ord_less_num(X4),T)) ) ).

% pinf(5)
tff(fact_3604_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X4))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),T)) ) ).

% pinf(5)
tff(fact_3605_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X4))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),T)) ) ).

% pinf(5)
tff(fact_3606_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X4))
     => ( X4 != T ) ) ).

% pinf(4)
tff(fact_3607_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X4))
     => ( X4 != T ) ) ).

% pinf(4)
tff(fact_3608_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(ord_less_num(Z4),X4))
     => ( X4 != T ) ) ).

% pinf(4)
tff(fact_3609_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X4))
     => ( X4 != T ) ) ).

% pinf(4)
tff(fact_3610_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X4))
     => ( X4 != T ) ) ).

% pinf(4)
tff(fact_3611_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X4))
     => ( X4 != T ) ) ).

% pinf(3)
tff(fact_3612_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X4))
     => ( X4 != T ) ) ).

% pinf(3)
tff(fact_3613_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(ord_less_num(Z4),X4))
     => ( X4 != T ) ) ).

% pinf(3)
tff(fact_3614_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X4))
     => ( X4 != T ) ) ).

% pinf(3)
tff(fact_3615_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X4))
     => ( X4 != T ) ) ).

% pinf(3)
tff(fact_3616_pinf_I2_J,axiom,
    ! [P: fun_real_bool,P3: fun_real_bool,Q: fun_real_bool,Q6: fun_real_bool] :
      ( ? [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(P3,X2)) ) )
     => ( ? [Z3: real] :
          ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X2))
           => ( pp(aa_real_bool(Q,X2))
            <=> pp(aa_real_bool(Q6,X2)) ) )
       => ? [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(Q,X4)) )
            <=> ( pp(aa_real_bool(P3,X4))
                | pp(aa_real_bool(Q6,X4)) ) ) ) ) ) ).

% pinf(2)
tff(fact_3617_pinf_I2_J,axiom,
    ! [P: fun_rat_bool,P3: fun_rat_bool,Q: fun_rat_bool,Q6: fun_rat_bool] :
      ( ? [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(P3,X2)) ) )
     => ( ? [Z3: rat] :
          ! [X2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X2))
           => ( pp(aa_rat_bool(Q,X2))
            <=> pp(aa_rat_bool(Q6,X2)) ) )
       => ? [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(Q,X4)) )
            <=> ( pp(aa_rat_bool(P3,X4))
                | pp(aa_rat_bool(Q6,X4)) ) ) ) ) ) ).

% pinf(2)
tff(fact_3618_pinf_I2_J,axiom,
    ! [P: fun_num_bool,P3: fun_num_bool,Q: fun_num_bool,Q6: fun_num_bool] :
      ( ? [Z3: num] :
        ! [X2: num] :
          ( pp(aa_num_bool(ord_less_num(Z3),X2))
         => ( pp(aa_num_bool(P,X2))
          <=> pp(aa_num_bool(P3,X2)) ) )
     => ( ? [Z3: num] :
          ! [X2: num] :
            ( pp(aa_num_bool(ord_less_num(Z3),X2))
           => ( pp(aa_num_bool(Q,X2))
            <=> pp(aa_num_bool(Q6,X2)) ) )
       => ? [Z4: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(ord_less_num(Z4),X4))
           => ( ( pp(aa_num_bool(P,X4))
                | pp(aa_num_bool(Q,X4)) )
            <=> ( pp(aa_num_bool(P3,X4))
                | pp(aa_num_bool(Q6,X4)) ) ) ) ) ) ).

% pinf(2)
tff(fact_3619_pinf_I2_J,axiom,
    ! [P: fun_nat_bool,P3: fun_nat_bool,Q: fun_nat_bool,Q6: fun_nat_bool] :
      ( ? [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(P3,X2)) ) )
     => ( ? [Z3: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X2))
           => ( pp(aa_nat_bool(Q,X2))
            <=> pp(aa_nat_bool(Q6,X2)) ) )
       => ? [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(Q,X4)) )
            <=> ( pp(aa_nat_bool(P3,X4))
                | pp(aa_nat_bool(Q6,X4)) ) ) ) ) ) ).

% pinf(2)
tff(fact_3620_pinf_I2_J,axiom,
    ! [P: fun_int_bool,P3: fun_int_bool,Q: fun_int_bool,Q6: fun_int_bool] :
      ( ? [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(P3,X2)) ) )
     => ( ? [Z3: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X2))
           => ( pp(aa_int_bool(Q,X2))
            <=> pp(aa_int_bool(Q6,X2)) ) )
       => ? [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(Q,X4)) )
            <=> ( pp(aa_int_bool(P3,X4))
                | pp(aa_int_bool(Q6,X4)) ) ) ) ) ) ).

% pinf(2)
tff(fact_3621_pinf_I1_J,axiom,
    ! [P: fun_real_bool,P3: fun_real_bool,Q: fun_real_bool,Q6: fun_real_bool] :
      ( ? [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(P3,X2)) ) )
     => ( ? [Z3: real] :
          ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X2))
           => ( pp(aa_real_bool(Q,X2))
            <=> pp(aa_real_bool(Q6,X2)) ) )
       => ? [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(Q,X4)) )
            <=> ( pp(aa_real_bool(P3,X4))
                & pp(aa_real_bool(Q6,X4)) ) ) ) ) ) ).

% pinf(1)
tff(fact_3622_pinf_I1_J,axiom,
    ! [P: fun_rat_bool,P3: fun_rat_bool,Q: fun_rat_bool,Q6: fun_rat_bool] :
      ( ? [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(P3,X2)) ) )
     => ( ? [Z3: rat] :
          ! [X2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X2))
           => ( pp(aa_rat_bool(Q,X2))
            <=> pp(aa_rat_bool(Q6,X2)) ) )
       => ? [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(Q,X4)) )
            <=> ( pp(aa_rat_bool(P3,X4))
                & pp(aa_rat_bool(Q6,X4)) ) ) ) ) ) ).

% pinf(1)
tff(fact_3623_pinf_I1_J,axiom,
    ! [P: fun_num_bool,P3: fun_num_bool,Q: fun_num_bool,Q6: fun_num_bool] :
      ( ? [Z3: num] :
        ! [X2: num] :
          ( pp(aa_num_bool(ord_less_num(Z3),X2))
         => ( pp(aa_num_bool(P,X2))
          <=> pp(aa_num_bool(P3,X2)) ) )
     => ( ? [Z3: num] :
          ! [X2: num] :
            ( pp(aa_num_bool(ord_less_num(Z3),X2))
           => ( pp(aa_num_bool(Q,X2))
            <=> pp(aa_num_bool(Q6,X2)) ) )
       => ? [Z4: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(ord_less_num(Z4),X4))
           => ( ( pp(aa_num_bool(P,X4))
                & pp(aa_num_bool(Q,X4)) )
            <=> ( pp(aa_num_bool(P3,X4))
                & pp(aa_num_bool(Q6,X4)) ) ) ) ) ) ).

% pinf(1)
tff(fact_3624_pinf_I1_J,axiom,
    ! [P: fun_nat_bool,P3: fun_nat_bool,Q: fun_nat_bool,Q6: fun_nat_bool] :
      ( ? [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(P3,X2)) ) )
     => ( ? [Z3: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X2))
           => ( pp(aa_nat_bool(Q,X2))
            <=> pp(aa_nat_bool(Q6,X2)) ) )
       => ? [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(Q,X4)) )
            <=> ( pp(aa_nat_bool(P3,X4))
                & pp(aa_nat_bool(Q6,X4)) ) ) ) ) ) ).

% pinf(1)
tff(fact_3625_pinf_I1_J,axiom,
    ! [P: fun_int_bool,P3: fun_int_bool,Q: fun_int_bool,Q6: fun_int_bool] :
      ( ? [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(P3,X2)) ) )
     => ( ? [Z3: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X2))
           => ( pp(aa_int_bool(Q,X2))
            <=> pp(aa_int_bool(Q6,X2)) ) )
       => ? [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(Q,X4)) )
            <=> ( pp(aa_int_bool(P3,X4))
                & pp(aa_int_bool(Q6,X4)) ) ) ) ) ) ).

% pinf(1)
tff(fact_3626_bounded__Max__nat,axiom,
    ! [P: fun_nat_bool,X: nat,M4: nat] :
      ( pp(aa_nat_bool(P,X))
     => ( ! [X2: nat] :
            ( pp(aa_nat_bool(P,X2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),M4)) )
       => ~ ! [M3: nat] :
              ( pp(aa_nat_bool(P,M3))
             => ~ ! [X4: nat] :
                    ( pp(aa_nat_bool(P,X4))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),M3)) ) ) ) ) ).

% bounded_Max_nat
tff(fact_3627_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))
     => ( ! [X2: int,K3: int] :
            ( pp(aa_int_bool(P,X2))
          <=> pp(aa_int_bool(P,aa_int_int(minus_minus_int(X2),aa_int_int(times_times_int(K3),D)))) )
       => ( ? [X_12: int] : pp(aa_int_bool(P,X_12))
        <=> ? [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D)))
              & pp(aa_int_bool(P,X3)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_3628_aset_I7_J,axiom,
    ! [D3: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),A2))
                 => ( X4 != aa_int_int(minus_minus_int(Xb3),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(plus_plus_int(X4),D3))) ) ) ) ).

% aset(7)
tff(fact_3629_aset_I5_J,axiom,
    ! [D3: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => ( pp(aa_set_int_bool(member_int(T),A2))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A2))
                   => ( X4 != aa_int_int(minus_minus_int(Xb3),Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(X4),D3)),T)) ) ) ) ) ).

% aset(5)
tff(fact_3630_aset_I4_J,axiom,
    ! [D3: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => ( pp(aa_set_int_bool(member_int(T),A2))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A2))
                   => ( X4 != aa_int_int(minus_minus_int(Xb3),Xa3) ) ) )
           => ( ( X4 != T )
             => ( aa_int_int(plus_plus_int(X4),D3) != T ) ) ) ) ) ).

% aset(4)
tff(fact_3631_aset_I3_J,axiom,
    ! [D3: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(plus_plus_int(T),one_one_int)),A2))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A2))
                   => ( X4 != aa_int_int(minus_minus_int(Xb3),Xa3) ) ) )
           => ( ( X4 = T )
             => ( aa_int_int(plus_plus_int(X4),D3) = T ) ) ) ) ) ).

% aset(3)
tff(fact_3632_bset_I7_J,axiom,
    ! [D3: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => ( pp(aa_set_int_bool(member_int(T),B2))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B2))
                   => ( X4 != aa_int_int(plus_plus_int(Xb3),Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(minus_minus_int(X4),D3))) ) ) ) ) ).

% bset(7)
tff(fact_3633_bset_I5_J,axiom,
    ! [D3: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),B2))
                 => ( X4 != aa_int_int(plus_plus_int(Xb3),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(minus_minus_int(X4),D3)),T)) ) ) ) ).

% bset(5)
tff(fact_3634_bset_I4_J,axiom,
    ! [D3: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => ( pp(aa_set_int_bool(member_int(T),B2))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B2))
                   => ( X4 != aa_int_int(plus_plus_int(Xb3),Xa3) ) ) )
           => ( ( X4 != T )
             => ( aa_int_int(minus_minus_int(X4),D3) != T ) ) ) ) ) ).

% bset(4)
tff(fact_3635_bset_I3_J,axiom,
    ! [D3: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(minus_minus_int(T),one_one_int)),B2))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B2))
                   => ( X4 != aa_int_int(plus_plus_int(Xb3),Xa3) ) ) )
           => ( ( X4 = T )
             => ( aa_int_int(minus_minus_int(X4),D3) = T ) ) ) ) ) ).

% bset(3)
tff(fact_3636_aset_I8_J,axiom,
    ! [D3: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),A2))
                 => ( X4 != aa_int_int(minus_minus_int(Xb3),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(plus_plus_int(X4),D3))) ) ) ) ).

% aset(8)
tff(fact_3637_aset_I6_J,axiom,
    ! [D3: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(plus_plus_int(T),one_one_int)),A2))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A2))
                   => ( X4 != aa_int_int(minus_minus_int(Xb3),Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(X4),D3)),T)) ) ) ) ) ).

% aset(6)
tff(fact_3638_bset_I8_J,axiom,
    ! [D3: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(minus_minus_int(T),one_one_int)),B2))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B2))
                   => ( X4 != aa_int_int(plus_plus_int(Xb3),Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(minus_minus_int(X4),D3))) ) ) ) ) ).

% bset(8)
tff(fact_3639_bset_I6_J,axiom,
    ! [D3: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),B2))
                 => ( X4 != aa_int_int(plus_plus_int(Xb3),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(minus_minus_int(X4),D3)),T)) ) ) ) ).

% bset(6)
tff(fact_3640_cppi,axiom,
    ! [D3: 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),D3))
     => ( ? [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(P3,X2)) ) )
       => ( ! [X2: int] :
              ( ! [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
                 => ! [Xb: int] :
                      ( pp(aa_set_int_bool(member_int(Xb),A2))
                     => ( X2 != aa_int_int(minus_minus_int(Xb),Xa) ) ) )
             => ( pp(aa_int_bool(P,X2))
               => pp(aa_int_bool(P,aa_int_int(plus_plus_int(X2),D3))) ) )
         => ( ! [X2: int,K3: int] :
                ( pp(aa_int_bool(P3,X2))
              <=> pp(aa_int_bool(P3,aa_int_int(minus_minus_int(X2),aa_int_int(times_times_int(K3),D3)))) )
           => ( ? [X_12: int] : pp(aa_int_bool(P,X_12))
            <=> ( ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D3)))
                    & pp(aa_int_bool(P3,X3)) )
                | ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D3)))
                    & ? [Xa4: int] :
                        ( pp(aa_set_int_bool(member_int(Xa4),A2))
                        & pp(aa_int_bool(P,aa_int_int(minus_minus_int(Xa4),X3))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_3641_cpmi,axiom,
    ! [D3: 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),D3))
     => ( ? [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(P3,X2)) ) )
       => ( ! [X2: int] :
              ( ! [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
                 => ! [Xb: int] :
                      ( pp(aa_set_int_bool(member_int(Xb),B2))
                     => ( X2 != aa_int_int(plus_plus_int(Xb),Xa) ) ) )
             => ( pp(aa_int_bool(P,X2))
               => pp(aa_int_bool(P,aa_int_int(minus_minus_int(X2),D3))) ) )
         => ( ! [X2: int,K3: int] :
                ( pp(aa_int_bool(P3,X2))
              <=> pp(aa_int_bool(P3,aa_int_int(minus_minus_int(X2),aa_int_int(times_times_int(K3),D3)))) )
           => ( ? [X_12: int] : pp(aa_int_bool(P,X_12))
            <=> ( ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D3)))
                    & pp(aa_int_bool(P3,X3)) )
                | ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D3)))
                    & ? [Xa4: int] :
                        ( pp(aa_set_int_bool(member_int(Xa4),B2))
                        & pp(aa_int_bool(P,aa_int_int(plus_plus_int(Xa4),X3))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_3642_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z4))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X4)) ) ).

% minf(8)
tff(fact_3643_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z4))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X4)) ) ).

% minf(8)
tff(fact_3644_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(ord_less_num(X4),Z4))
     => ~ pp(aa_num_bool(ord_less_eq_num(T),X4)) ) ).

% minf(8)
tff(fact_3645_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z4))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X4)) ) ).

% minf(8)
tff(fact_3646_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z4))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X4)) ) ).

% minf(8)
tff(fact_3647_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z4))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),T)) ) ).

% minf(6)
tff(fact_3648_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z4))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),T)) ) ).

% minf(6)
tff(fact_3649_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(ord_less_num(X4),Z4))
     => pp(aa_num_bool(ord_less_eq_num(X4),T)) ) ).

% minf(6)
tff(fact_3650_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z4))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),T)) ) ).

% minf(6)
tff(fact_3651_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z4))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),T)) ) ).

% minf(6)
tff(fact_3652_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X4))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X4)) ) ).

% pinf(8)
tff(fact_3653_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X4))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X4)) ) ).

% pinf(8)
tff(fact_3654_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(ord_less_num(Z4),X4))
     => pp(aa_num_bool(ord_less_eq_num(T),X4)) ) ).

% pinf(8)
tff(fact_3655_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X4))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X4)) ) ).

% pinf(8)
tff(fact_3656_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X4))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X4)) ) ).

% pinf(8)
tff(fact_3657_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X4))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),T)) ) ).

% pinf(6)
tff(fact_3658_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X4))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),T)) ) ).

% pinf(6)
tff(fact_3659_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(ord_less_num(Z4),X4))
     => ~ pp(aa_num_bool(ord_less_eq_num(X4),T)) ) ).

% pinf(6)
tff(fact_3660_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X4))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),T)) ) ).

% pinf(6)
tff(fact_3661_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X4))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),T)) ) ).

% pinf(6)
tff(fact_3662_inf__period_I1_J,axiom,
    ! [P: fun_real_bool,D3: real,Q: fun_real_bool] :
      ( ! [X2: real,K3: real] :
          ( pp(aa_real_bool(P,X2))
        <=> pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K3),D3)))) )
     => ( ! [X2: real,K3: real] :
            ( pp(aa_real_bool(Q,X2))
          <=> pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K3),D3)))) )
       => ! [X4: real,K4: real] :
            ( ( pp(aa_real_bool(P,X4))
              & pp(aa_real_bool(Q,X4)) )
          <=> ( pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3))))
              & pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3)))) ) ) ) ) ).

% inf_period(1)
tff(fact_3663_inf__period_I1_J,axiom,
    ! [P: fun_rat_bool,D3: rat,Q: fun_rat_bool] :
      ( ! [X2: rat,K3: rat] :
          ( pp(aa_rat_bool(P,X2))
        <=> pp(aa_rat_bool(P,minus_minus_rat(X2,aa_rat_rat(times_times_rat(K3),D3)))) )
     => ( ! [X2: rat,K3: rat] :
            ( pp(aa_rat_bool(Q,X2))
          <=> pp(aa_rat_bool(Q,minus_minus_rat(X2,aa_rat_rat(times_times_rat(K3),D3)))) )
       => ! [X4: rat,K4: rat] :
            ( ( pp(aa_rat_bool(P,X4))
              & pp(aa_rat_bool(Q,X4)) )
          <=> ( pp(aa_rat_bool(P,minus_minus_rat(X4,aa_rat_rat(times_times_rat(K4),D3))))
              & pp(aa_rat_bool(Q,minus_minus_rat(X4,aa_rat_rat(times_times_rat(K4),D3)))) ) ) ) ) ).

% inf_period(1)
tff(fact_3664_inf__period_I1_J,axiom,
    ! [P: fun_int_bool,D3: int,Q: fun_int_bool] :
      ( ! [X2: int,K3: int] :
          ( pp(aa_int_bool(P,X2))
        <=> pp(aa_int_bool(P,aa_int_int(minus_minus_int(X2),aa_int_int(times_times_int(K3),D3)))) )
     => ( ! [X2: int,K3: int] :
            ( pp(aa_int_bool(Q,X2))
          <=> pp(aa_int_bool(Q,aa_int_int(minus_minus_int(X2),aa_int_int(times_times_int(K3),D3)))) )
       => ! [X4: int,K4: int] :
            ( ( pp(aa_int_bool(P,X4))
              & pp(aa_int_bool(Q,X4)) )
          <=> ( pp(aa_int_bool(P,aa_int_int(minus_minus_int(X4),aa_int_int(times_times_int(K4),D3))))
              & pp(aa_int_bool(Q,aa_int_int(minus_minus_int(X4),aa_int_int(times_times_int(K4),D3)))) ) ) ) ) ).

% inf_period(1)
tff(fact_3665_inf__period_I2_J,axiom,
    ! [P: fun_real_bool,D3: real,Q: fun_real_bool] :
      ( ! [X2: real,K3: real] :
          ( pp(aa_real_bool(P,X2))
        <=> pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K3),D3)))) )
     => ( ! [X2: real,K3: real] :
            ( pp(aa_real_bool(Q,X2))
          <=> pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K3),D3)))) )
       => ! [X4: real,K4: real] :
            ( ( pp(aa_real_bool(P,X4))
              | pp(aa_real_bool(Q,X4)) )
          <=> ( pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3))))
              | pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3)))) ) ) ) ) ).

% inf_period(2)
tff(fact_3666_inf__period_I2_J,axiom,
    ! [P: fun_rat_bool,D3: rat,Q: fun_rat_bool] :
      ( ! [X2: rat,K3: rat] :
          ( pp(aa_rat_bool(P,X2))
        <=> pp(aa_rat_bool(P,minus_minus_rat(X2,aa_rat_rat(times_times_rat(K3),D3)))) )
     => ( ! [X2: rat,K3: rat] :
            ( pp(aa_rat_bool(Q,X2))
          <=> pp(aa_rat_bool(Q,minus_minus_rat(X2,aa_rat_rat(times_times_rat(K3),D3)))) )
       => ! [X4: rat,K4: rat] :
            ( ( pp(aa_rat_bool(P,X4))
              | pp(aa_rat_bool(Q,X4)) )
          <=> ( pp(aa_rat_bool(P,minus_minus_rat(X4,aa_rat_rat(times_times_rat(K4),D3))))
              | pp(aa_rat_bool(Q,minus_minus_rat(X4,aa_rat_rat(times_times_rat(K4),D3)))) ) ) ) ) ).

% inf_period(2)
tff(fact_3667_inf__period_I2_J,axiom,
    ! [P: fun_int_bool,D3: int,Q: fun_int_bool] :
      ( ! [X2: int,K3: int] :
          ( pp(aa_int_bool(P,X2))
        <=> pp(aa_int_bool(P,aa_int_int(minus_minus_int(X2),aa_int_int(times_times_int(K3),D3)))) )
     => ( ! [X2: int,K3: int] :
            ( pp(aa_int_bool(Q,X2))
          <=> pp(aa_int_bool(Q,aa_int_int(minus_minus_int(X2),aa_int_int(times_times_int(K3),D3)))) )
       => ! [X4: int,K4: int] :
            ( ( pp(aa_int_bool(P,X4))
              | pp(aa_int_bool(Q,X4)) )
          <=> ( pp(aa_int_bool(P,aa_int_int(minus_minus_int(X4),aa_int_int(times_times_int(K4),D3))))
              | pp(aa_int_bool(Q,aa_int_int(minus_minus_int(X4),aa_int_int(times_times_int(K4),D3)))) ) ) ) ) ).

% inf_period(2)
tff(fact_3668_bounded__nat__set__is__finite,axiom,
    ! [N3: set_nat,N: nat] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),N3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),N)) )
     => pp(aa_set_nat_bool(finite_finite_nat,N3)) ) ).

% bounded_nat_set_is_finite
tff(fact_3669_finite__nat__set__iff__bounded,axiom,
    ! [N3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N3))
    <=> ? [M7: nat] :
        ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),N3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),M7)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_3670_finite__nat__set__iff__bounded__le,axiom,
    ! [N3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N3))
    <=> ? [M7: nat] :
        ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),N3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),M7)) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_3671_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_gb(P),I)))) ).

% finite_M_bounded_by_nat
tff(fact_3672_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_gc(F),U)))) ) ).

% finite_less_ub
tff(fact_3673_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ~ finite_finite_rat(set_or633870826150836451st_rat(A,B)) ) ).

% infinite_Icc
tff(fact_3674_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ~ finite_finite_real(set_or1222579329274155063t_real(A,B)) ) ).

% infinite_Icc
tff(fact_3675_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ord_less_set_set_nat(set_or4548717258645045905et_nat(A,B),set_or4548717258645045905et_nat(C,D))
    <=> ( ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
          | ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C),A))
            & pp(aa_set_nat_bool(ord_less_eq_set_nat(B),D))
            & ( pp(aa_set_nat_bool(ord_less_set_nat(C),A))
              | pp(aa_set_nat_bool(ord_less_set_nat(B),D)) ) ) )
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3676_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( 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_3677_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ord_less_set_num(set_or7049704709247886629st_num(A,B),set_or7049704709247886629st_num(C,D))
    <=> ( ( ~ pp(aa_num_bool(ord_less_eq_num(A),B))
          | ( pp(aa_num_bool(ord_less_eq_num(C),A))
            & pp(aa_num_bool(ord_less_eq_num(B),D))
            & ( pp(aa_num_bool(ord_less_num(C),A))
              | pp(aa_num_bool(ord_less_num(B),D)) ) ) )
        & pp(aa_num_bool(ord_less_eq_num(C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3678_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_nat_set_nat(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_3679_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(set_or1266510415728281911st_int(A,B)),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_3680_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( 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_3681_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))
     => ( ! [X2: int,K3: int] :
            ( pp(aa_int_bool(P1,X2))
          <=> pp(aa_int_bool(P1,aa_int_int(minus_minus_int(X2),aa_int_int(times_times_int(K3),D)))) )
       => ( ? [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(P1,X2)) ) )
         => ( ? [X_13: int] : pp(aa_int_bool(P1,X_13))
           => ? [X_1: int] : pp(aa_int_bool(P,X_1)) ) ) ) ) ).

% minusinfinity
tff(fact_3682_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))
     => ( ! [X2: int,K3: int] :
            ( pp(aa_int_bool(P3,X2))
          <=> pp(aa_int_bool(P3,aa_int_int(minus_minus_int(X2),aa_int_int(times_times_int(K3),D)))) )
       => ( ? [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(P3,X2)) ) )
         => ( ? [X_13: int] : pp(aa_int_bool(P3,X_13))
           => ? [X_1: int] : pp(aa_int_bool(P,X_1)) ) ) ) ) ).

% plusinfinity
tff(fact_3683_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N3: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(N3),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N3)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_3684_Bolzano,axiom,
    ! [A: real,B: real,P: fun_re1255401033344146673l_bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ! [A4: real,B4: real,C2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A4),B4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,B4),C2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A4),B4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B4),C2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A4),C2)) ) ) ) )
       => ( ! [X2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B))
               => ? [D5: real] :
                    ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D5))
                    & ! [A4: real,B4: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A4),X2))
                          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B4))
                          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B4),A4)),D5)) )
                       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A4),B4)) ) ) ) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),B)) ) ) ) ).

% Bolzano
tff(fact_3685_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_3686_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(times_times_rat(Z),X)),aa_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_3687_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(times_times_int(Z),X)),aa_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_3688_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_3689_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(times_times_rat(X),Z)),aa_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_3690_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(times_times_int(X),Z)),aa_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_3691_divides__aux__eq,axiom,
    ! [Q2: nat,R: nat] :
      ( unique6322359934112328802ux_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R))
    <=> ( R = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_3692_divides__aux__eq,axiom,
    ! [Q2: int,R: int] :
      ( unique6319869463603278526ux_int(product_Pair_int_int(Q2,R))
    <=> ( R = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_3693_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
     => ( eucl_rel_int(aa_int_int(plus_plus_int(A),one_one_int),B,product_Pair_int_int(Q2,R))
       => eucl_rel_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B),product_Pair_int_int(Q2,aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_3694_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
     => ( eucl_rel_int(A,B,product_Pair_int_int(Q2,R))
       => eucl_rel_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B),product_Pair_int_int(Q2,aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_3695_unique__remainder,axiom,
    ! [A: int,B: int,Q2: int,R: int,Q5: int,R4: int] :
      ( eucl_rel_int(A,B,product_Pair_int_int(Q2,R))
     => ( eucl_rel_int(A,B,product_Pair_int_int(Q5,R4))
       => ( R = R4 ) ) ) ).

% unique_remainder
tff(fact_3696_unique__quotient,axiom,
    ! [A: int,B: int,Q2: int,R: int,Q5: int,R4: int] :
      ( eucl_rel_int(A,B,product_Pair_int_int(Q2,R))
     => ( eucl_rel_int(A,B,product_Pair_int_int(Q5,R4))
       => ( Q2 = Q5 ) ) ) ).

% unique_quotient
tff(fact_3697_eucl__rel__int__by0,axiom,
    ! [K: int] : eucl_rel_int(K,zero_zero_int,product_Pair_int_int(zero_zero_int,K)) ).

% eucl_rel_int_by0
tff(fact_3698_div__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( eucl_rel_int(K,L,product_Pair_int_int(Q2,R))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = Q2 ) ) ).

% div_int_unique
tff(fact_3699_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( eucl_rel_int(K,L,product_Pair_int_int(Q2,R))
     => ( aa_int_int(modulo_modulo_int(K),L) = R ) ) ).

% mod_int_unique
tff(fact_3700_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero_int )
     => ( ( K = aa_int_int(times_times_int(Q2),L) )
       => eucl_rel_int(K,L,product_Pair_int_int(Q2,zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_3701_eucl__rel__int,axiom,
    ! [K: int,L: int] : eucl_rel_int(K,L,product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L),aa_int_int(modulo_modulo_int(K),L))) ).

% eucl_rel_int
tff(fact_3702_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( eucl_rel_int(K,L,product_Pair_int_int(Q2,R))
    <=> ( ( K = aa_int_int(plus_plus_int(aa_int_int(times_times_int(L),Q2)),R) )
        & ( 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),R))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R),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),R))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R),zero_zero_int)) ) )
            & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
             => ( Q2 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_3703_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_3704_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(times_times_rat(X),Z)),aa_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_3705_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(times_times_int(X),Z)),aa_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_3706_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
      ( ! [M3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M3),zero_zero_nat))
     => ( ! [M3: 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,M3),N2)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M3),N2)) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ).

% gcd_nat_induct
tff(fact_3707_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_concat_bit(aa_nat_nat(suc,N),K),L) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_concat_bit(N,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_3708_dbl__simps_I3_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_3709_dbl__simps_I3_J,axiom,
    aa_real_real(neg_numeral_dbl_real,one_one_real) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_3710_dbl__simps_I3_J,axiom,
    neg_numeral_dbl_rat(one_one_rat) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_3711_dbl__simps_I3_J,axiom,
    aa_int_int(neg_numeral_dbl_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_3712_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_3713_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(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(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_3714_even__succ__mod__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_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(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3715_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_3716_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_3717_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_3718_even__succ__div__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_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(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_3719_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_3720_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(aa_int_fun_int_int(divide_divide_int,aa_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(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_3721_signed__take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(N,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_3722_signed__take__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),A) = aa_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(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_3723_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_3724_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_3725_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_3726_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_3727_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_3728_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_3729_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_3730_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_3731_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_3732_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_3733_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_3734_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_3735_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_3736_dvd__add__triv__left__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(A,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_3737_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_3738_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(A,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_3739_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_3740_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(plus_plus_int(A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_3741_dvd__add__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_3742_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_3743_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(B,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_3744_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_3745_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(plus_plus_int(B),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_3746_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_3747_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_3748_div__dvd__div,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( 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,A),C))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,divide6298287555418463151nteger(B,A)),divide6298287555418463151nteger(C,A)))
        <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),C)) ) ) ) ).

% div_dvd_div
tff(fact_3749_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: 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,A),C))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ) ).

% div_dvd_div
tff(fact_3750_div__dvd__div,axiom,
    ! [A: int,B: int,C: 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,A),C))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ) ).

% div_dvd_div
tff(fact_3751_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_3752_signed__take__bit__of__0,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_3753_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_3754_dbl__simps_I2_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,zero_zero_complex) = zero_zero_complex ).

% dbl_simps(2)
tff(fact_3755_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_3756_dbl__simps_I2_J,axiom,
    neg_numeral_dbl_rat(zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_3757_dbl__simps_I2_J,axiom,
    aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_3758_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_3759_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_3760_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(times_times_int(B),A)),aa_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_3761_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_3762_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_3763_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(times_times_int(A),B)),aa_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_3764_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_3765_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_3766_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_3767_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(times_times_rat(A),C)),aa_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_3768_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(times_times_int(A),C)),aa_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_3769_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_3770_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_3771_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_3772_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(times_times_rat(C),A)),aa_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_3773_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(times_times_int(C),A)),aa_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_3774_unit__prod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer)) ) ) ).

% unit_prod
tff(fact_3775_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)) ) ) ).

% unit_prod
tff(fact_3776_unit__prod,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(A),B)),one_one_int)) ) ) ).

% unit_prod
tff(fact_3777_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A))))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3778_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3779_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(B,aa_rat_rat(times_times_rat(C),A))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3780_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),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,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3781_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(plus_plus_int(B),aa_int_int(times_times_int(C),A))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3782_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3783_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3784_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(aa_rat_rat(times_times_rat(C),A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3785_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3786_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(plus_plus_int(aa_int_int(times_times_int(C),A)),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3787_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_3788_dvd__mult__div__cancel,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(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_3789_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( aa_int_int(times_times_int(A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_3790_dvd__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_3791_dvd__div__mult__self,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(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_3792_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_3793_unit__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),one_one_Code_integer)) ) ) ).

% unit_div
tff(fact_3794_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat)) ) ) ).

% unit_div
tff(fact_3795_unit__div,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int)) ) ) ).

% unit_div
tff(fact_3796_unit__div__1__unit,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,A)),one_one_Code_integer)) ) ).

% unit_div_1_unit
tff(fact_3797_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)),one_one_nat)) ) ).

% unit_div_1_unit
tff(fact_3798_unit__div__1__unit,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)),one_one_int)) ) ).

% unit_div_1_unit
tff(fact_3799_unit__div__1__div__1,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ( divide6298287555418463151nteger(one_one_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_3800_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_3801_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_3802_div__add,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ) ).

% div_add
tff(fact_3803_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_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,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ) ).

% div_add
tff(fact_3804_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).

% div_add
tff(fact_3805_div__diff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
       => ( divide6298287555418463151nteger(minus_8373710615458151222nteger(A,B),C) = minus_8373710615458151222nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ) ).

% div_diff
tff(fact_3806_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(minus_minus_int(A),B)),C) = aa_int_int(minus_minus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).

% div_diff
tff(fact_3807_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_3808_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_3809_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_3810_signed__take__bit__Suc__1,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_3811_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,K)),one_one_int) = one_one_int ).

% signed_take_bit_numeral_of_1
tff(fact_3812_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_concat_bit(N,K),L)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ).

% concat_bit_nonnegative_iff
tff(fact_3813_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_3814_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_3815_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,numeral_numeral_real(K)) = numeral_numeral_real(aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_3816_dbl__simps_I5_J,axiom,
    ! [K: num] : neg_numeral_dbl_rat(numeral_numeral_rat(K)) = numeral_numeral_rat(aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_3817_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_3818_unit__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_3819_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_3820_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_3821_unit__mult__div__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(one_one_Code_integer,A)) = divide6298287555418463151nteger(B,A) ) ) ).

% unit_mult_div_div
tff(fact_3822_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) ) ) ).

% unit_mult_div_div
tff(fact_3823_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( aa_int_int(times_times_int(B),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% unit_mult_div_div
tff(fact_3824_even__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% even_Suc
tff(fact_3825_even__Suc__Suc__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% even_Suc_Suc_iff
tff(fact_3826_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_3827_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_3828_even__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)) ) ) ).

% even_mult_iff
tff(fact_3829_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)))
    <=> ( 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(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% even_mult_iff
tff(fact_3830_even__mult__iff,axiom,
    ! [A: int,B: 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(times_times_int(A),B)))
    <=> ( 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_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) ) ) ).

% even_mult_iff
tff(fact_3831_odd__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)))
    <=> ~ ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
        <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)) ) ) ).

% odd_add
tff(fact_3832_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> ~ ( ~ 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(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% odd_add
tff(fact_3833_odd__add,axiom,
    ! [A: int,B: 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(plus_plus_int(A),B)))
    <=> ~ ( ~ 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_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) ) ) ).

% odd_add
tff(fact_3834_even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)) ) ) ).

% even_add
tff(fact_3835_even__add,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> ( 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(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% even_add
tff(fact_3836_even__add,axiom,
    ! [A: int,B: 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(plus_plus_int(A),B)))
    <=> ( 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_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) ) ) ).

% even_add
tff(fact_3837_even__mod__2__iff,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))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),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))),A)) ) ).

% even_mod_2_iff
tff(fact_3838_even__mod__2__iff,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))),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ).

% even_mod_2_iff
tff(fact_3839_even__mod__2__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)) ) ).

% even_mod_2_iff
tff(fact_3840_odd__Suc__div__two,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_3841_even__Suc__div__two,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_3842_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(times_times_int(aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_bit0
tff(fact_3843_zero__le__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3844_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3845_zero__le__power__eq__numeral,axiom,
    ! [A: int,W: 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,W))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3846_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_3847_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_3848_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_3849_power__less__zero__eq__numeral,axiom,
    ! [A: real,W: 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,W))),zero_zero_real))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3850_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W: 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,W))),zero_zero_rat))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3851_power__less__zero__eq__numeral,axiom,
    ! [A: int,W: 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,W))),zero_zero_int))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3852_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,one_one_Code_integer)))
    <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)) ) ).

% even_plus_one_iff
tff(fact_3853_even__plus__one__iff,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))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) ) ).

% even_plus_one_iff
tff(fact_3854_even__plus__one__iff,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))),aa_int_int(plus_plus_int(A),one_one_int)))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ).

% even_plus_one_iff
tff(fact_3855_even__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),minus_8373710615458151222nteger(A,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))) ) ).

% even_diff
tff(fact_3856_even__diff,axiom,
    ! [A: int,B: 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(minus_minus_int(A),B)))
    <=> 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(plus_plus_int(A),B))) ) ).

% even_diff
tff(fact_3857_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_3858_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_3859_zero__less__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & ( 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,W)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3860_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W: 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,W))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & ( 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,W)))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3861_zero__less__power__eq__numeral,axiom,
    ! [A: int,W: 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,W))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & ( 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,W)))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3862_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),one_one_Code_integer) ) ) ).

% odd_succ_div_two
tff(fact_3863_odd__succ__div__two,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(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_3864_odd__succ__div__two,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(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_3865_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_3866_even__succ__div__two,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(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_3867_even__succ__div__two,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(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_3868_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_3869_even__succ__div__2,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(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),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))) ) ) ).

% even_succ_div_2
tff(fact_3870_even__succ__div__2,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(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(one_one_int),A)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_3871_even__power,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_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,aa_num_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_3872_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_3873_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_3874_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_3875_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
     => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3876_odd__two__times__div__two__succ,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(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_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3877_odd__two__times__div__two__succ,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(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_int) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3878_power__le__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,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,W)))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3879_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W: 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,W))),zero_zero_rat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,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,W)))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3880_power__le__zero__eq__numeral,axiom,
    ! [A: int,W: 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,W))),zero_zero_int))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,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,W)))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3881_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N),one_one_Code_integer)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3882_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_3883_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(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_3884_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
     => ? [B7: nat,C5: nat] :
          ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B7),C5) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B7),B))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C5),C)) ) ) ).

% division_decomp
tff(fact_3885_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(times_times_int(B),C)))
     => ? [B7: int,C5: int] :
          ( ( A = aa_int_int(times_times_int(B7),C5) )
          & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B7),B))
          & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C5),C)) ) ) ).

% division_decomp
tff(fact_3886_dvd__productE,axiom,
    ! [P2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)))
     => ~ ! [X2: nat,Y2: nat] :
            ( ( P2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X2),Y2) )
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X2),A))
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Y2),B)) ) ) ) ).

% dvd_productE
tff(fact_3887_dvd__productE,axiom,
    ! [P2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,P2),aa_int_int(times_times_int(A),B)))
     => ~ ! [X2: int,Y2: int] :
            ( ( P2 = aa_int_int(times_times_int(X2),Y2) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X2),A))
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Y2),B)) ) ) ) ).

% dvd_productE
tff(fact_3888_dvd__refl,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),A)) ).

% dvd_refl
tff(fact_3889_dvd__refl,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),A)) ).

% dvd_refl
tff(fact_3890_dvd__refl,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),A)) ).

% dvd_refl
tff(fact_3891_dvd__trans,axiom,
    ! [A: nat,B: nat,C: 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,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_trans
tff(fact_3892_dvd__trans,axiom,
    ! [A: int,B: int,C: 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,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_trans
tff(fact_3893_dvd__trans,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( 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),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C)) ) ) ).

% dvd_trans
tff(fact_3894_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_3895_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_3896_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_3897_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_3898_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_3899_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_3900_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_3901_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_3902_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_3903_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A))) ).

% dvd_triv_right
tff(fact_3904_dvd__triv__right,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A))) ).

% dvd_triv_right
tff(fact_3905_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),A))) ).

% dvd_triv_right
tff(fact_3906_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))) ).

% dvd_triv_right
tff(fact_3907_dvd__triv__right,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(times_times_int(B),A))) ).

% dvd_triv_right
tff(fact_3908_dvd__mult__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),C)) ) ).

% dvd_mult_right
tff(fact_3909_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),C)) ) ).

% dvd_mult_right
tff(fact_3910_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),C)) ) ).

% dvd_mult_right
tff(fact_3911_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: 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)),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ).

% dvd_mult_right
tff(fact_3912_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(A),B)),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ).

% dvd_mult_right
tff(fact_3913_mult__dvd__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( 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))
       => 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),D))) ) ) ).

% mult_dvd_mono
tff(fact_3914_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),D))
       => 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),D))) ) ) ).

% mult_dvd_mono
tff(fact_3915_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ).

% mult_dvd_mono
tff(fact_3916_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: 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))
       => 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)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ).

% mult_dvd_mono
tff(fact_3917_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: 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))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D))) ) ) ).

% mult_dvd_mono
tff(fact_3918_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))) ).

% dvd_triv_left
tff(fact_3919_dvd__triv__left,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ).

% dvd_triv_left
tff(fact_3920_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(A),B))) ).

% dvd_triv_left
tff(fact_3921_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))) ).

% dvd_triv_left
tff(fact_3922_dvd__triv__left,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(times_times_int(A),B))) ).

% dvd_triv_left
tff(fact_3923_dvd__mult__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C)) ) ).

% dvd_mult_left
tff(fact_3924_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C)) ) ).

% dvd_mult_left
tff(fact_3925_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C)) ) ).

% dvd_mult_left
tff(fact_3926_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: 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)),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ).

% dvd_mult_left
tff(fact_3927_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(A),B)),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ).

% dvd_mult_left
tff(fact_3928_dvd__mult2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( 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,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))) ) ).

% dvd_mult2
tff(fact_3929_dvd__mult2,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ).

% dvd_mult2
tff(fact_3930_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C))) ) ).

% dvd_mult2
tff(fact_3931_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: 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,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ).

% dvd_mult2
tff(fact_3932_dvd__mult2,axiom,
    ! [A: int,B: int,C: 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,A),aa_int_int(times_times_int(B),C))) ) ).

% dvd_mult2
tff(fact_3933_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))) ) ).

% dvd_mult
tff(fact_3934_dvd__mult,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ).

% dvd_mult
tff(fact_3935_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C))) ) ).

% dvd_mult
tff(fact_3936_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),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,B),C))) ) ).

% dvd_mult
tff(fact_3937_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(times_times_int(B),C))) ) ).

% dvd_mult
tff(fact_3938_dvd__def,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
    <=> ? [K2: code_integer] : A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K2) ) ).

% dvd_def
tff(fact_3939_dvd__def,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
    <=> ? [K2: real] : A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K2) ) ).

% dvd_def
tff(fact_3940_dvd__def,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
    <=> ? [K2: rat] : A = aa_rat_rat(times_times_rat(B),K2) ) ).

% dvd_def
tff(fact_3941_dvd__def,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
    <=> ? [K2: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) ) ).

% dvd_def
tff(fact_3942_dvd__def,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
    <=> ? [K2: int] : A = aa_int_int(times_times_int(B),K2) ) ).

% dvd_def
tff(fact_3943_dvdI,axiom,
    ! [A: code_integer,B: code_integer,K: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A)) ) ).

% dvdI
tff(fact_3944_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A)) ) ).

% dvdI
tff(fact_3945_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A = aa_rat_rat(times_times_rat(B),K) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A)) ) ).

% dvdI
tff(fact_3946_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% dvdI
tff(fact_3947_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A = aa_int_int(times_times_int(B),K) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% dvdI
tff(fact_3948_dvdE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
     => ~ ! [K3: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K3) ) ).

% dvdE
tff(fact_3949_dvdE,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
     => ~ ! [K3: real] : A != aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K3) ) ).

% dvdE
tff(fact_3950_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ~ ! [K3: rat] : A != aa_rat_rat(times_times_rat(B),K3) ) ).

% dvdE
tff(fact_3951_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
     => ~ ! [K3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K3) ) ).

% dvdE
tff(fact_3952_dvdE,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ~ ! [K3: int] : A != aa_int_int(times_times_int(B),K3) ) ).

% dvdE
tff(fact_3953_one__dvd,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,one_one_Code_integer),A)) ).

% one_dvd
tff(fact_3954_one__dvd,axiom,
    ! [A: complex] : pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,one_one_complex),A)) ).

% one_dvd
tff(fact_3955_one__dvd,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,one_one_real),A)) ).

% one_dvd
tff(fact_3956_one__dvd,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,one_one_rat),A)) ).

% one_dvd
tff(fact_3957_one__dvd,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,one_one_nat),A)) ).

% one_dvd
tff(fact_3958_one__dvd,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,one_one_int),A)) ).

% one_dvd
tff(fact_3959_unit__imp__dvd,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A)) ) ).

% unit_imp_dvd
tff(fact_3960_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% unit_imp_dvd
tff(fact_3961_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% unit_imp_dvd
tff(fact_3962_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B: code_integer] :
      ( 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),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer)) ) ) ).

% dvd_unit_imp_unit
tff(fact_3963_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: 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,B),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat)) ) ) ).

% dvd_unit_imp_unit
tff(fact_3964_dvd__unit__imp__unit,axiom,
    ! [A: int,B: 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,B),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int)) ) ) ).

% dvd_unit_imp_unit
tff(fact_3965_dvd__add,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( 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,A),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C))) ) ) ).

% dvd_add
tff(fact_3966_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))) ) ) ).

% dvd_add
tff(fact_3967_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(B,C))) ) ) ).

% dvd_add
tff(fact_3968_dvd__add,axiom,
    ! [A: nat,B: nat,C: 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,A),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))) ) ) ).

% dvd_add
tff(fact_3969_dvd__add,axiom,
    ! [A: int,B: int,C: 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,A),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(plus_plus_int(B),C))) ) ) ).

% dvd_add
tff(fact_3970_dvd__add__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_3971_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_3972_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(B,C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_3973_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_3974_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(plus_plus_int(B),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_3975_dvd__add__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( 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,A),plus_p5714425477246183910nteger(B,C)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_3976_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_3977_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(B,C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_3978_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: 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,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_3979_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: 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,A),aa_int_int(plus_plus_int(B),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_3980_dvd__diff,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Y))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Z))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),minus_8373710615458151222nteger(Y,Z))) ) ) ).

% dvd_diff
tff(fact_3981_dvd__diff,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))) ) ) ).

% dvd_diff
tff(fact_3982_dvd__diff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),minus_minus_rat(Y,Z))) ) ) ).

% dvd_diff
tff(fact_3983_dvd__diff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_int_int(minus_minus_int(Y),Z))) ) ) ).

% dvd_diff
tff(fact_3984_dvd__diff__commute,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),minus_8373710615458151222nteger(C,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),minus_8373710615458151222nteger(B,C))) ) ).

% dvd_diff_commute
tff(fact_3985_dvd__diff__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(minus_minus_int(C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(minus_minus_int(B),C))) ) ).

% dvd_diff_commute
tff(fact_3986_dvd__div__eq__iff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
       => ( ( divide6298287555418463151nteger(A,C) = divide6298287555418463151nteger(B,C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3987_dvd__div__eq__iff,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C),A))
     => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C),B))
       => ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3988_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),B))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3989_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),B))
       => ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3990_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3991_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3992_dvd__div__eq__cancel,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger(A,C) = divide6298287555418463151nteger(B,C) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),A))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3993_dvd__div__eq__cancel,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
     => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C),A))
       => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3994_dvd__div__eq__cancel,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) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3995_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3996_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = 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,C),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3997_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3998_div__div__div__same,axiom,
    ! [D: code_integer,B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
       => ( divide6298287555418463151nteger(divide6298287555418463151nteger(A,D),divide6298287555418463151nteger(B,D)) = divide6298287555418463151nteger(A,B) ) ) ) ).

% div_div_div_same
tff(fact_3999_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),B))
     => ( 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,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% div_div_div_same
tff(fact_4000_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),D)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% div_div_div_same
tff(fact_4001_dvd__power__same,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Y))
     => 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),N))) ) ).

% dvd_power_same
tff(fact_4002_dvd__power__same,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),Y))
     => 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),N))) ) ).

% dvd_power_same
tff(fact_4003_dvd__power__same,axiom,
    ! [X: real,Y: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y))
     => 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),N))) ) ).

% dvd_power_same
tff(fact_4004_dvd__power__same,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),Y))
     => 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),N))) ) ).

% dvd_power_same
tff(fact_4005_dvd__power__same,axiom,
    ! [X: int,Y: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
     => 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),N))) ) ).

% dvd_power_same
tff(fact_4006_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)))
     => ( 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,C),A)) ) ) ).

% dvd_mod_imp_dvd
tff(fact_4007_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B)))
     => ( 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,C),A)) ) ) ).

% dvd_mod_imp_dvd
tff(fact_4008_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B)))
     => ( 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,C),A)) ) ) ).

% dvd_mod_imp_dvd
tff(fact_4009_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: 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,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A)) ) ) ).

% dvd_mod_iff
tff(fact_4010_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: 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,C),aa_int_int(modulo_modulo_int(A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A)) ) ) ).

% dvd_mod_iff
tff(fact_4011_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( 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,C),modulo364778990260209775nteger(A,B)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),A)) ) ) ).

% dvd_mod_iff
tff(fact_4012_dvd__mod,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))) ) ) ).

% dvd_mod
tff(fact_4013_dvd__mod,axiom,
    ! [K: int,M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(modulo_modulo_int(M),N))) ) ) ).

% dvd_mod
tff(fact_4014_dvd__mod,axiom,
    ! [K: code_integer,M: code_integer,N: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,K),M))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,K),N))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,K),modulo364778990260209775nteger(M,N))) ) ) ).

% dvd_mod
tff(fact_4015_mod__mod__cancel,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) ) ) ).

% mod_mod_cancel
tff(fact_4016_mod__mod__cancel,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( aa_int_int(modulo_modulo_int(aa_int_int(modulo_modulo_int(A),B)),C) = aa_int_int(modulo_modulo_int(A),C) ) ) ).

% mod_mod_cancel
tff(fact_4017_mod__mod__cancel,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
     => ( modulo364778990260209775nteger(modulo364778990260209775nteger(A,B),C) = modulo364778990260209775nteger(A,C) ) ) ).

% mod_mod_cancel
tff(fact_4018_signed__take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(times_times_int(aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),L))) = aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(times_times_int(K),L)) ).

% signed_take_bit_mult
tff(fact_4019_signed__take__bit__add,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(plus_plus_int(aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),L))) = aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(plus_plus_int(K),L)) ).

% signed_take_bit_add
tff(fact_4020_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_4021_signed__take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(minus_minus_int(aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),L))) = aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(minus_minus_int(K),L)) ).

% signed_take_bit_diff
tff(fact_4022_zdvd__zdiffD,axiom,
    ! [K: int,M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(minus_minus_int(M),N)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),M)) ) ) ).

% zdvd_zdiffD
tff(fact_4023_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_4024_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),B))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)),D) ) )
         => ? [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
              & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Y2)),D) )
                | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_4025_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D2: nat,X2: nat,Y2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),A))
      & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),B))
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),D2) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)),D2) ) ) ) ).

% bezout_add_nat
tff(fact_4026_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D2: nat,X2: nat,Y2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),A))
      & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),B))
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)) = D2 )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)) = D2 ) ) ) ).

% bezout1_nat
tff(fact_4027_subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ord_less_eq_set_real(collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_gd,A)),collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_gd,B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% subset_divisors_dvd
tff(fact_4028_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_ge,A))),collect_int(aa_int_fun_int_bool(aTP_Lamm_ge,B))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% subset_divisors_dvd
tff(fact_4029_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ord_le7084787975880047091nteger(collect_Code_integer(aa_Cod7250484176784686200r_bool(aTP_Lamm_gf,A)),collect_Code_integer(aa_Cod7250484176784686200r_bool(aTP_Lamm_gf,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B)) ) ).

% subset_divisors_dvd
tff(fact_4030_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gg,A))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gg,B))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% subset_divisors_dvd
tff(fact_4031_concat__bit__assoc,axiom,
    ! [N: nat,K: int,M: nat,L: int,R: int] : aa_int_int(bit_concat_bit(N,K),aa_int_int(bit_concat_bit(M,L),R)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N),aa_int_int(bit_concat_bit(N,K),L)),R) ).

% concat_bit_assoc
tff(fact_4032_strict__subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ord_less_set_real(collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_gd,A)),collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_gd,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_4033_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gg,A))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gg,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_4034_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_ge,A))),collect_int(aa_int_fun_int_bool(aTP_Lamm_ge,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_4035_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ord_le1307284697595431911nteger(collect_Code_integer(aa_Cod7250484176784686200r_bool(aTP_Lamm_gf,A)),collect_Code_integer(aa_Cod7250484176784686200r_bool(aTP_Lamm_gf,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_4036_even__signed__take__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(M,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)) ) ).

% even_signed_take_bit_iff
tff(fact_4037_even__signed__take__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(aa_nat_fun_int_int(bit_ri631733984087533419it_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)) ) ).

% even_signed_take_bit_iff
tff(fact_4038_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_4039_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_4040_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_4041_minf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z4: code_integer] :
    ! [X4: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X4),Z4))
     => ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S)))
      <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S))) ) ) ).

% minf(10)
tff(fact_4042_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z4))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S)))
      <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S))) ) ) ).

% minf(10)
tff(fact_4043_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z4))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X4,S)))
      <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X4,S))) ) ) ).

% minf(10)
tff(fact_4044_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z4))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),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,X4),S))) ) ) ).

% minf(10)
tff(fact_4045_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z4))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),S)))
      <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),S))) ) ) ).

% minf(10)
tff(fact_4046_minf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z4: code_integer] :
    ! [X4: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X4),Z4))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S))) ) ) ).

% minf(9)
tff(fact_4047_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z4))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S))) ) ) ).

% minf(9)
tff(fact_4048_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z4))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X4,S)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X4,S))) ) ) ).

% minf(9)
tff(fact_4049_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),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,X4),S))) ) ) ).

% minf(9)
tff(fact_4050_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),S)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),S))) ) ) ).

% minf(9)
tff(fact_4051_pinf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z4: code_integer] :
    ! [X4: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z4),X4))
     => ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S)))
      <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S))) ) ) ).

% pinf(10)
tff(fact_4052_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X4))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S)))
      <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S))) ) ) ).

% pinf(10)
tff(fact_4053_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X4))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X4,S)))
      <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X4,S))) ) ) ).

% pinf(10)
tff(fact_4054_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X4))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),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,X4),S))) ) ) ).

% pinf(10)
tff(fact_4055_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X4))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),S)))
      <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),S))) ) ) ).

% pinf(10)
tff(fact_4056_pinf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z4: code_integer] :
    ! [X4: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z4),X4))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S))) ) ) ).

% pinf(9)
tff(fact_4057_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X4))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S))) ) ) ).

% pinf(9)
tff(fact_4058_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X4))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X4,S)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X4,S))) ) ) ).

% pinf(9)
tff(fact_4059_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),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,X4),S))) ) ) ).

% pinf(9)
tff(fact_4060_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),S)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),S))) ) ) ).

% pinf(9)
tff(fact_4061_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_4062_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_4063_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_4064_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ( ( divide_divide_rat(A,B) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_4065_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_4066_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(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_4067_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_4068_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_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) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_4069_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( ( aa_int_int(times_times_int(B),A) = aa_int_int(times_times_int(C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_4070_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_4071_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_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) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_4072_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( ( aa_int_int(times_times_int(A),B) = aa_int_int(times_times_int(A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_4073_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),C)) ) ) ).

% mult_unit_dvd_iff'
tff(fact_4074_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ).

% mult_unit_dvd_iff'
tff(fact_4075_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(A),B)),C))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ).

% mult_unit_dvd_iff'
tff(fact_4076_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C)) ) ) ).

% dvd_mult_unit_iff'
tff(fact_4077_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_mult_unit_iff'
tff(fact_4078_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(times_times_int(B),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_mult_unit_iff'
tff(fact_4079_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C)) ) ) ).

% mult_unit_dvd_iff
tff(fact_4080_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% mult_unit_dvd_iff
tff(fact_4081_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(A),B)),C))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% mult_unit_dvd_iff
tff(fact_4082_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C)) ) ) ).

% dvd_mult_unit_iff
tff(fact_4083_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( 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)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_mult_unit_iff
tff(fact_4084_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(times_times_int(C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_mult_unit_iff
tff(fact_4085_is__unit__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer)) ) ) ).

% is_unit_mult_iff
tff(fact_4086_is__unit__mult__iff,axiom,
    ! [A: nat,B: 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)),one_one_nat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat)) ) ) ).

% is_unit_mult_iff
tff(fact_4087_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(A),B)),one_one_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int)) ) ) ).

% is_unit_mult_iff
tff(fact_4088_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
      ( 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,D),C))
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),divide6298287555418463151nteger(C,D)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_4089_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: 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,D),C))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),D)) = 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),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_4090_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: 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,D),C))
       => ( aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_4091_dvd__mult__imp__div,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)),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C))) ) ).

% dvd_mult_imp_div
tff(fact_4092_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: 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),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))) ) ).

% dvd_mult_imp_div
tff(fact_4093_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(A),C)),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))) ) ).

% dvd_mult_imp_div
tff(fact_4094_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A))
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_4095_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: 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),C)),A))
     => ( 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) ) ) ).

% dvd_div_mult2_eq
tff(fact_4096_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(B),C)),A))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(times_times_int(B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_4097_div__div__eq__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( 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,B),A))
       => ( divide6298287555418463151nteger(A,divide6298287555418463151nteger(B,C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_4098_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: 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,B),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_4099_div__div__eq__right,axiom,
    ! [C: int,B: int,A: 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,B),A))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_4100_div__mult__swap,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ) ) ).

% div_mult_swap
tff(fact_4101_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_4102_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( aa_int_int(times_times_int(A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A),B)),C) ) ) ).

% div_mult_swap
tff(fact_4103_dvd__div__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,C)),A) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),C) ) ) ).

% dvd_div_mult
tff(fact_4104_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_4105_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_4106_unit__div__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(C,A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_4107_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_4108_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_4109_div__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( 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),C)) ) ) ).

% div_unit_dvd_iff
tff(fact_4110_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% div_unit_dvd_iff
tff(fact_4111_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% div_unit_dvd_iff
tff(fact_4112_dvd__div__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(C,B)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C)) ) ) ).

% dvd_div_unit_iff
tff(fact_4113_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_div_unit_iff
tff(fact_4114_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_div_unit_iff
tff(fact_4115_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),A))
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_4116_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_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,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_4117_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_4118_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_4119_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_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,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_4120_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_4121_div__power,axiom,
    ! [B: code_integer,A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(divide6298287555418463151nteger(A,B)),N) = divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),N),aa_nat_Code_integer(power_8256067586552552935nteger(B),N)) ) ) ).

% div_power
tff(fact_4122_div__power,axiom,
    ! [B: nat,A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_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)) ) ) ).

% div_power
tff(fact_4123_div__power,axiom,
    ! [B: int,A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ( aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),N) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)) ) ) ).

% div_power
tff(fact_4124_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_4125_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_4126_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_4127_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_4128_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_4129_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_4130_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_4131_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_4132_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_4133_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_4134_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_4135_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_4136_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_4137_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_4138_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_4139_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_4140_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_4141_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_4142_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_4143_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_4144_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_4145_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_4146_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_4147_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_4148_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)))) ).

% dvd_minus_mod
tff(fact_4149_dvd__minus__mod,axiom,
    ! [B: int,A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),aa_int_int(minus_minus_int(A),aa_int_int(modulo_modulo_int(A),B)))) ).

% dvd_minus_mod
tff(fact_4150_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)))) ).

% dvd_minus_mod
tff(fact_4151_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(B),C) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),aa_int_int(minus_minus_int(A),B))) ) ).

% mod_eq_dvd_iff
tff(fact_4152_mod__eq__dvd__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),minus_8373710615458151222nteger(A,B))) ) ).

% mod_eq_dvd_iff
tff(fact_4153_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_4154_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D2: nat,X2: nat,Y2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),B))
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),D2) ) ) ) ).

% bezout_add_strong_nat
tff(fact_4155_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_4156_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_4157_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_4158_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_4159_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_4160_zdvd__mono,axiom,
    ! [K: int,M: int,T: int] :
      ( ( K != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),T))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(K),M)),aa_int_int(times_times_int(K),T))) ) ) ).

% zdvd_mono
tff(fact_4161_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(K),M)),aa_int_int(times_times_int(K),N)))
     => ( ( K != zero_zero_int )
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N)) ) ) ).

% zdvd_mult_cancel
tff(fact_4162_dbl__def,axiom,
    ! [X: real] : aa_real_real(neg_numeral_dbl_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),X) ).

% dbl_def
tff(fact_4163_dbl__def,axiom,
    ! [X: rat] : neg_numeral_dbl_rat(X) = plus_plus_rat(X,X) ).

% dbl_def
tff(fact_4164_dbl__def,axiom,
    ! [X: int] : aa_int_int(neg_numeral_dbl_int,X) = aa_int_int(plus_plus_int(X),X) ).

% dbl_def
tff(fact_4165_zdvd__period,axiom,
    ! [A: int,D: int,X: int,T: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),D))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(plus_plus_int(X),T)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(X),aa_int_int(times_times_int(C),D))),T))) ) ) ).

% zdvd_period
tff(fact_4166_zdvd__reduce,axiom,
    ! [K: int,N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(plus_plus_int(N),aa_int_int(times_times_int(K),M))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N)) ) ).

% zdvd_reduce
tff(fact_4167_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_gg,M)))) ) ).

% finite_divisors_nat
tff(fact_4168_div2__even__ext__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Y),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))),X))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Y)) )
       => ( X = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_4169_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 )
         => ! [C2: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2) ) ) ).

% unit_dvdE
tff(fact_4170_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 )
         => ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) ) ) ).

% unit_dvdE
tff(fact_4171_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 )
         => ! [C2: int] : B != aa_int_int(times_times_int(A),C2) ) ) ).

% unit_dvdE
tff(fact_4172_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_4173_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_4174_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_4175_unity__coeff__ex,axiom,
    ! [P: fun_rat_bool,L: rat] :
      ( ? [X3: rat] : pp(aa_rat_bool(P,aa_rat_rat(times_times_rat(L),X3)))
    <=> ? [X3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,L),plus_plus_rat(X3,zero_zero_rat)))
          & pp(aa_rat_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_4176_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_4177_unity__coeff__ex,axiom,
    ! [P: fun_int_bool,L: int] :
      ( ? [X3: int] : pp(aa_int_bool(P,aa_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(plus_plus_int(X3),zero_zero_int)))
          & pp(aa_int_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_4178_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_4179_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_4180_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(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,D),C) )
            <=> ( aa_int_int(times_times_int(B),C) = aa_int_int(times_times_int(A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_4181_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_4182_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_4183_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(aa_int_fun_int_int(divide_divide_int,B),C)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(A),C)),B)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_4184_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_4185_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_4186_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(aa_int_fun_int_int(divide_divide_int,A),B)),C))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(times_times_int(C),B))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_4187_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_4188_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_4189_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(aa_int_fun_int_int(divide_divide_int,B),A) = C )
        <=> ( B = aa_int_int(times_times_int(C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_4190_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_4191_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_4192_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(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_4193_even__numeral,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,N)))) ).

% even_numeral
tff(fact_4194_even__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)))) ).

% even_numeral
tff(fact_4195_even__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) ).

% even_numeral
tff(fact_4196_inf__period_I3_J,axiom,
    ! [D: code_integer,D3: code_integer,T: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),D3))
     => ! [X4: code_integer,K4: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,T)))
        <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X4,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D3)),T))) ) ) ).

% inf_period(3)
tff(fact_4197_inf__period_I3_J,axiom,
    ! [D: real,D3: real,T: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),D3))
     => ! [X4: real,K4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),T)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3))),T))) ) ) ).

% inf_period(3)
tff(fact_4198_inf__period_I3_J,axiom,
    ! [D: rat,D3: rat,T: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),D3))
     => ! [X4: rat,K4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X4,T)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(minus_minus_rat(X4,aa_rat_rat(times_times_rat(K4),D3)),T))) ) ) ).

% inf_period(3)
tff(fact_4199_inf__period_I3_J,axiom,
    ! [D: int,D3: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D3))
     => ! [X4: int,K4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),T)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X4),aa_int_int(times_times_int(K4),D3))),T))) ) ) ).

% inf_period(3)
tff(fact_4200_inf__period_I4_J,axiom,
    ! [D: code_integer,D3: code_integer,T: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),D3))
     => ! [X4: code_integer,K4: code_integer] :
          ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,T)))
        <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X4,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D3)),T))) ) ) ).

% inf_period(4)
tff(fact_4201_inf__period_I4_J,axiom,
    ! [D: real,D3: real,T: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),D3))
     => ! [X4: real,K4: real] :
          ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),T)))
        <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3))),T))) ) ) ).

% inf_period(4)
tff(fact_4202_inf__period_I4_J,axiom,
    ! [D: rat,D3: rat,T: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),D3))
     => ! [X4: rat,K4: rat] :
          ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X4,T)))
        <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(minus_minus_rat(X4,aa_rat_rat(times_times_rat(K4),D3)),T))) ) ) ).

% inf_period(4)
tff(fact_4203_inf__period_I4_J,axiom,
    ! [D: int,D3: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D3))
     => ! [X4: int,K4: int] :
          ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),T)))
        <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X4),aa_int_int(times_times_int(K4),D3))),T))) ) ) ).

% inf_period(4)
tff(fact_4204_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),one_one_Code_integer))
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_4205_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),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),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) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_4206_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),one_one_int))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(times_times_int(B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_4207_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),one_one_Code_integer))
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ) ) ).

% unit_div_mult_swap
tff(fact_4208_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_4209_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),one_one_int))
     => ( aa_int_int(times_times_int(A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_4210_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),C) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),B) ) ) ).

% unit_div_commute
tff(fact_4211_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: 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(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_4212_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A),C)),B) ) ) ).

% unit_div_commute
tff(fact_4213_div__mult__unit2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ) ).

% div_mult_unit2
tff(fact_4214_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),one_one_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),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) ) ) ) ).

% div_mult_unit2
tff(fact_4215_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(times_times_int(B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_4216_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( ( A = divide6298287555418463151nteger(C,B) )
      <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_4217_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_4218_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( ( A = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B) )
      <=> ( aa_int_int(times_times_int(A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_4219_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( ( divide6298287555418463151nteger(A,B) = C )
      <=> ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_4220_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: 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) = C )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_4221_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = C )
      <=> ( A = aa_int_int(times_times_int(C),B) ) ) ) ).

% unit_eq_div1
tff(fact_4222_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_4223_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_4224_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_4225_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_4226_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_4227_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_4228_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_4229_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_4230_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_4231_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_4232_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_4233_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q2: 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),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_4234_even__zero,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_z3403309356797280102nteger)) ).

% even_zero
tff(fact_4235_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_4236_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_4237_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_4238_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_4239_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(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(times_times_int(B),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4240_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_4241_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_4242_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(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(times_times_int(A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4243_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 )
         => ! [B4: code_integer] :
              ( ( B4 != zero_z3403309356797280102nteger )
             => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B4),one_one_Code_integer))
               => ( ( divide6298287555418463151nteger(one_one_Code_integer,A) = B4 )
                 => ( ( divide6298287555418463151nteger(one_one_Code_integer,B4) = A )
                   => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B4) = one_one_Code_integer )
                     => ( divide6298287555418463151nteger(C,A) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4244_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 )
         => ! [B4: nat] :
              ( ( B4 != zero_zero_nat )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B4),one_one_nat))
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B4 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B4) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B4) = 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),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4245_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 )
         => ! [B4: int] :
              ( ( B4 != zero_zero_int )
             => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B4),one_one_int))
               => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A) = B4 )
                 => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B4) = A )
                   => ( ( aa_int_int(times_times_int(A),B4) = one_one_int )
                     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) != aa_int_int(times_times_int(C),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4246_evenE,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
     => ~ ! [B4: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B4) ) ).

% evenE
tff(fact_4247_evenE,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))
     => ~ ! [B4: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B4) ) ).

% evenE
tff(fact_4248_evenE,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))
     => ~ ! [B4: int] : A != aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B4) ) ).

% evenE
tff(fact_4249_odd__one,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),one_one_Code_integer)) ).

% odd_one
tff(fact_4250_odd__one,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),one_one_nat)) ).

% odd_one
tff(fact_4251_odd__one,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),one_one_int)) ).

% odd_one
tff(fact_4252_odd__even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
     => ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))) ) ) ).

% odd_even_add
tff(fact_4253_odd__even__add,axiom,
    ! [A: nat,B: 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(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
       => 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,A),B))) ) ) ).

% odd_even_add
tff(fact_4254_odd__even__add,axiom,
    ! [A: int,B: 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))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))
       => 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(plus_plus_int(A),B))) ) ) ).

% odd_even_add
tff(fact_4255_bit__eq__rec,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
        <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)) )
        & ( divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = divide6298287555418463151nteger(B,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_4256_bit__eq__rec,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( ( 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(dvd_dvd_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_4257_bit__eq__rec,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( ( 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_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) )
        & ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_4258_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_4259_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_4260_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_4261_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_4262_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_4263_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_4264_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_4265_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_4266_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_4267_even__even__mod__4__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))))) ) ).

% even_even_mod_4_iff
tff(fact_4268_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_4269_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_4270_dvd__minus__add,axiom,
    ! [Q2: nat,N: nat,R: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),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),Q2)))
        <=> 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,R),M)),Q2)))) ) ) ) ).

% dvd_minus_add
tff(fact_4271_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_4272_mod__nat__eqI,axiom,
    ! [R: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,R),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R),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),R)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = R ) ) ) ) ).

% mod_nat_eqI
tff(fact_4273_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_4274_bset_I9_J,axiom,
    ! [D: int,D3: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D3))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),B2))
                 => ( X4 != aa_int_int(plus_plus_int(Xb3),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),T)))
           => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X4),D3)),T))) ) ) ) ).

% bset(9)
tff(fact_4275_bset_I10_J,axiom,
    ! [D: int,D3: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D3))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),B2))
                 => ( X4 != aa_int_int(plus_plus_int(Xb3),Xa3) ) ) )
         => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),T)))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X4),D3)),T))) ) ) ) ).

% bset(10)
tff(fact_4276_aset_I9_J,axiom,
    ! [D: int,D3: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D3))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),A2))
                 => ( X4 != aa_int_int(minus_minus_int(Xb3),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),T)))
           => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(X4),D3)),T))) ) ) ) ).

% aset(9)
tff(fact_4277_aset_I10_J,axiom,
    ! [D: int,D3: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D3))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D3)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),A2))
                 => ( X4 != aa_int_int(minus_minus_int(Xb3),Xa3) ) ) )
         => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),T)))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(X4),D3)),T))) ) ) ) ).

% aset(10)
tff(fact_4278_even__two__times__div__two,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_4279_even__two__times__div__two,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(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)))) = A ) ) ).

% even_two_times_div_two
tff(fact_4280_even__two__times__div__two,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(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_4281_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_4282_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_4283_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
    <=> ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_4284_odd__iff__mod__2__eq__one,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))) = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_4285_odd__iff__mod__2__eq__one,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))) = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_4286_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
    <=> ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_4287_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_4288_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_4289_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_4290_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_4291_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_4292_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% signed_take_bit_int_less_exp
tff(fact_4293_even__unset__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se8260200283734997820nteger(M,A)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_4294_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(aa_nat_fun_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_4295_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_4296_even__set__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se2793503036327961859nteger(M,A)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_4297_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(aa_nat_fun_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_4298_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_4299_even__flip__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se1345352211410354436nteger(M,A)))
    <=> ~ ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_4300_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(aa_nat_fun_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_4301_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_4302_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(minus_minus_int(K),L)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(K),L))) ) ).

% even_diff_iff
tff(fact_4303_oddE,axiom,
    ! [A: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
     => ~ ! [B4: code_integer] : A != plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B4),one_one_Code_integer) ) ).

% oddE
tff(fact_4304_oddE,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))
     => ~ ! [B4: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B4)),one_one_nat) ) ).

% oddE
tff(fact_4305_oddE,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))
     => ~ ! [B4: int] : A != aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B4)),one_one_int) ) ).

% oddE
tff(fact_4306_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_4307_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_4308_mod2__eq__if,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
       => ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
       => ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ) ) ) ).

% mod2_eq_if
tff(fact_4309_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_4310_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_4311_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
       => ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
         => ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_4312_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_4313_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_4314_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_4315_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_4316_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_4317_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_4318_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_4319_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_4320_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_4321_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(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_4322_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(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_4323_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_4324_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_4325_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_4326_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( ! [A4: nat,B4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),B4))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B4),A4)) )
     => ( ! [A4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),zero_zero_nat))
       => ( ! [A4: nat,B4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),B4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ) ).

% Euclid_induct
tff(fact_4327_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)),aa_int_int(minus_minus_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N))))) ) ).

% signed_take_bit_int_less_eq
tff(fact_4328_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(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_4329_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_4330_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(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_4331_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_4332_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_4333_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_4334_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_4335_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_4336_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_4337_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_4338_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N))))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(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_4339_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_4340_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(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_4341_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,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(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(aa_num_Code_integer(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,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(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_4342_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_4343_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(aa_int_fun_int_int(divide_divide_int,aa_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(aa_int_fun_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_4344_triangle__def,axiom,
    ! [N: nat] : aa_nat_nat(nat_triangle,N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_4345_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) ) )
         => ~ ! [Va3: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va3)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))))
                     => ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))))
                     => ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_4346_signed__take__bit__rec,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( N = zero_zero_nat )
       => ( bit_ri6519982836138164636nteger(N,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) )
      & ( ( N != zero_zero_nat )
       => ( bit_ri6519982836138164636nteger(N,A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(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,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ) ) ) ).

% signed_take_bit_rec
tff(fact_4347_signed__take__bit__rec,axiom,
    ! [N: nat,A: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_int_int(aa_nat_fun_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(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),A) = aa_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(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% signed_take_bit_rec
tff(fact_4348_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,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_4349_flip__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,zero_zero_nat),A) = aa_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(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_4350_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_4351_set__decode__Suc,axiom,
    ! [N: nat,X: nat] :
      ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,N)),aa_nat_set_nat(nat_set_decode,X)))
    <=> pp(aa_set_nat_bool(member_nat(N),aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% set_decode_Suc
tff(fact_4352_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_4353_diff__shunt__var,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( minus_minus_set_real(X,Y) = bot_bot_set_real )
    <=> ord_less_eq_set_real(X,Y) ) ).

% diff_shunt_var
tff(fact_4354_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(ord_less_eq_set_nat(X),Y)) ) ).

% diff_shunt_var
tff(fact_4355_add__scale__eq__noteq,axiom,
    ! [R: complex,A: complex,B: complex,C: complex,D: complex] :
      ( ( R != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(R),C)) != aa_complex_complex(plus_plus_complex(B),aa_complex_complex(times_times_complex(R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4356_add__scale__eq__noteq,axiom,
    ! [R: real,A: real,B: real,C: real,D: real] :
      ( ( R != 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,R),C)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4357_add__scale__eq__noteq,axiom,
    ! [R: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( plus_plus_rat(A,aa_rat_rat(times_times_rat(R),C)) != plus_plus_rat(B,aa_rat_rat(times_times_rat(R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4358_add__scale__eq__noteq,axiom,
    ! [R: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R != 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,R),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4359_add__scale__eq__noteq,axiom,
    ! [R: int,A: int,B: int,C: int,D: int] :
      ( ( R != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(R),C)) != aa_int_int(plus_plus_int(B),aa_int_int(times_times_int(R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4360_artanh__def,axiom,
    ! [X: real] : aa_real_real(artanh_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_4361_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(nth_nat(replicate_nat(N,X)),I))) ) ) ).

% intind
tff(fact_4362_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_4363_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_4364_verit__minus__simplify_I4_J,axiom,
    ! [B: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,B)) = B ).

% verit_minus_simplify(4)
tff(fact_4365_verit__minus__simplify_I4_J,axiom,
    ! [B: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,B)) = B ).

% verit_minus_simplify(4)
tff(fact_4366_verit__minus__simplify_I4_J,axiom,
    ! [B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,B)) = B ).

% verit_minus_simplify(4)
tff(fact_4367_verit__minus__simplify_I4_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = B ).

% verit_minus_simplify(4)
tff(fact_4368_verit__minus__simplify_I4_J,axiom,
    ! [B: rat] : uminus_uminus_rat(uminus_uminus_rat(B)) = B ).

% verit_minus_simplify(4)
tff(fact_4369_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_4370_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_4371_neg__equal__iff__equal,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4372_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_4373_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat(A) = uminus_uminus_rat(B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4374_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_4375_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_4376_add_Oinverse__inverse,axiom,
    ! [A: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = A ).

% add.inverse_inverse
tff(fact_4377_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).

% add.inverse_inverse
tff(fact_4378_add_Oinverse__inverse,axiom,
    ! [A: rat] : uminus_uminus_rat(uminus_uminus_rat(A)) = A ).

% add.inverse_inverse
tff(fact_4379_compl__le__compl__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(X)),uminus5710092332889474511et_nat(Y)))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X)) ) ).

% compl_le_compl_iff
tff(fact_4380_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_4381_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_4382_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(B)),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_4383_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_4384_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_4385_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_4386_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_4387_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat(A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_4388_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_4389_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_4390_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_4391_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = uminus_uminus_rat(A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_4392_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_4393_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_4394_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_4395_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_4396_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
tff(fact_4397_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_4398_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_4399_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_4400_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_4401_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = uminus_uminus_rat(A) )
    <=> ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4402_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_4403_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_4404_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_4405_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_4406_add_Oinverse__neutral,axiom,
    uminus_uminus_rat(zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_4407_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_4408_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_4409_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_4410_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% neg_less_iff_less
tff(fact_4411_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(M)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_4412_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_4413_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_4414_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_4415_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( uminus_uminus_rat(numeral_numeral_rat(M)) = uminus_uminus_rat(numeral_numeral_rat(N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_4416_mult__minus__left,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% mult_minus_left
tff(fact_4417_mult__minus__left,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(times_times_int(A),B)) ).

% mult_minus_left
tff(fact_4418_mult__minus__left,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B)) ).

% mult_minus_left
tff(fact_4419_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_left
tff(fact_4420_mult__minus__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(A)),B) = uminus_uminus_rat(aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_left
tff(fact_4421_minus__mult__minus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) ).

% minus_mult_minus
tff(fact_4422_minus__mult__minus,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(times_times_int(A),B) ).

% minus_mult_minus
tff(fact_4423_minus__mult__minus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(times_times_complex(A),B) ).

% minus_mult_minus
tff(fact_4424_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) ).

% minus_mult_minus
tff(fact_4425_minus__mult__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(A)),uminus_uminus_rat(B)) = aa_rat_rat(times_times_rat(A),B) ).

% minus_mult_minus
tff(fact_4426_mult__minus__right,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% mult_minus_right
tff(fact_4427_mult__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(times_times_int(A),B)) ).

% mult_minus_right
tff(fact_4428_mult__minus__right,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B)) ).

% mult_minus_right
tff(fact_4429_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_right
tff(fact_4430_mult__minus__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),uminus_uminus_rat(B)) = uminus_uminus_rat(aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_right
tff(fact_4431_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_4432_minus__add__distrib,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(A),B)) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) ).

% minus_add_distrib
tff(fact_4433_minus__add__distrib,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_add_distrib
tff(fact_4434_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_4435_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(plus_plus_rat(A,B)) = plus_plus_rat(uminus_uminus_rat(A),uminus_uminus_rat(B)) ).

% minus_add_distrib
tff(fact_4436_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_4437_minus__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(plus_plus_int(A),B)) = B ).

% minus_add_cancel
tff(fact_4438_minus__add__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(plus_plus_complex(A),B)) = B ).

% minus_add_cancel
tff(fact_4439_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_4440_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : plus_plus_rat(uminus_uminus_rat(A),plus_plus_rat(A,B)) = B ).

% minus_add_cancel
tff(fact_4441_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_4442_add__minus__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),B)) = B ).

% add_minus_cancel
tff(fact_4443_add__minus__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B)) = B ).

% add_minus_cancel
tff(fact_4444_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_4445_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : plus_plus_rat(A,plus_plus_rat(uminus_uminus_rat(A),B)) = B ).

% add_minus_cancel
tff(fact_4446_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_4447_minus__diff__eq,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(minus_minus_int(A),B)) = aa_int_int(minus_minus_int(B),A) ).

% minus_diff_eq
tff(fact_4448_minus__diff__eq,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A),B)) = aa_complex_complex(minus_minus_complex(B),A) ).

% minus_diff_eq
tff(fact_4449_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_4450_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(minus_minus_rat(A,B)) = minus_minus_rat(B,A) ).

% minus_diff_eq
tff(fact_4451_div__minus__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ).

% div_minus_minus
tff(fact_4452_div__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = divide6298287555418463151nteger(A,B) ).

% div_minus_minus
tff(fact_4453_minus__dvd__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(uminus_uminus_real,X)),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y)) ) ).

% minus_dvd_iff
tff(fact_4454_minus__dvd__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(uminus_uminus_int,X)),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y)) ) ).

% minus_dvd_iff
tff(fact_4455_minus__dvd__iff,axiom,
    ! [X: complex,Y: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(uminus1482373934393186551omplex,X)),Y))
    <=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),Y)) ) ).

% minus_dvd_iff
tff(fact_4456_minus__dvd__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),Y))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Y)) ) ).

% minus_dvd_iff
tff(fact_4457_minus__dvd__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,uminus_uminus_rat(X)),Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Y)) ) ).

% minus_dvd_iff
tff(fact_4458_dvd__minus__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_real_real(uminus_uminus_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y)) ) ).

% dvd_minus_iff
tff(fact_4459_dvd__minus__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_int_int(uminus_uminus_int,Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y)) ) ).

% dvd_minus_iff
tff(fact_4460_dvd__minus__iff,axiom,
    ! [X: complex,Y: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),aa_complex_complex(uminus1482373934393186551omplex,Y)))
    <=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),Y)) ) ).

% dvd_minus_iff
tff(fact_4461_dvd__minus__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Y)) ) ).

% dvd_minus_iff
tff(fact_4462_dvd__minus__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),uminus_uminus_rat(Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Y)) ) ).

% dvd_minus_iff
tff(fact_4463_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_4464_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_4465_mod__minus__minus,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),B)) ).

% mod_minus_minus
tff(fact_4466_mod__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,B)) ).

% mod_minus_minus
tff(fact_4467_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_4468_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_4469_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_4470_of__bool__less__eq__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_n356916108424825756nteger(P)),zero_n356916108424825756nteger(Q)))
    <=> ( pp(P)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_4471_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n1201886186963655149omplex(P) = zero_zero_complex )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_4472_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n3304061248610475627l_real(P) = zero_zero_real )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_4473_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n2052037380579107095ol_rat(P) = zero_zero_rat )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_4474_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n2687167440665602831ol_nat(P) = zero_zero_nat )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_4475_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_4476_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n356916108424825756nteger(P) = zero_z3403309356797280102nteger )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_4477_of__bool__eq_I1_J,axiom,
    zero_n1201886186963655149omplex(fFalse) = zero_zero_complex ).

% of_bool_eq(1)
tff(fact_4478_of__bool__eq_I1_J,axiom,
    zero_n3304061248610475627l_real(fFalse) = zero_zero_real ).

% of_bool_eq(1)
tff(fact_4479_of__bool__eq_I1_J,axiom,
    zero_n2052037380579107095ol_rat(fFalse) = zero_zero_rat ).

% of_bool_eq(1)
tff(fact_4480_of__bool__eq_I1_J,axiom,
    zero_n2687167440665602831ol_nat(fFalse) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_4481_of__bool__eq_I1_J,axiom,
    aa_bool_int(zero_n2684676970156552555ol_int,fFalse) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_4482_of__bool__eq_I1_J,axiom,
    zero_n356916108424825756nteger(fFalse) = zero_z3403309356797280102nteger ).

% of_bool_eq(1)
tff(fact_4483_real__add__minus__iff,axiom,
    ! [X: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(uminus_uminus_real,A)) = zero_zero_real )
    <=> ( X = A ) ) ).

% real_add_minus_iff
tff(fact_4484_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_4485_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_4486_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_4487_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_4488_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_n356916108424825756nteger(P)),zero_n356916108424825756nteger(Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_4489_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( zero_n1201886186963655149omplex(P) = one_one_complex )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_4490_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( zero_n3304061248610475627l_real(P) = one_one_real )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_4491_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( zero_n2052037380579107095ol_rat(P) = one_one_rat )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_4492_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( zero_n2687167440665602831ol_nat(P) = one_one_nat )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_4493_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( aa_bool_int(zero_n2684676970156552555ol_int,P) = one_one_int )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_4494_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( zero_n356916108424825756nteger(P) = one_one_Code_integer )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_4495_of__bool__eq_I2_J,axiom,
    zero_n1201886186963655149omplex(fTrue) = one_one_complex ).

% of_bool_eq(2)
tff(fact_4496_of__bool__eq_I2_J,axiom,
    zero_n3304061248610475627l_real(fTrue) = one_one_real ).

% of_bool_eq(2)
tff(fact_4497_of__bool__eq_I2_J,axiom,
    zero_n2052037380579107095ol_rat(fTrue) = one_one_rat ).

% of_bool_eq(2)
tff(fact_4498_of__bool__eq_I2_J,axiom,
    zero_n2687167440665602831ol_nat(fTrue) = one_one_nat ).

% of_bool_eq(2)
tff(fact_4499_of__bool__eq_I2_J,axiom,
    aa_bool_int(zero_n2684676970156552555ol_int,fTrue) = one_one_int ).

% of_bool_eq(2)
tff(fact_4500_of__bool__eq_I2_J,axiom,
    zero_n356916108424825756nteger(fTrue) = one_one_Code_integer ).

% of_bool_eq(2)
tff(fact_4501_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_4502_length__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(N,X)) = N ).

% length_replicate
tff(fact_4503_length__replicate,axiom,
    ! [N: nat,X: bool] : size_size_list_bool(replicate_bool(N,X)) = N ).

% length_replicate
tff(fact_4504_length__replicate,axiom,
    ! [N: nat,X: nat] : size_size_list_nat(replicate_nat(N,X)) = N ).

% length_replicate
tff(fact_4505_length__replicate,axiom,
    ! [N: nat,X: int] : size_size_list_int(replicate_int(N,X)) = N ).

% length_replicate
tff(fact_4506_of__bool__or__iff,axiom,
    ! [P: bool,Q: bool] : zero_n2687167440665602831ol_nat(aa_bool_bool(fdisj(P),Q)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_n2687167440665602831ol_nat(P)),zero_n2687167440665602831ol_nat(Q)) ).

% of_bool_or_iff
tff(fact_4507_of__bool__or__iff,axiom,
    ! [P: bool,Q: bool] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fdisj(P),Q)) = aa_int_int(ord_max_int(aa_bool_int(zero_n2684676970156552555ol_int,P)),aa_bool_int(zero_n2684676970156552555ol_int,Q)) ).

% of_bool_or_iff
tff(fact_4508_of__bool__or__iff,axiom,
    ! [P: bool,Q: bool] : zero_n356916108424825756nteger(aa_bool_bool(fdisj(P),Q)) = ord_max_Code_integer(zero_n356916108424825756nteger(P),zero_n356916108424825756nteger(Q)) ).

% of_bool_or_iff
tff(fact_4509_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_4510_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_4511_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_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_4512_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_4513_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_4514_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_4515_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_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_4516_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_4517_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_4518_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_4519_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),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_4520_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_4521_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_4522_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_4523_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_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_4524_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_4525_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_4526_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_4527_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_4528_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_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_4529_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_4530_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_4531_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_4532_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_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_4533_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_4534_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_4535_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_4536_neg__less__pos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_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_4537_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_4538_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_4539_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_4540_less__neg__neg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),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_4541_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_4542_ab__left__minus,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_left_minus
tff(fact_4543_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_4544_ab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_4545_ab__left__minus,axiom,
    ! [A: rat] : plus_plus_rat(uminus_uminus_rat(A),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_4546_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_4547_add_Oright__inverse,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(A),aa_int_int(uminus_uminus_int,A)) = zero_zero_int ).

% add.right_inverse
tff(fact_4548_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_4549_add_Oright__inverse,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_4550_add_Oright__inverse,axiom,
    ! [A: rat] : plus_plus_rat(A,uminus_uminus_rat(A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_4551_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_4552_diff__0,axiom,
    ! [A: int] : aa_int_int(minus_minus_int(zero_zero_int),A) = aa_int_int(uminus_uminus_int,A) ).

% diff_0
tff(fact_4553_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_4554_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% diff_0
tff(fact_4555_diff__0,axiom,
    ! [A: rat] : minus_minus_rat(zero_zero_rat,A) = uminus_uminus_rat(A) ).

% diff_0
tff(fact_4556_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_4557_verit__minus__simplify_I3_J,axiom,
    ! [B: int] : aa_int_int(minus_minus_int(zero_zero_int),B) = aa_int_int(uminus_uminus_int,B) ).

% verit_minus_simplify(3)
tff(fact_4558_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_4559_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% verit_minus_simplify(3)
tff(fact_4560_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : minus_minus_rat(zero_zero_rat,B) = uminus_uminus_rat(B) ).

% verit_minus_simplify(3)
tff(fact_4561_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(M)),numeral_numeral_real(N))) ).

% add_neg_numeral_simps(3)
tff(fact_4562_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ).

% add_neg_numeral_simps(3)
tff(fact_4563_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(N))) ).

% add_neg_numeral_simps(3)
tff(fact_4564_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_num_Code_integer(numera6620942414471956472nteger,N))) ).

% add_neg_numeral_simps(3)
tff(fact_4565_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(N))) = uminus_uminus_rat(plus_plus_rat(numeral_numeral_rat(M),numeral_numeral_rat(N))) ).

% add_neg_numeral_simps(3)
tff(fact_4566_mult__minus1,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,one_one_real)),Z) = aa_real_real(uminus_uminus_real,Z) ).

% mult_minus1
tff(fact_4567_mult__minus1,axiom,
    ! [Z: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,one_one_int)),Z) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1
tff(fact_4568_mult__minus1,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Z) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1
tff(fact_4569_mult__minus1,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Z) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1
tff(fact_4570_mult__minus1,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(one_one_rat)),Z) = uminus_uminus_rat(Z) ).

% mult_minus1
tff(fact_4571_mult__minus1__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Z) ).

% mult_minus1_right
tff(fact_4572_mult__minus1__right,axiom,
    ! [Z: int] : aa_int_int(times_times_int(Z),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1_right
tff(fact_4573_mult__minus1__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1_right
tff(fact_4574_mult__minus1__right,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1_right
tff(fact_4575_mult__minus1__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(Z) ).

% mult_minus1_right
tff(fact_4576_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_4577_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(minus_minus_int(B),A) ).

% uminus_add_conv_diff
tff(fact_4578_uminus__add__conv__diff,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(minus_minus_complex(B),A) ).

% uminus_add_conv_diff
tff(fact_4579_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_4580_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : plus_plus_rat(uminus_uminus_rat(A),B) = minus_minus_rat(B,A) ).

% uminus_add_conv_diff
tff(fact_4581_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_4582_diff__minus__eq__add,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(plus_plus_int(A),B) ).

% diff_minus_eq_add
tff(fact_4583_diff__minus__eq__add,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(plus_plus_complex(A),B) ).

% diff_minus_eq_add
tff(fact_4584_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_4585_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(A,uminus_uminus_rat(B)) = plus_plus_rat(A,B) ).

% diff_minus_eq_add
tff(fact_4586_div__minus1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A) ).

% div_minus1_right
tff(fact_4587_div__minus1__right,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% div_minus1_right
tff(fact_4588_divide__minus1,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,X) ).

% divide_minus1
tff(fact_4589_divide__minus1,axiom,
    ! [X: complex] : aa_complex_complex(divide1717551699836669952omplex(X),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,X) ).

% divide_minus1
tff(fact_4590_divide__minus1,axiom,
    ! [X: rat] : divide_divide_rat(X,uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(X) ).

% divide_minus1
tff(fact_4591_minus__mod__self1,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(B),A)),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) ).

% minus_mod_self1
tff(fact_4592_minus__mod__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(B,A),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% minus_mod_self1
tff(fact_4593_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_4594_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_4595_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_4596_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_4597_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_4598_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger(P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_4599_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_4600_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_4601_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_4602_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_4603_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_4604_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_4605_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_4606_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_n356916108424825756nteger(P)),one_one_Code_integer))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_4607_of__bool__not__iff,axiom,
    ! [P: bool] : zero_n1201886186963655149omplex(aa_bool_bool(fNot,P)) = aa_complex_complex(minus_minus_complex(one_one_complex),zero_n1201886186963655149omplex(P)) ).

% of_bool_not_iff
tff(fact_4608_of__bool__not__iff,axiom,
    ! [P: bool] : zero_n3304061248610475627l_real(aa_bool_bool(fNot,P)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),zero_n3304061248610475627l_real(P)) ).

% of_bool_not_iff
tff(fact_4609_of__bool__not__iff,axiom,
    ! [P: bool] : zero_n2052037380579107095ol_rat(aa_bool_bool(fNot,P)) = minus_minus_rat(one_one_rat,zero_n2052037380579107095ol_rat(P)) ).

% of_bool_not_iff
tff(fact_4610_of__bool__not__iff,axiom,
    ! [P: bool] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,P)) = aa_int_int(minus_minus_int(one_one_int),aa_bool_int(zero_n2684676970156552555ol_int,P)) ).

% of_bool_not_iff
tff(fact_4611_of__bool__not__iff,axiom,
    ! [P: bool] : zero_n356916108424825756nteger(aa_bool_bool(fNot,P)) = minus_8373710615458151222nteger(one_one_Code_integer,zero_n356916108424825756nteger(P)) ).

% of_bool_not_iff
tff(fact_4612_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_4613_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] : bit_ri6519982836138164636nteger(N,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% signed_take_bit_of_minus_1
tff(fact_4614_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% signed_take_bit_of_minus_1
tff(fact_4615_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_4616_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_4617_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_4618_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_4619_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_4620_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_4621_in__set__replicate,axiom,
    ! [X: real,N: nat,Y: real] :
      ( member_real(X,set_real2(replicate_real(N,Y)))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4622_in__set__replicate,axiom,
    ! [X: complex,N: nat,Y: complex] :
      ( pp(aa_set_complex_bool(member_complex(X),set_complex2(replicate_complex(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4623_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_4624_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_4625_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_4626_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(nth_nat(replicate_nat(N,X)),I) = X ) ) ).

% nth_replicate
tff(fact_4627_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_4628_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_4629_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_numeral_dbl_real,numeral_numeral_real(K))) ).

% dbl_simps(1)
tff(fact_4630_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_simps(1)
tff(fact_4631_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K))) ).

% dbl_simps(1)
tff(fact_4632_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu8804712462038260780nteger(aa_num_Code_integer(numera6620942414471956472nteger,K))) ).

% dbl_simps(1)
tff(fact_4633_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_numeral_dbl_rat(uminus_uminus_rat(numeral_numeral_rat(K))) = uminus_uminus_rat(neg_numeral_dbl_rat(numeral_numeral_rat(K))) ).

% dbl_simps(1)
tff(fact_4634_triangle__Suc,axiom,
    ! [N: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,N)),aa_nat_nat(suc,N)) ).

% triangle_Suc
tff(fact_4635_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_4636_add__neg__numeral__special_I7_J,axiom,
    aa_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_4637_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_4638_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_4639_add__neg__numeral__special_I7_J,axiom,
    plus_plus_rat(one_one_rat,uminus_uminus_rat(one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_4640_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_4641_add__neg__numeral__special_I8_J,axiom,
    aa_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_4642_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_4643_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_4644_add__neg__numeral__special_I8_J,axiom,
    plus_plus_rat(uminus_uminus_rat(one_one_rat),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_4645_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_4646_diff__numeral__special_I12_J,axiom,
    aa_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_4647_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_4648_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_4649_diff__numeral__special_I12_J,axiom,
    minus_minus_rat(uminus_uminus_rat(one_one_rat),uminus_uminus_rat(one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_4650_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4651_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4652_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4653_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4654_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( uminus_uminus_rat(numeral_numeral_rat(N)) = uminus_uminus_rat(one_one_rat) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4655_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4656_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4657_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4658_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4659_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( uminus_uminus_rat(one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4660_minus__one__mult__self,axiom,
    ! [N: nat] : 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)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)) = one_one_real ).

% minus_one_mult_self
tff(fact_4661_minus__one__mult__self,axiom,
    ! [N: nat] : aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)) = one_one_int ).

% minus_one_mult_self
tff(fact_4662_minus__one__mult__self,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)) = one_one_complex ).

% minus_one_mult_self
tff(fact_4663_minus__one__mult__self,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_4664_minus__one__mult__self,axiom,
    ! [N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)),aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)) = one_one_rat ).

% minus_one_mult_self
tff(fact_4665_left__minus__one__mult__self,axiom,
    ! [N: nat,A: 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)),N)),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)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4666_left__minus__one__mult__self,axiom,
    ! [N: nat,A: int] : aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4667_left__minus__one__mult__self,axiom,
    ! [N: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4668_left__minus__one__mult__self,axiom,
    ! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4669_left__minus__one__mult__self,axiom,
    ! [N: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4670_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_4671_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_4672_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(V)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = numeral_numeral_real(U) ) ) ) ).

% max_number_of(2)
tff(fact_4673_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))))
       => ( ord_max_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))))
       => ( ord_max_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))) = aa_num_Code_integer(numera6620942414471956472nteger,U) ) ) ) ).

% max_number_of(2)
tff(fact_4674_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(U)),uminus_uminus_rat(numeral_numeral_rat(V))))
       => ( ord_max_rat(numeral_numeral_rat(U),uminus_uminus_rat(numeral_numeral_rat(V))) = uminus_uminus_rat(numeral_numeral_rat(V)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(U)),uminus_uminus_rat(numeral_numeral_rat(V))))
       => ( ord_max_rat(numeral_numeral_rat(U),uminus_uminus_rat(numeral_numeral_rat(V))) = numeral_numeral_rat(U) ) ) ) ).

% max_number_of(2)
tff(fact_4675_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_4676_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,numeral_numeral_real(U))),numeral_numeral_real(V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)) = numeral_numeral_real(V) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(U)) ) ) ) ).

% max_number_of(3)
tff(fact_4677_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))),aa_num_Code_integer(numera6620942414471956472nteger,V)))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_num_Code_integer(numera6620942414471956472nteger,V)) = aa_num_Code_integer(numera6620942414471956472nteger,V) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))),aa_num_Code_integer(numera6620942414471956472nteger,V)))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_num_Code_integer(numera6620942414471956472nteger,V)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)) ) ) ) ).

% max_number_of(3)
tff(fact_4678_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(U))),numeral_numeral_rat(V)))
       => ( ord_max_rat(uminus_uminus_rat(numeral_numeral_rat(U)),numeral_numeral_rat(V)) = numeral_numeral_rat(V) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(U))),numeral_numeral_rat(V)))
       => ( ord_max_rat(uminus_uminus_rat(numeral_numeral_rat(U)),numeral_numeral_rat(V)) = uminus_uminus_rat(numeral_numeral_rat(U)) ) ) ) ).

% max_number_of(3)
tff(fact_4679_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_4680_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,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(V)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(U)) ) ) ) ).

% max_number_of(4)
tff(fact_4681_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)) ) ) ) ).

% max_number_of(4)
tff(fact_4682_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(U))),uminus_uminus_rat(numeral_numeral_rat(V))))
       => ( ord_max_rat(uminus_uminus_rat(numeral_numeral_rat(U)),uminus_uminus_rat(numeral_numeral_rat(V))) = uminus_uminus_rat(numeral_numeral_rat(V)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(U))),uminus_uminus_rat(numeral_numeral_rat(V))))
       => ( ord_max_rat(uminus_uminus_rat(numeral_numeral_rat(U)),uminus_uminus_rat(numeral_numeral_rat(V))) = uminus_uminus_rat(numeral_numeral_rat(U)) ) ) ) ).

% max_number_of(4)
tff(fact_4683_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_4684_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_4685_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_4686_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_4687_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_4688_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_4689_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),Y)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Y) ).

% semiring_norm(168)
tff(fact_4690_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: rat] : plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(V)),plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(W)),Y)) = plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Y) ).

% semiring_norm(168)
tff(fact_4691_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(N)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% diff_numeral_simps(3)
tff(fact_4692_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% diff_numeral_simps(3)
tff(fact_4693_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% diff_numeral_simps(3)
tff(fact_4694_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% diff_numeral_simps(3)
tff(fact_4695_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(M)),numeral_numeral_rat(N)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% diff_numeral_simps(3)
tff(fact_4696_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% diff_numeral_simps(2)
tff(fact_4697_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% diff_numeral_simps(2)
tff(fact_4698_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% diff_numeral_simps(2)
tff(fact_4699_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : minus_8373710615458151222nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% diff_numeral_simps(2)
tff(fact_4700_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : minus_minus_rat(numeral_numeral_rat(M),uminus_uminus_rat(numeral_numeral_rat(N))) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% diff_numeral_simps(2)
tff(fact_4701_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_4702_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_4703_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_4704_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_4705_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(V))),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),Y)) = aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_4706_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_4707_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_4708_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_4709_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_4710_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),Y)) = aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_4711_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_4712_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,W)),Y)) = aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_4713_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_4714_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_4715_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(V))),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),Y)) = aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_4716_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_4717_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_4718_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_4719_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_4720_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(N))) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_4721_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(N)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_4722_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_4723_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_4724_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_4725_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(M))),numeral_numeral_rat(N)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_4726_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_4727_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_4728_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_4729_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_4730_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(numeral_numeral_rat(N))) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_4731_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,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))))
    <=> pp(aa_num_bool(ord_less_eq_num(N),M)) ) ).

% neg_numeral_le_iff
tff(fact_4732_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))))
    <=> pp(aa_num_bool(ord_less_eq_num(N),M)) ) ).

% neg_numeral_le_iff
tff(fact_4733_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(numeral_numeral_rat(N))))
    <=> pp(aa_num_bool(ord_less_eq_num(N),M)) ) ).

% neg_numeral_le_iff
tff(fact_4734_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(ord_less_eq_num(N),M)) ) ).

% neg_numeral_le_iff
tff(fact_4735_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,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))))
    <=> pp(aa_num_bool(ord_less_num(N),M)) ) ).

% neg_numeral_less_iff
tff(fact_4736_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(ord_less_num(N),M)) ) ).

% neg_numeral_less_iff
tff(fact_4737_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))))
    <=> pp(aa_num_bool(ord_less_num(N),M)) ) ).

% neg_numeral_less_iff
tff(fact_4738_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(numeral_numeral_rat(N))))
    <=> pp(aa_num_bool(ord_less_num(N),M)) ) ).

% neg_numeral_less_iff
tff(fact_4739_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,numeral_numeral_real(M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4740_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,aa_num_Code_integer(numera6620942414471956472nteger,M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4741_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(numeral_numeral_rat(M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4742_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_4743_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: 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,numeral_numeral_real(W)))))
    <=> 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,numeral_numeral_real(W))))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4744_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W))))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4745_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W: 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,numeral_numeral_real(W)))),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,numeral_numeral_real(W)))),B)) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4746_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W)))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W)))),B)) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4747_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) )
    <=> ( ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = B ) )
        & ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4748_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) )
    <=> ( ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = B ) )
        & ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4749_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W))) )
    <=> ( ( ( uminus_uminus_rat(numeral_numeral_rat(W)) != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W))) = B ) )
        & ( ( uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4750_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = A )
    <=> ( ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) ) )
        & ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4751_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = A )
    <=> ( ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ) )
        & ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4752_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W))) = A )
    <=> ( ( ( uminus_uminus_rat(numeral_numeral_rat(W)) != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W))) ) )
        & ( ( uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4753_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,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,one_one_real)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4754_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_4755_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4756_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(one_one_rat)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4757_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: 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,numeral_numeral_real(W)))))
    <=> 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,numeral_numeral_real(W))))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_4758_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W))))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_4759_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W: 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,numeral_numeral_real(W)))),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,numeral_numeral_real(W)))),B)) ) ).

% divide_less_eq_numeral1(2)
tff(fact_4760_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W)))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W)))),B)) ) ).

% divide_less_eq_numeral1(2)
tff(fact_4761_power2__minus,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_4762_power2__minus,axiom,
    ! [A: int] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_4763_power2__minus,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_4764_power2__minus,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_4765_power2__minus,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_4766_odd__of__bool__self,axiom,
    ! [P2: bool] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_n2687167440665602831ol_nat(P2)))
    <=> pp(P2) ) ).

% odd_of_bool_self
tff(fact_4767_odd__of__bool__self,axiom,
    ! [P2: bool] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_bool_int(zero_n2684676970156552555ol_int,P2)))
    <=> pp(P2) ) ).

% odd_of_bool_self
tff(fact_4768_odd__of__bool__self,axiom,
    ! [P2: bool] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger(P2)))
    <=> pp(P2) ) ).

% odd_of_bool_self
tff(fact_4769_add__neg__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4770_add__neg__numeral__special_I9_J,axiom,
    aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4771_add__neg__numeral__special_I9_J,axiom,
    aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4772_add__neg__numeral__special_I9_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4773_add__neg__numeral__special_I9_J,axiom,
    plus_plus_rat(uminus_uminus_rat(one_one_rat),uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4774_diff__numeral__special_I10_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4775_diff__numeral__special_I10_J,axiom,
    aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4776_diff__numeral__special_I10_J,axiom,
    aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4777_diff__numeral__special_I10_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4778_diff__numeral__special_I10_J,axiom,
    minus_minus_rat(uminus_uminus_rat(one_one_rat),one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4779_diff__numeral__special_I11_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4780_diff__numeral__special_I11_J,axiom,
    aa_int_int(minus_minus_int(one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4781_diff__numeral__special_I11_J,axiom,
    aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4782_diff__numeral__special_I11_J,axiom,
    minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4783_diff__numeral__special_I11_J,axiom,
    minus_minus_rat(one_one_rat,uminus_uminus_rat(one_one_rat)) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4784_minus__1__div__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_2_eq
tff(fact_4785_minus__1__div__2__eq,axiom,
    divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% minus_1_div_2_eq
tff(fact_4786_bits__minus__1__mod__2__eq,axiom,
    aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_minus_1_mod_2_eq
tff(fact_4787_bits__minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_minus_1_mod_2_eq
tff(fact_4788_minus__1__mod__2__eq,axiom,
    aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% minus_1_mod_2_eq
tff(fact_4789_minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% minus_1_mod_2_eq
tff(fact_4790_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_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)) = 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)) ).

% Power.ring_1_class.power_minus_even
tff(fact_4791_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_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)) = 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)) ).

% Power.ring_1_class.power_minus_even
tff(fact_4792_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_4793_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_4794_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(uminus_uminus_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)) = 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)) ).

% Power.ring_1_class.power_minus_even
tff(fact_4795_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_4796_of__bool__half__eq__0,axiom,
    ! [B: bool] : aa_int_int(aa_int_fun_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_4797_of__bool__half__eq__0,axiom,
    ! [B: bool] : divide6298287555418463151nteger(zero_n356916108424825756nteger(B),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% of_bool_half_eq_0
tff(fact_4798_Parity_Oring__1__class_Opower__minus__even,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))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_4799_Parity_Oring__1__class_Opower__minus__even,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))
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_nat_int(power_power_int(A),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_4800_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: complex] :
      ( 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_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_nat_complex(power_power_complex(A),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_4801_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: 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))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_4802_Parity_Oring__1__class_Opower__minus__even,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))
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N) = aa_nat_rat(power_power_rat(A),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_4803_power__minus__odd,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))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ) ) ).

% power_minus_odd
tff(fact_4804_power__minus__odd,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))
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),N)) ) ) ).

% power_minus_odd
tff(fact_4805_power__minus__odd,axiom,
    ! [N: nat,A: complex] :
      ( ~ 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_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A),N)) ) ) ).

% power_minus_odd
tff(fact_4806_power__minus__odd,axiom,
    ! [N: nat,A: 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))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ).

% power_minus_odd
tff(fact_4807_power__minus__odd,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))
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N) = uminus_uminus_rat(aa_nat_rat(power_power_rat(A),N)) ) ) ).

% power_minus_odd
tff(fact_4808_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_4809_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_int_int(minus_minus_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,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_4810_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_4811_diff__numeral__special_I4_J,axiom,
    ! [M: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_4812_diff__numeral__special_I4_J,axiom,
    ! [M: num] : minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(M)),one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_4813_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_4814_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_int_int(minus_minus_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_4815_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_4816_diff__numeral__special_I3_J,axiom,
    ! [N: num] : minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_4817_diff__numeral__special_I3_J,axiom,
    ! [N: num] : minus_minus_rat(one_one_rat,uminus_uminus_rat(numeral_numeral_rat(N))) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_4818_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(times_times_int(aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_4819_set__decode__0,axiom,
    ! [X: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),aa_nat_set_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_4820_dbl__simps_I4_J,axiom,
    aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4821_dbl__simps_I4_J,axiom,
    aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4822_dbl__simps_I4_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4823_dbl__simps_I4_J,axiom,
    neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4824_dbl__simps_I4_J,axiom,
    neg_numeral_dbl_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4825_power__minus1__even,axiom,
    ! [N: nat] : 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(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_real ).

% power_minus1_even
tff(fact_4826_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),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_int ).

% power_minus1_even
tff(fact_4827_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),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_complex ).

% power_minus1_even
tff(fact_4828_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),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_Code_integer ).

% power_minus1_even
tff(fact_4829_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),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_rat ).

% power_minus1_even
tff(fact_4830_neg__one__even__power,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_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) = one_one_real ) ) ).

% neg_one_even_power
tff(fact_4831_neg__one__even__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = one_one_int ) ) ).

% neg_one_even_power
tff(fact_4832_neg__one__even__power,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_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) = one_one_complex ) ) ).

% neg_one_even_power
tff(fact_4833_neg__one__even__power,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_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_4834_neg__one__even__power,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_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_4835_neg__one__odd__power,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_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% neg_one_odd_power
tff(fact_4836_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% neg_one_odd_power
tff(fact_4837_neg__one__odd__power,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_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% neg_one_odd_power
tff(fact_4838_neg__one__odd__power,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_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_4839_neg__one__odd__power,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_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N) = uminus_uminus_rat(one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_4840_signed__take__bit__0,axiom,
    ! [A: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_4841_signed__take__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_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_4842_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_4843_one__div__2__pow__eq,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_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_4844_one__div__2__pow__eq,axiom,
    ! [N: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(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_4845_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_4846_bits__1__div__exp,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_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_4847_bits__1__div__exp,axiom,
    ! [N: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(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_4848_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_4849_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_4850_one__mod__2__pow__eq,axiom,
    ! [N: nat] : modulo364778990260209775nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(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_4851_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_4852_signed__take__bit__minus,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(uminus_uminus_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K))) = aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(uminus_uminus_int,K)) ).

% signed_take_bit_minus
tff(fact_4853_compl__le__swap2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),X))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(X)),Y)) ) ).

% compl_le_swap2
tff(fact_4854_compl__le__swap1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),uminus5710092332889474511et_nat(X)))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(X),uminus5710092332889474511et_nat(Y))) ) ).

% compl_le_swap1
tff(fact_4855_compl__mono,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),uminus5710092332889474511et_nat(X))) ) ).

% compl_mono
tff(fact_4856_of__bool__eq__iff,axiom,
    ! [P2: bool,Q2: bool] :
      ( ( zero_n2687167440665602831ol_nat(P2) = zero_n2687167440665602831ol_nat(Q2) )
    <=> ( pp(P2)
      <=> pp(Q2) ) ) ).

% of_bool_eq_iff
tff(fact_4857_of__bool__eq__iff,axiom,
    ! [P2: bool,Q2: bool] :
      ( ( aa_bool_int(zero_n2684676970156552555ol_int,P2) = aa_bool_int(zero_n2684676970156552555ol_int,Q2) )
    <=> ( pp(P2)
      <=> pp(Q2) ) ) ).

% of_bool_eq_iff
tff(fact_4858_of__bool__eq__iff,axiom,
    ! [P2: bool,Q2: bool] :
      ( ( zero_n356916108424825756nteger(P2) = zero_n356916108424825756nteger(Q2) )
    <=> ( pp(P2)
      <=> pp(Q2) ) ) ).

% of_bool_eq_iff
tff(fact_4859_verit__negate__coefficient_I3_J,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_4860_verit__negate__coefficient_I3_J,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_4861_verit__negate__coefficient_I3_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_4862_verit__negate__coefficient_I3_J,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
     => ( uminus_uminus_rat(A) = uminus_uminus_rat(B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_4863_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_4864_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_4865_minus__equation__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B )
    <=> ( aa_complex_complex(uminus1482373934393186551omplex,B) = A ) ) ).

% minus_equation_iff
tff(fact_4866_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_4867_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat(A) = B )
    <=> ( uminus_uminus_rat(B) = A ) ) ).

% minus_equation_iff
tff(fact_4868_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_4869_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_4870_equation__minus__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( B = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% equation_minus_iff
tff(fact_4871_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_4872_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = uminus_uminus_rat(B) )
    <=> ( B = uminus_uminus_rat(A) ) ) ).

% equation_minus_iff
tff(fact_4873_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : zero_n3304061248610475627l_real(aa_bool_bool(fconj(P),Q)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_n3304061248610475627l_real(P)),zero_n3304061248610475627l_real(Q)) ).

% of_bool_conj
tff(fact_4874_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : zero_n2052037380579107095ol_rat(aa_bool_bool(fconj(P),Q)) = aa_rat_rat(times_times_rat(zero_n2052037380579107095ol_rat(P)),zero_n2052037380579107095ol_rat(Q)) ).

% of_bool_conj
tff(fact_4875_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : zero_n2687167440665602831ol_nat(aa_bool_bool(fconj(P),Q)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(P)),zero_n2687167440665602831ol_nat(Q)) ).

% of_bool_conj
tff(fact_4876_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(P),Q)) = aa_int_int(times_times_int(aa_bool_int(zero_n2684676970156552555ol_int,P)),aa_bool_int(zero_n2684676970156552555ol_int,Q)) ).

% of_bool_conj
tff(fact_4877_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : zero_n356916108424825756nteger(aa_bool_bool(fconj(P),Q)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(P)),zero_n356916108424825756nteger(Q)) ).

% of_bool_conj
tff(fact_4878_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_4879_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_4880_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_4881_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),uminus_uminus_rat(B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),uminus_uminus_rat(A))) ) ).

% le_minus_iff
tff(fact_4882_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_4883_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_4884_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_4885_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(B)),A)) ) ).

% minus_le_iff
tff(fact_4886_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_4887_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_4888_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_4889_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,uminus_uminus_rat(B)),uminus_uminus_rat(A))) ) ).

% le_imp_neg_le
tff(fact_4890_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_4891_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_4892_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_4893_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_4894_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(B)),A)) ) ).

% minus_less_iff
tff(fact_4895_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_4896_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_4897_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_4898_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),uminus_uminus_rat(B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),uminus_uminus_rat(A))) ) ).

% less_minus_iff
tff(fact_4899_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_4900_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_4901_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_4902_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,uminus_uminus_rat(B)),uminus_uminus_rat(A))) ) ).

% verit_negate_coefficient(2)
tff(fact_4903_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : numeral_numeral_real(M) != aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) ).

% numeral_neq_neg_numeral
tff(fact_4904_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : aa_num_int(numeral_numeral_int,M) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% numeral_neq_neg_numeral
tff(fact_4905_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : numera6690914467698888265omplex(M) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% numeral_neq_neg_numeral
tff(fact_4906_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : aa_num_Code_integer(numera6620942414471956472nteger,M) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) ).

% numeral_neq_neg_numeral
tff(fact_4907_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : numeral_numeral_rat(M) != uminus_uminus_rat(numeral_numeral_rat(N)) ).

% numeral_neq_neg_numeral
tff(fact_4908_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(uminus_uminus_real,numeral_numeral_real(M)) != numeral_numeral_real(N) ).

% neg_numeral_neq_numeral
tff(fact_4909_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) != aa_num_int(numeral_numeral_int,N) ).

% neg_numeral_neq_numeral
tff(fact_4910_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M)) != numera6690914467698888265omplex(N) ).

% neg_numeral_neq_numeral
tff(fact_4911_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)) != aa_num_Code_integer(numera6620942414471956472nteger,N) ).

% neg_numeral_neq_numeral
tff(fact_4912_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : uminus_uminus_rat(numeral_numeral_rat(M)) != numeral_numeral_rat(N) ).

% neg_numeral_neq_numeral
tff(fact_4913_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),B) )
    <=> ( ( A = B )
        | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% square_eq_iff
tff(fact_4914_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(times_times_int(A),A) = aa_int_int(times_times_int(B),B) )
    <=> ( ( A = B )
        | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ).

% square_eq_iff
tff(fact_4915_square__eq__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),A) = aa_complex_complex(times_times_complex(B),B) )
    <=> ( ( A = B )
        | ( A = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).

% square_eq_iff
tff(fact_4916_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),B) )
    <=> ( ( A = B )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ).

% square_eq_iff
tff(fact_4917_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),A) = aa_rat_rat(times_times_rat(B),B) )
    <=> ( ( A = B )
        | ( A = uminus_uminus_rat(B) ) ) ) ).

% square_eq_iff
tff(fact_4918_minus__mult__commute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B)) ).

% minus_mult_commute
tff(fact_4919_minus__mult__commute,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(times_times_int(A),aa_int_int(uminus_uminus_int,B)) ).

% minus_mult_commute
tff(fact_4920_minus__mult__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_mult_commute
tff(fact_4921_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_mult_commute
tff(fact_4922_minus__mult__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(A)),B) = aa_rat_rat(times_times_rat(A),uminus_uminus_rat(B)) ).

% minus_mult_commute
tff(fact_4923_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_4924_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_4925_one__neq__neg__one,axiom,
    one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% one_neq_neg_one
tff(fact_4926_one__neq__neg__one,axiom,
    one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_4927_one__neq__neg__one,axiom,
    one_one_rat != uminus_uminus_rat(one_one_rat) ).

% one_neq_neg_one
tff(fact_4928_is__num__normalize_I8_J,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)) ).

% is_num_normalize(8)
tff(fact_4929_is__num__normalize_I8_J,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(A),B)) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ).

% is_num_normalize(8)
tff(fact_4930_is__num__normalize_I8_J,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% is_num_normalize(8)
tff(fact_4931_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% is_num_normalize(8)
tff(fact_4932_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(plus_plus_rat(A,B)) = plus_plus_rat(uminus_uminus_rat(B),uminus_uminus_rat(A)) ).

% is_num_normalize(8)
tff(fact_4933_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_4934_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(A),B)) = aa_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_4935_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% add.inverse_distrib_swap
tff(fact_4936_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_4937_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(plus_plus_rat(A,B)) = plus_plus_rat(uminus_uminus_rat(B),uminus_uminus_rat(A)) ).

% add.inverse_distrib_swap
tff(fact_4938_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_4939_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2 = aa_int_int(plus_plus_int(K),A) )
     => ( aa_int_int(uminus_uminus_int,A2) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,K)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_4940_group__cancel_Oneg1,axiom,
    ! [A2: complex,K: complex,A: complex] :
      ( ( A2 = aa_complex_complex(plus_plus_complex(K),A) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ) ).

% group_cancel.neg1
tff(fact_4941_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_4942_group__cancel_Oneg1,axiom,
    ! [A2: rat,K: rat,A: rat] :
      ( ( A2 = plus_plus_rat(K,A) )
     => ( uminus_uminus_rat(A2) = plus_plus_rat(uminus_uminus_rat(K),uminus_uminus_rat(A)) ) ) ).

% group_cancel.neg1
tff(fact_4943_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_4944_minus__diff__commute,axiom,
    ! [B: int,A: int] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,B)),A) = aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,A)),B) ).

% minus_diff_commute
tff(fact_4945_minus__diff__commute,axiom,
    ! [B: complex,A: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),A) = aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).

% minus_diff_commute
tff(fact_4946_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_4947_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : minus_minus_rat(uminus_uminus_rat(B),A) = minus_minus_rat(uminus_uminus_rat(A),B) ).

% minus_diff_commute
tff(fact_4948_minus__diff__minus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ).

% minus_diff_minus
tff(fact_4949_minus__diff__minus,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(minus_minus_int(A),B)) ).

% minus_diff_minus
tff(fact_4950_minus__diff__minus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A),B)) ).

% minus_diff_minus
tff(fact_4951_minus__diff__minus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A,B)) ).

% minus_diff_minus
tff(fact_4952_minus__diff__minus,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(uminus_uminus_rat(A),uminus_uminus_rat(B)) = uminus_uminus_rat(minus_minus_rat(A,B)) ).

% minus_diff_minus
tff(fact_4953_div__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) ).

% div_minus_right
tff(fact_4954_div__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% div_minus_right
tff(fact_4955_minus__divide__left,axiom,
    ! [A: real,B: 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,aa_real_real(uminus_uminus_real,A)),B) ).

% minus_divide_left
tff(fact_4956_minus__divide__left,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).

% minus_divide_left
tff(fact_4957_minus__divide__left,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(divide_divide_rat(A,B)) = divide_divide_rat(uminus_uminus_rat(A),B) ).

% minus_divide_left
tff(fact_4958_minus__divide__divide,axiom,
    ! [A: real,B: 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) ).

% minus_divide_divide
tff(fact_4959_minus__divide__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ).

% minus_divide_divide
tff(fact_4960_minus__divide__divide,axiom,
    ! [A: rat,B: rat] : divide_divide_rat(uminus_uminus_rat(A),uminus_uminus_rat(B)) = divide_divide_rat(A,B) ).

% minus_divide_divide
tff(fact_4961_minus__divide__right,axiom,
    ! [A: real,B: 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)) ).

% minus_divide_right
tff(fact_4962_minus__divide__right,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_divide_right
tff(fact_4963_minus__divide__right,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(divide_divide_rat(A,B)) = divide_divide_rat(A,uminus_uminus_rat(B)) ).

% minus_divide_right
tff(fact_4964_mod__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B)) ).

% mod_minus_right
tff(fact_4965_mod__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B)) ).

% mod_minus_right
tff(fact_4966_mod__minus__cong,axiom,
    ! [A: int,B: int,A3: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = aa_int_int(modulo_modulo_int(A3),B) )
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A3)),B) ) ) ).

% mod_minus_cong
tff(fact_4967_mod__minus__cong,axiom,
    ! [A: code_integer,B: code_integer,A3: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = modulo364778990260209775nteger(A3,B) )
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A3),B) ) ) ).

% mod_minus_cong
tff(fact_4968_mod__minus__eq,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),B))),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) ).

% mod_minus_eq
tff(fact_4969_mod__minus__eq,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,B)),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% mod_minus_eq
tff(fact_4970_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_4971_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_4972_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_4973_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_4974_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_4975_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_4976_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_4977_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger(P))) ).

% zero_less_eq_of_bool
tff(fact_4978_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_4979_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_4980_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_4981_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_4982_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_n356916108424825756nteger(P)),one_one_Code_integer)) ).

% of_bool_less_eq_one
tff(fact_4983_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_4984_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_4985_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_4986_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_4987_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_4988_of__bool__def,axiom,
    ! [P2: bool] :
      ( ( pp(P2)
       => ( zero_n356916108424825756nteger(P2) = one_one_Code_integer ) )
      & ( ~ pp(P2)
       => ( zero_n356916108424825756nteger(P2) = zero_z3403309356797280102nteger ) ) ) ).

% of_bool_def
tff(fact_4989_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_4990_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_4991_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_4992_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_4993_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_4994_split__of__bool,axiom,
    ! [P: fun_Co8822979462182099027r_bool,P2: bool] :
      ( pp(aa_Code_integer_bool(P,zero_n356916108424825756nteger(P2)))
    <=> ( ( pp(P2)
         => pp(aa_Code_integer_bool(P,one_one_Code_integer)) )
        & ( ~ pp(P2)
         => pp(aa_Code_integer_bool(P,zero_z3403309356797280102nteger)) ) ) ) ).

% split_of_bool
tff(fact_4995_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_4996_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_4997_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_4998_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_4999_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_5000_split__of__bool__asm,axiom,
    ! [P: fun_Co8822979462182099027r_bool,P2: bool] :
      ( pp(aa_Code_integer_bool(P,zero_n356916108424825756nteger(P2)))
    <=> ~ ( ( pp(P2)
            & ~ pp(aa_Code_integer_bool(P,one_one_Code_integer)) )
          | ( ~ pp(P2)
            & ~ pp(aa_Code_integer_bool(P,zero_z3403309356797280102nteger)) ) ) ) ).

% split_of_bool_asm
tff(fact_5001_finite__set__decode,axiom,
    ! [N: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(nat_set_decode,N))) ).

% finite_set_decode
tff(fact_5002_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,numeral_numeral_real(M))),numeral_numeral_real(N))) ).

% neg_numeral_le_numeral
tff(fact_5003_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N))) ).

% neg_numeral_le_numeral
tff(fact_5004_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),numeral_numeral_rat(N))) ).

% neg_numeral_le_numeral
tff(fact_5005_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_5006_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(N)))) ).

% not_numeral_le_neg_numeral
tff(fact_5007_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_5008_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(N)))) ).

% not_numeral_le_neg_numeral
tff(fact_5009_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_5010_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_real != aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) ).

% zero_neq_neg_numeral
tff(fact_5011_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_5012_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% zero_neq_neg_numeral
tff(fact_5013_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) ).

% zero_neq_neg_numeral
tff(fact_5014_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_rat != uminus_uminus_rat(numeral_numeral_rat(N)) ).

% zero_neq_neg_numeral
tff(fact_5015_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(N)))) ).

% not_numeral_less_neg_numeral
tff(fact_5016_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_5017_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_5018_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(N)))) ).

% not_numeral_less_neg_numeral
tff(fact_5019_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,numeral_numeral_real(M))),numeral_numeral_real(N))) ).

% neg_numeral_less_numeral
tff(fact_5020_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_5021_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N))) ).

% neg_numeral_less_numeral
tff(fact_5022_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),numeral_numeral_rat(N))) ).

% neg_numeral_less_numeral
tff(fact_5023_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_5024_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_5025_le__minus__one__simps_I2_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),one_one_rat)) ).

% le_minus_one_simps(2)
tff(fact_5026_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_5027_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_5028_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_5029_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),uminus_uminus_rat(one_one_rat))) ).

% le_minus_one_simps(4)
tff(fact_5030_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_5031_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_5032_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_5033_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_5034_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_5035_zero__neq__neg__one,axiom,
    zero_zero_rat != uminus_uminus_rat(one_one_rat) ).

% zero_neq_neg_one
tff(fact_5036_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_5037_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = zero_zero_int )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% add_eq_0_iff
tff(fact_5038_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_5039_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_5040_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat(A,B) = zero_zero_rat )
    <=> ( B = uminus_uminus_rat(A) ) ) ).

% add_eq_0_iff
tff(fact_5041_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_5042_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] : aa_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_5043_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_5044_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_5045_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : plus_plus_rat(uminus_uminus_rat(A),A) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_5046_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_5047_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,A) = B ) ) ).

% add.inverse_unique
tff(fact_5048_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_5049_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_5050_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat(A,B) = zero_zero_rat )
     => ( uminus_uminus_rat(A) = B ) ) ).

% add.inverse_unique
tff(fact_5051_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_5052_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( aa_int_int(plus_plus_int(A),B) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5053_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_5054_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_5055_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = uminus_uminus_rat(B) )
    <=> ( plus_plus_rat(A,B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5056_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_5057_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(plus_plus_int(A),B) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5058_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_5059_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_5060_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat(A) = B )
    <=> ( plus_plus_rat(A,B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5061_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_5062_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_5063_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_5064_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),uminus_uminus_rat(one_one_rat))) ).

% less_minus_one_simps(4)
tff(fact_5065_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_5066_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_5067_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_5068_less__minus__one__simps_I2_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(one_one_rat)),one_one_rat)) ).

% less_minus_one_simps(2)
tff(fact_5069_numeral__times__minus__swap,axiom,
    ! [W: num,X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),aa_real_real(uminus_uminus_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) ).

% numeral_times_minus_swap
tff(fact_5070_numeral__times__minus__swap,axiom,
    ! [W: num,X: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,W)),aa_int_int(uminus_uminus_int,X)) = aa_int_int(times_times_int(X),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))) ).

% numeral_times_minus_swap
tff(fact_5071_numeral__times__minus__swap,axiom,
    ! [W: num,X: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(times_times_complex(X),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).

% numeral_times_minus_swap
tff(fact_5072_numeral__times__minus__swap,axiom,
    ! [W: num,X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))) ).

% numeral_times_minus_swap
tff(fact_5073_numeral__times__minus__swap,axiom,
    ! [W: num,X: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),uminus_uminus_rat(X)) = aa_rat_rat(times_times_rat(X),uminus_uminus_rat(numeral_numeral_rat(W))) ).

% numeral_times_minus_swap
tff(fact_5074_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_5075_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_5076_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( uminus_uminus_rat(divide_divide_rat(A,B)) = divide_divide_rat(A,uminus_uminus_rat(B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5077_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_5078_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_5079_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( divide_divide_rat(uminus_uminus_rat(A),uminus_uminus_rat(B)) = divide_divide_rat(A,B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5080_numeral__neq__neg__one,axiom,
    ! [N: num] : numeral_numeral_real(N) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_5081_numeral__neq__neg__one,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,N) != aa_int_int(uminus_uminus_int,one_one_int) ).

% numeral_neq_neg_one
tff(fact_5082_numeral__neq__neg__one,axiom,
    ! [N: num] : numera6690914467698888265omplex(N) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_5083_numeral__neq__neg__one,axiom,
    ! [N: num] : aa_num_Code_integer(numera6620942414471956472nteger,N) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_5084_numeral__neq__neg__one,axiom,
    ! [N: num] : numeral_numeral_rat(N) != uminus_uminus_rat(one_one_rat) ).

% numeral_neq_neg_one
tff(fact_5085_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_real != aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) ).

% one_neq_neg_numeral
tff(fact_5086_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% one_neq_neg_numeral
tff(fact_5087_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% one_neq_neg_numeral
tff(fact_5088_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) ).

% one_neq_neg_numeral
tff(fact_5089_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_rat != uminus_uminus_rat(numeral_numeral_rat(N)) ).

% one_neq_neg_numeral
tff(fact_5090_square__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X) = one_one_real )
    <=> ( ( X = one_one_real )
        | ( X = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_5091_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( aa_int_int(times_times_int(X),X) = one_one_int )
    <=> ( ( X = one_one_int )
        | ( X = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_5092_square__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(times_times_complex(X),X) = one_one_complex )
    <=> ( ( X = one_one_complex )
        | ( X = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_5093_square__eq__1__iff,axiom,
    ! [X: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X) = one_one_Code_integer )
    <=> ( ( X = one_one_Code_integer )
        | ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_5094_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(times_times_rat(X),X) = one_one_rat )
    <=> ( ( X = one_one_rat )
        | ( X = uminus_uminus_rat(one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_5095_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_5096_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),B) = aa_int_int(plus_plus_int(A),aa_int_int(uminus_uminus_int,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5097_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5098_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_5099_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(A,B) = plus_plus_rat(A,uminus_uminus_rat(B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5100_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_5101_diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),B) = aa_int_int(plus_plus_int(A),aa_int_int(uminus_uminus_int,B)) ).

% diff_conv_add_uminus
tff(fact_5102_diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% diff_conv_add_uminus
tff(fact_5103_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_5104_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(A,B) = plus_plus_rat(A,uminus_uminus_rat(B)) ).

% diff_conv_add_uminus
tff(fact_5105_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_5106_group__cancel_Osub2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(plus_plus_int(K),B) )
     => ( aa_int_int(minus_minus_int(A),B2) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,K)),aa_int_int(minus_minus_int(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5107_group__cancel_Osub2,axiom,
    ! [B2: complex,K: complex,B: complex,A: complex] :
      ( ( B2 = aa_complex_complex(plus_plus_complex(K),B) )
     => ( aa_complex_complex(minus_minus_complex(A),B2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(minus_minus_complex(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5108_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_5109_group__cancel_Osub2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = plus_plus_rat(K,B) )
     => ( minus_minus_rat(A,B2) = plus_plus_rat(uminus_uminus_rat(K),minus_minus_rat(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_5110_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(Xs)))
         => ( X2 = X ) )
     => ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_5111_replicate__length__same,axiom,
    ! [Xs: list_bool,X: bool] :
      ( ! [X2: bool] :
          ( member_bool(X2,set_bool2(Xs))
         => ( pp(X2)
          <=> pp(X) ) )
     => ( replicate_bool(size_size_list_bool(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_5112_replicate__length__same,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
         => ( X2 = X ) )
     => ( replicate_nat(size_size_list_nat(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_5113_replicate__length__same,axiom,
    ! [Xs: list_int,X: int] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),set_int2(Xs)))
         => ( X2 = X ) )
     => ( replicate_int(size_size_list_int(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_5114_replicate__eqI,axiom,
    ! [Xs: list_real,N: nat,X: real] :
      ( ( size_size_list_real(Xs) = N )
     => ( ! [Y2: real] :
            ( member_real(Y2,set_real2(Xs))
           => ( Y2 = X ) )
       => ( Xs = replicate_real(N,X) ) ) ) ).

% replicate_eqI
tff(fact_5115_replicate__eqI,axiom,
    ! [Xs: list_complex,N: nat,X: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = N )
     => ( ! [Y2: complex] :
            ( pp(aa_set_complex_bool(member_complex(Y2),set_complex2(Xs)))
           => ( Y2 = X ) )
       => ( Xs = replicate_complex(N,X) ) ) ) ).

% replicate_eqI
tff(fact_5116_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_5117_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_5118_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_5119_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_5120_dvd__div__neg,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),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ) ) ).

% dvd_div_neg
tff(fact_5121_dvd__div__neg,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% dvd_div_neg
tff(fact_5122_dvd__div__neg,axiom,
    ! [B: complex,A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B),A))
     => ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ) ) ).

% dvd_div_neg
tff(fact_5123_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_div_neg
tff(fact_5124_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ( divide_divide_rat(A,uminus_uminus_rat(B)) = uminus_uminus_rat(divide_divide_rat(A,B)) ) ) ).

% dvd_div_neg
tff(fact_5125_dvd__neg__div,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,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ) ) ).

% dvd_neg_div
tff(fact_5126_dvd__neg__div,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% dvd_neg_div
tff(fact_5127_dvd__neg__div,axiom,
    ! [B: complex,A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B),A))
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ) ) ).

% dvd_neg_div
tff(fact_5128_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_neg_div
tff(fact_5129_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ( divide_divide_rat(uminus_uminus_rat(A),B) = uminus_uminus_rat(divide_divide_rat(A,B)) ) ) ).

% dvd_neg_div
tff(fact_5130_real__minus__mult__self__le,axiom,
    ! [U: 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(times_times_real,U),U))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X))) ).

% real_minus_mult_self_le
tff(fact_5131_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(times_times_int(M),N) = one_one_int )
     => ( ( M = one_one_int )
        | ( M = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_5132_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(times_times_int(M),N) = one_one_int )
    <=> ( ( ( M = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( N = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_5133_minus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(minus_minus_int(zero_zero_int),L) = aa_int_int(uminus_uminus_int,L) ).

% minus_int_code(2)
tff(fact_5134_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,K)),L) != zero_zero_int )
     => ( aa_int_int(modulo_modulo_int(K),L) != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
tff(fact_5135_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(modulo_modulo_int(K),aa_int_int(uminus_uminus_int,L)) != zero_zero_int )
     => ( aa_int_int(modulo_modulo_int(K),L) != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
tff(fact_5136_minus__real__def,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(uminus_uminus_real,Y)) ).

% minus_real_def
tff(fact_5137_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_5138_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_5139_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_5140_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_5141_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_5142_ln__ge__zero,axiom,
    ! [X: 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_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X))) ) ).

% ln_ge_zero
tff(fact_5143_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,numeral_numeral_real(N)))) ).

% not_zero_le_neg_numeral
tff(fact_5144_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)))) ).

% not_zero_le_neg_numeral
tff(fact_5145_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(N)))) ).

% not_zero_le_neg_numeral
tff(fact_5146_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_5147_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(N))),zero_zero_real)) ).

% neg_numeral_le_zero
tff(fact_5148_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))),zero_z3403309356797280102nteger)) ).

% neg_numeral_le_zero
tff(fact_5149_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(N))),zero_zero_rat)) ).

% neg_numeral_le_zero
tff(fact_5150_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_5151_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,numeral_numeral_real(N)))) ).

% not_zero_less_neg_numeral
tff(fact_5152_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_5153_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)))) ).

% not_zero_less_neg_numeral
tff(fact_5154_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(N)))) ).

% not_zero_less_neg_numeral
tff(fact_5155_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(N))),zero_zero_real)) ).

% neg_numeral_less_zero
tff(fact_5156_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_5157_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))),zero_z3403309356797280102nteger)) ).

% neg_numeral_less_zero
tff(fact_5158_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(N))),zero_zero_rat)) ).

% neg_numeral_less_zero
tff(fact_5159_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_5160_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_5161_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(one_one_rat))) ).

% le_minus_one_simps(3)
tff(fact_5162_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_5163_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_5164_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_5165_le__minus__one__simps_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),zero_zero_rat)) ).

% le_minus_one_simps(1)
tff(fact_5166_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_5167_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_5168_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_5169_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_5170_less__minus__one__simps_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(one_one_rat)),zero_zero_rat)) ).

% less_minus_one_simps(1)
tff(fact_5171_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_5172_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_5173_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_5174_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),uminus_uminus_rat(one_one_rat))) ).

% less_minus_one_simps(3)
tff(fact_5175_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,numeral_numeral_real(M)))) ).

% not_one_le_neg_numeral
tff(fact_5176_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,aa_num_Code_integer(numera6620942414471956472nteger,M)))) ).

% not_one_le_neg_numeral
tff(fact_5177_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),uminus_uminus_rat(numeral_numeral_rat(M)))) ).

% not_one_le_neg_numeral
tff(fact_5178_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_5179_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,one_one_real))) ).

% not_numeral_le_neg_one
tff(fact_5180_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% not_numeral_le_neg_one
tff(fact_5181_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(M)),uminus_uminus_rat(one_one_rat))) ).

% not_numeral_le_neg_one
tff(fact_5182_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_5183_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,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,one_one_real))) ).

% neg_numeral_le_neg_one
tff(fact_5184_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% neg_numeral_le_neg_one
tff(fact_5185_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(one_one_rat))) ).

% neg_numeral_le_neg_one
tff(fact_5186_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_5187_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)),numeral_numeral_real(M))) ).

% neg_one_le_numeral
tff(fact_5188_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,M))) ).

% neg_one_le_numeral
tff(fact_5189_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),numeral_numeral_rat(M))) ).

% neg_one_le_numeral
tff(fact_5190_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_5191_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),one_one_real)) ).

% neg_numeral_le_one
tff(fact_5192_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),one_one_Code_integer)) ).

% neg_numeral_le_one
tff(fact_5193_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),one_one_rat)) ).

% neg_numeral_le_one
tff(fact_5194_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_5195_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,numeral_numeral_real(M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5196_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_5197_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,aa_num_Code_integer(numera6620942414471956472nteger,M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5198_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(numeral_numeral_rat(M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5199_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,numeral_numeral_real(M)))) ).

% not_one_less_neg_numeral
tff(fact_5200_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_5201_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,aa_num_Code_integer(numera6620942414471956472nteger,M)))) ).

% not_one_less_neg_numeral
tff(fact_5202_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),uminus_uminus_rat(numeral_numeral_rat(M)))) ).

% not_one_less_neg_numeral
tff(fact_5203_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,one_one_real))) ).

% not_numeral_less_neg_one
tff(fact_5204_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_5205_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% not_numeral_less_neg_one
tff(fact_5206_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(M)),uminus_uminus_rat(one_one_rat))) ).

% not_numeral_less_neg_one
tff(fact_5207_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)),numeral_numeral_real(M))) ).

% neg_one_less_numeral
tff(fact_5208_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_5209_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,M))) ).

% neg_one_less_numeral
tff(fact_5210_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(one_one_rat)),numeral_numeral_rat(M))) ).

% neg_one_less_numeral
tff(fact_5211_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),one_one_real)) ).

% neg_numeral_less_one
tff(fact_5212_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_5213_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),one_one_Code_integer)) ).

% neg_numeral_less_one
tff(fact_5214_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),one_one_rat)) ).

% neg_numeral_less_one
tff(fact_5215_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_5216_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_5217_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = uminus_uminus_rat(divide_divide_rat(B,C)) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),C) = uminus_uminus_rat(B) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_5218_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_5219_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_5220_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( uminus_uminus_rat(divide_divide_rat(B,C)) = A )
    <=> ( ( ( C != zero_zero_rat )
         => ( uminus_uminus_rat(B) = aa_rat_rat(times_times_rat(A),C) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_5221_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_5222_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_5223_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( uminus_uminus_rat(divide_divide_rat(A,B)) = C )
      <=> ( uminus_uminus_rat(A) = aa_rat_rat(times_times_rat(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5224_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_5225_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_5226_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C = uminus_uminus_rat(divide_divide_rat(A,B)) )
      <=> ( aa_rat_rat(times_times_rat(C),B) = uminus_uminus_rat(A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5227_mult__1s__ring__1_I1_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(one))),B) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(1)
tff(fact_5228_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))),B) = aa_int_int(uminus_uminus_int,B) ).

% mult_1s_ring_1(1)
tff(fact_5229_mult__1s__ring__1_I1_J,axiom,
    ! [B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% mult_1s_ring_1(1)
tff(fact_5230_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one))),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(1)
tff(fact_5231_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(one))),B) = uminus_uminus_rat(B) ).

% mult_1s_ring_1(1)
tff(fact_5232_mult__1s__ring__1_I2_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(one))) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(2)
tff(fact_5233_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] : aa_int_int(times_times_int(B),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) = aa_int_int(uminus_uminus_int,B) ).

% mult_1s_ring_1(2)
tff(fact_5234_mult__1s__ring__1_I2_J,axiom,
    ! [B: complex] : aa_complex_complex(times_times_complex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% mult_1s_ring_1(2)
tff(fact_5235_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(2)
tff(fact_5236_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(B),uminus_uminus_rat(numeral_numeral_rat(one))) = uminus_uminus_rat(B) ).

% mult_1s_ring_1(2)
tff(fact_5237_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_5238_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_5239_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat(A,B) = uminus_uminus_rat(one_one_rat) )
    <=> ( ( B != zero_zero_rat )
        & ( A = uminus_uminus_rat(B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_5240_uminus__numeral__One,axiom,
    aa_real_real(uminus_uminus_real,numeral_numeral_real(one)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% uminus_numeral_One
tff(fact_5241_uminus__numeral__One,axiom,
    aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% uminus_numeral_One
tff(fact_5242_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_5243_uminus__numeral__One,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_5244_uminus__numeral__One,axiom,
    uminus_uminus_rat(numeral_numeral_rat(one)) = uminus_uminus_rat(one_one_rat) ).

% uminus_numeral_One
tff(fact_5245_power__minus,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N) = 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)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).

% power_minus
tff(fact_5246_power__minus,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(power_power_int(A),N)) ).

% power_minus
tff(fact_5247_power__minus,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(power_power_complex(A),N)) ).

% power_minus
tff(fact_5248_power__minus,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).

% power_minus
tff(fact_5249_power__minus,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)),aa_nat_rat(power_power_rat(A),N)) ).

% power_minus
tff(fact_5250_power__minus__Bit0,axiom,
    ! [X: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_5251_power__minus__Bit0,axiom,
    ! [X: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_5252_power__minus__Bit0,axiom,
    ! [X: complex,K: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_5253_power__minus__Bit0,axiom,
    ! [X: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_5254_power__minus__Bit0,axiom,
    ! [X: rat,K: num] : aa_nat_rat(power_power_rat(uminus_uminus_rat(X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_5255_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_5256_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_5257_real__add__le__0__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,X),Y)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(uminus_uminus_real,X))) ) ).

% real_add_le_0_iff
tff(fact_5258_real__0__le__add__iff,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,X),Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,X)),Y)) ) ).

% real_0_le_add_iff
tff(fact_5259_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) = zero_zero_int ) )
      & ( ( aa_int_int(modulo_modulo_int(A),B) != zero_zero_int )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(minus_minus_int(B),aa_int_int(modulo_modulo_int(A),B)) ) ) ) ).

% zmod_zminus1_eq_if
tff(fact_5260_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
       => ( aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,B)) = zero_zero_int ) )
      & ( ( aa_int_int(modulo_modulo_int(A),B) != zero_zero_int )
       => ( aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(A),B)),B) ) ) ) ).

% zmod_zminus2_eq_if
tff(fact_5261_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_5262_ln__add__one__self__le__self,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_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_self
tff(fact_5263_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_5264_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_5265_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_5266_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_5267_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,uminus_uminus_rat(divide_divide_rat(B,C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% pos_minus_divide_less_eq
tff(fact_5268_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_5269_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),uminus_uminus_rat(divide_divide_rat(B,C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B))) ) ) ).

% pos_less_minus_divide_eq
tff(fact_5270_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_5271_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,uminus_uminus_rat(divide_divide_rat(B,C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B))) ) ) ).

% neg_minus_divide_less_eq
tff(fact_5272_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_5273_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),uminus_uminus_rat(divide_divide_rat(B,C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% neg_less_minus_divide_eq
tff(fact_5274_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_5275_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_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,uminus_uminus_rat(B)),aa_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(times_times_rat(A),C)),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_5276_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_5277_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),uminus_uminus_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(times_times_rat(A),C)),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,uminus_uminus_rat(B)),aa_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_5278_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = 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,numeral_numeral_real(W))),C) = B ) )
        & ( ( C = zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5279_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C) = B ) )
        & ( ( C = zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5280_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( uminus_uminus_rat(numeral_numeral_rat(W)) = divide_divide_rat(B,C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C) = B ) )
        & ( ( C = zero_zero_rat )
         => ( uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5281_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) )
    <=> ( ( ( C != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C) ) )
        & ( ( C = zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5282_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> ( ( ( C != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C) ) )
        & ( ( C = zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5283_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( divide_divide_rat(B,C) = uminus_uminus_rat(numeral_numeral_rat(W)) )
    <=> ( ( ( C != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C) ) )
        & ( ( C = zero_zero_rat )
         => ( uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5284_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(nat_set_decode,M)),aa_nat_set_nat(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_5285_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_5286_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_5287_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( plus_plus_rat(uminus_uminus_rat(divide_divide_rat(X,Z)),Y) = divide_divide_rat(plus_plus_rat(uminus_uminus_rat(X),aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5288_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_5289_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_5290_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( plus_plus_rat(uminus_uminus_rat(divide_divide_rat(A,Z)),B) = B ) )
      & ( ( Z != zero_zero_rat )
       => ( plus_plus_rat(uminus_uminus_rat(divide_divide_rat(A,Z)),B) = divide_divide_rat(plus_plus_rat(uminus_uminus_rat(A),aa_rat_rat(times_times_rat(B),Z)),Z) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_5291_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_5292_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_5293_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( minus_minus_rat(uminus_uminus_rat(divide_divide_rat(A,Z)),B) = uminus_uminus_rat(B) ) )
      & ( ( Z != zero_zero_rat )
       => ( minus_minus_rat(uminus_uminus_rat(divide_divide_rat(A,Z)),B) = divide_divide_rat(minus_minus_rat(uminus_uminus_rat(A),aa_rat_rat(times_times_rat(B),Z)),Z) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_5294_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_5295_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_5296_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( minus_minus_rat(divide_divide_rat(A,Z),B) = uminus_uminus_rat(B) ) )
      & ( ( Z != zero_zero_rat )
       => ( minus_minus_rat(divide_divide_rat(A,Z),B) = divide_divide_rat(minus_minus_rat(A,aa_rat_rat(times_times_rat(B),Z)),Z) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_5297_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_5298_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_5299_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(uminus_uminus_rat(divide_divide_rat(X,Z)),Y) = divide_divide_rat(minus_minus_rat(uminus_uminus_rat(X),aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5300_even__minus,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))),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ).

% even_minus
tff(fact_5301_even__minus,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)) ) ).

% even_minus
tff(fact_5302_power2__eq__iff,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))) )
    <=> ( ( X = Y )
        | ( X = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5303_power2__eq__iff,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))) )
    <=> ( ( X = Y )
        | ( X = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5304_power2__eq__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( X = Y )
        | ( X = aa_complex_complex(uminus1482373934393186551omplex,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5305_power2__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( 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))) )
    <=> ( ( X = Y )
        | ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5306_power2__eq__iff,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))) )
    <=> ( ( X = Y )
        | ( X = uminus_uminus_rat(Y) ) ) ) ).

% power2_eq_iff
tff(fact_5307_uminus__power__if,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))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N) = 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))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ) ) ) ).

% uminus_power_if
tff(fact_5308_uminus__power__if,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))
       => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = 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))
       => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),N)) ) ) ) ).

% uminus_power_if
tff(fact_5309_uminus__power__if,axiom,
    ! [N: nat,A: complex] :
      ( ( 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_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_nat_complex(power_power_complex(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))
       => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A),N)) ) ) ) ).

% uminus_power_if
tff(fact_5310_uminus__power__if,axiom,
    ! [N: nat,A: 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))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_nat_Code_integer(power_8256067586552552935nteger(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))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ) ).

% uminus_power_if
tff(fact_5311_uminus__power__if,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))
       => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N) = 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))
       => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N) = uminus_uminus_rat(aa_nat_rat(power_power_rat(A),N)) ) ) ) ).

% uminus_power_if
tff(fact_5312_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(aa_int_fun_int_int(divide_divide_int,B2),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),N))) ) ) ).

% verit_less_mono_div_int2
tff(fact_5313_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(aa_int_fun_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_5314_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_5315_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_5316_of__bool__odd__eq__mod__2,axiom,
    ! [A: nat] : zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_5317_of__bool__odd__eq__mod__2,axiom,
    ! [A: int] : aa_bool_int(zero_n2684676970156552555ol_int,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))),A))) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_5318_of__bool__odd__eq__mod__2,axiom,
    ! [A: code_integer] : zero_n356916108424825756nteger(aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))) = modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_5319_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_5320_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),uminus_uminus_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(times_times_rat(A),C)),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,uminus_uminus_rat(B)),aa_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_5321_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_5322_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_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,uminus_uminus_rat(B)),aa_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(times_times_rat(A),C)),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_5323_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_5324_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),uminus_uminus_rat(divide_divide_rat(B,C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% neg_le_minus_divide_eq
tff(fact_5325_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_5326_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,uminus_uminus_rat(divide_divide_rat(B,C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B))) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5327_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_5328_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),uminus_uminus_rat(divide_divide_rat(B,C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B))) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5329_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_5330_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,uminus_uminus_rat(divide_divide_rat(B,C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5331_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),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,numeral_numeral_real(W))),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,numeral_numeral_real(W))),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,numeral_numeral_real(W))),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5332_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(W))),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(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),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(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),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,uminus_uminus_rat(numeral_numeral_rat(W))),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5333_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: 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,numeral_numeral_real(W))))
    <=> ( ( 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,numeral_numeral_real(W))),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,numeral_numeral_real(W))),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,numeral_numeral_real(W)))) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5334_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,C)),uminus_uminus_rat(numeral_numeral_rat(W))))
    <=> ( ( 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(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),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(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),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),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5335_power2__eq__1__iff,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))) = one_one_real )
    <=> ( ( A = one_one_real )
        | ( A = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% power2_eq_1_iff
tff(fact_5336_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( ( A = one_one_int )
        | ( A = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% power2_eq_1_iff
tff(fact_5337_power2__eq__1__iff,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
    <=> ( ( A = one_one_complex )
        | ( A = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% power2_eq_1_iff
tff(fact_5338_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( ( A = one_one_Code_integer )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% power2_eq_1_iff
tff(fact_5339_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( ( A = one_one_rat )
        | ( A = uminus_uminus_rat(one_one_rat) ) ) ) ).

% power2_eq_1_iff
tff(fact_5340_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) = one_one_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% minus_one_power_iff
tff(fact_5341_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = one_one_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))
       => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% minus_one_power_iff
tff(fact_5342_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) = one_one_complex ) )
      & ( ~ 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_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% minus_one_power_iff
tff(fact_5343_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = one_one_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))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% minus_one_power_iff
tff(fact_5344_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N) = one_one_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))
       => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N) = uminus_uminus_rat(one_one_rat) ) ) ) ).

% minus_one_power_iff
tff(fact_5345_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_5346_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_5347_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_5348_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_5349_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(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(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_5350_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% realpow_square_minus_le
tff(fact_5351_ln__one__minus__pos__lower__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_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_5352_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K)) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_5353_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K))) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_5354_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_5355_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L))
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,K)),L) = aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(L),one_one_int)),aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(K),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_5356_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(minus_minus_int(B),one_one_int) ) ) ).

% zmod_minus1
tff(fact_5357_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) )
        & ( ( aa_int_int(modulo_modulo_int(A),B) != zero_zero_int )
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int) ) ) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_5358_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) )
        & ( ( aa_int_int(modulo_modulo_int(A),B) != zero_zero_int )
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int) ) ) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_5359_zminus1__lemma,axiom,
    ! [A: int,B: int,Q2: int,R: int] :
      ( eucl_rel_int(A,B,product_Pair_int_int(Q2,R))
     => ( ( B != zero_zero_int )
       => eucl_rel_int(aa_int_int(uminus_uminus_int,A),B,product_Pair_int_int(aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R),zero_zero_int),aa_int_int(uminus_uminus_int,Q2)),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,Q2)),one_one_int)),aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R),zero_zero_int),zero_zero_int),aa_int_int(minus_minus_int(B),R)))) ) ) ).

% zminus1_lemma
tff(fact_5360_bits__induct,axiom,
    ! [P: fun_nat_bool,A: nat] :
      ( ! [A4: nat] :
          ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A4 )
         => pp(aa_nat_bool(P,A4)) )
     => ( ! [A4: nat,B4: bool] :
            ( pp(aa_nat_bool(P,A4))
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(B4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A4))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A4 )
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(B4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A4)))) ) )
       => pp(aa_nat_bool(P,A)) ) ) ).

% bits_induct
tff(fact_5361_bits__induct,axiom,
    ! [P: fun_int_bool,A: int] :
      ( ! [A4: int] :
          ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A4 )
         => pp(aa_int_bool(P,A4)) )
     => ( ! [A4: int,B4: bool] :
            ( pp(aa_int_bool(P,A4))
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(aa_bool_int(zero_n2684676970156552555ol_int,B4)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A4))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A4 )
             => pp(aa_int_bool(P,aa_int_int(plus_plus_int(aa_bool_int(zero_n2684676970156552555ol_int,B4)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A4)))) ) )
       => pp(aa_int_bool(P,A)) ) ) ).

% bits_induct
tff(fact_5362_bits__induct,axiom,
    ! [P: fun_Co8822979462182099027r_bool,A: code_integer] :
      ( ! [A4: code_integer] :
          ( ( divide6298287555418463151nteger(A4,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A4 )
         => pp(aa_Code_integer_bool(P,A4)) )
     => ( ! [A4: code_integer,B4: bool] :
            ( pp(aa_Code_integer_bool(P,A4))
           => ( ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(zero_n356916108424825756nteger(B4),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A4)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A4 )
             => pp(aa_Code_integer_bool(P,plus_p5714425477246183910nteger(zero_n356916108424825756nteger(B4),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A4)))) ) )
       => pp(aa_Code_integer_bool(P,A)) ) ) ).

% bits_induct
tff(fact_5363_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),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,numeral_numeral_real(W))),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,numeral_numeral_real(W))),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,numeral_numeral_real(W))),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5364_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(W))),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(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),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(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),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,uminus_uminus_rat(numeral_numeral_rat(W))),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5365_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: 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,numeral_numeral_real(W))))
    <=> ( ( 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,numeral_numeral_real(W))),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,numeral_numeral_real(W))),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,numeral_numeral_real(W)))) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5366_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,C)),uminus_uminus_rat(numeral_numeral_rat(W))))
    <=> ( ( 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(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),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(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),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),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5367_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_5368_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_5369_square__le__1,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_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_5370_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_5371_minus__power__mult__self,axiom,
    ! [A: real,N: nat] : 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,A)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N)) = 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)) ).

% minus_power_mult_self
tff(fact_5372_minus__power__mult__self,axiom,
    ! [A: int,N: nat] : aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N)) = 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)) ).

% minus_power_mult_self
tff(fact_5373_minus__power__mult__self,axiom,
    ! [A: complex,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N)) = aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% minus_power_mult_self
tff(fact_5374_minus__power__mult__self,axiom,
    ! [A: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% minus_power_mult_self
tff(fact_5375_minus__power__mult__self,axiom,
    ! [A: rat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N)),aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N)) = 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)) ).

% minus_power_mult_self
tff(fact_5376_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(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_5377_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K) = K )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(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_5378_minus__1__div__exp__eq__int,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_exp_eq_int
tff(fact_5379_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(plus_plus_int(K),L)),zero_zero_int))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_5380_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_5381_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_5382_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B = plus_plus_rat(B,A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_5383_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_5384_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B = aa_int_int(plus_plus_int(B),A) )
    <=> ( A = zero_zero_int ) ) ).

% add_0_iff
tff(fact_5385_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_5386_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(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_5387_exp__mod__exp,axiom,
    ! [M: nat,N: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(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(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_5388_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_5389_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( plus_plus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),D)) != plus_plus_rat(aa_rat_rat(times_times_rat(A),D),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_5390_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_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)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_5391_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) != aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),D)),aa_int_int(times_times_int(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_5392_crossproduct__eq,axiom,
    ! [W: real,Y: real,X: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_5393_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,X: rat,Z: rat] :
      ( ( plus_plus_rat(aa_rat_rat(times_times_rat(W),Y),aa_rat_rat(times_times_rat(X),Z)) = plus_plus_rat(aa_rat_rat(times_times_rat(W),Z),aa_rat_rat(times_times_rat(X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_5394_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_5395_crossproduct__eq,axiom,
    ! [W: int,Y: int,X: int,Z: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(W),Y)),aa_int_int(times_times_int(X),Z)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(W),Z)),aa_int_int(times_times_int(X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_5396_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_5397_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_5398_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_5399_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% power_minus1_odd
tff(fact_5400_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),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))) = uminus_uminus_rat(one_one_rat) ).

% power_minus1_odd
tff(fact_5401_int__bit__induct,axiom,
    ! [P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(P,zero_zero_int))
     => ( pp(aa_int_bool(P,aa_int_int(uminus_uminus_int,one_one_int)))
       => ( ! [K3: int] :
              ( pp(aa_int_bool(P,K3))
             => ( ( K3 != zero_zero_int )
               => pp(aa_int_bool(P,aa_int_int(times_times_int(K3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
         => ( ! [K3: int] :
                ( pp(aa_int_bool(P,K3))
               => ( ( K3 != aa_int_int(uminus_uminus_int,one_one_int) )
                 => pp(aa_int_bool(P,aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(K3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
           => pp(aa_int_bool(P,K)) ) ) ) ) ).

% int_bit_induct
tff(fact_5402_ln__one__plus__pos__lower__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_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_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))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_5403_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(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(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_5404_set__decode__def,axiom,
    ! [X: nat] : aa_nat_set_nat(nat_set_decode,X) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gh,X)) ).

% set_decode_def
tff(fact_5405_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_5406_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] : aa_int_int(aa_int_fun_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(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_5407_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] : divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(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(aa_num_Code_integer(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(aa_num_Code_integer(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_5408_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))) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),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))) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),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_5409_ln__2__less__1,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,numeral_numeral_real(aa_num_num(bit0,one)))),one_one_real)) ).

% ln_2_less_1
tff(fact_5410_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [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,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),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_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_5411_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_5412_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R: int] : adjust_div(product_Pair_int_int(Q2,R)) = aa_int_int(plus_plus_int(Q2),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,R),zero_zero_int)))) ).

% Divides.adjust_div_eq
tff(fact_5413_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_5414_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_5415_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))) ) )
           => ~ ! [Va3: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va3)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))))
                       => ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))))
                       => ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va3)))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_5416_semiring__norm_I90_J,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_num(bit1,M) = aa_num_num(bit1,N) )
    <=> ( M = N ) ) ).

% semiring_norm(90)
tff(fact_5417_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( aa_num_num(bit1,X32) = aa_num_num(bit1,Y32) )
    <=> ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_5418_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_5419_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_5420_abs__idempotent,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_idempotent
tff(fact_5421_abs__idempotent,axiom,
    ! [A: rat] : abs_abs_rat(abs_abs_rat(A)) = abs_abs_rat(A) ).

% abs_idempotent
tff(fact_5422_abs__abs,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_abs
tff(fact_5423_abs__abs,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_abs
tff(fact_5424_abs__abs,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_abs
tff(fact_5425_abs__abs,axiom,
    ! [A: rat] : abs_abs_rat(abs_abs_rat(A)) = abs_abs_rat(A) ).

% abs_abs
tff(fact_5426_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_5427_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_5428_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = abs_abs_rat(A) )
    <=> ( A = zero_zero_rat ) ) ).

% abs_0_eq
tff(fact_5429_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_5430_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_5431_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_5432_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0
tff(fact_5433_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_5434_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_5435_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_5436_abs__zero,axiom,
    abs_abs_rat(zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_5437_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_5438_abs__0,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_5439_abs__0,axiom,
    aa_complex_complex(abs_abs_complex,zero_zero_complex) = zero_zero_complex ).

% abs_0
tff(fact_5440_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_5441_abs__0,axiom,
    abs_abs_rat(zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_5442_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_5443_semiring__norm_I89_J,axiom,
    ! [M: num,N: num] : aa_num_num(bit1,M) != aa_num_num(bit0,N) ).

% semiring_norm(89)
tff(fact_5444_semiring__norm_I88_J,axiom,
    ! [M: num,N: num] : aa_num_num(bit0,M) != aa_num_num(bit1,N) ).

% semiring_norm(88)
tff(fact_5445_semiring__norm_I86_J,axiom,
    ! [M: num] : aa_num_num(bit1,M) != one ).

% semiring_norm(86)
tff(fact_5446_semiring__norm_I84_J,axiom,
    ! [N: num] : one != aa_num_num(bit1,N) ).

% semiring_norm(84)
tff(fact_5447_abs__numeral,axiom,
    ! [N: num] : abs_abs_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_num_Code_integer(numera6620942414471956472nteger,N) ).

% abs_numeral
tff(fact_5448_abs__numeral,axiom,
    ! [N: num] : aa_real_real(abs_abs_real,numeral_numeral_real(N)) = numeral_numeral_real(N) ).

% abs_numeral
tff(fact_5449_abs__numeral,axiom,
    ! [N: num] : abs_abs_rat(numeral_numeral_rat(N)) = numeral_numeral_rat(N) ).

% abs_numeral
tff(fact_5450_abs__numeral,axiom,
    ! [N: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,N) ).

% abs_numeral
tff(fact_5451_abs__mult__self__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) ).

% abs_mult_self_eq
tff(fact_5452_abs__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) ).

% abs_mult_self_eq
tff(fact_5453_abs__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(abs_abs_rat(A)),abs_abs_rat(A)) = aa_rat_rat(times_times_rat(A),A) ).

% abs_mult_self_eq
tff(fact_5454_abs__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(times_times_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,A)) = aa_int_int(times_times_int(A),A) ).

% abs_mult_self_eq
tff(fact_5455_abs__1,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_5456_abs__1,axiom,
    aa_complex_complex(abs_abs_complex,one_one_complex) = one_one_complex ).

% abs_1
tff(fact_5457_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_5458_abs__1,axiom,
    abs_abs_rat(one_one_rat) = one_one_rat ).

% abs_1
tff(fact_5459_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_5460_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_5461_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_5462_abs__add__abs,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(plus_plus_rat(abs_abs_rat(A),abs_abs_rat(B))) = plus_plus_rat(abs_abs_rat(A),abs_abs_rat(B)) ).

% abs_add_abs
tff(fact_5463_abs__add__abs,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) = aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_add_abs
tff(fact_5464_abs__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(abs_abs_complex,B)) ).

% abs_divide
tff(fact_5465_abs__divide,axiom,
    ! [A: real,B: 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)) ).

% abs_divide
tff(fact_5466_abs__divide,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(divide_divide_rat(A,B)) = divide_divide_rat(abs_abs_rat(A),abs_abs_rat(B)) ).

% abs_divide
tff(fact_5467_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_5468_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_5469_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_5470_abs__minus__cancel,axiom,
    ! [A: rat] : abs_abs_rat(uminus_uminus_rat(A)) = abs_abs_rat(A) ).

% abs_minus_cancel
tff(fact_5471_abs__minus,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
tff(fact_5472_abs__minus,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
tff(fact_5473_abs__minus,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(abs_abs_complex,A) ).

% abs_minus
tff(fact_5474_abs__minus,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).

% abs_minus
tff(fact_5475_abs__minus,axiom,
    ! [A: rat] : abs_abs_rat(uminus_uminus_rat(A)) = abs_abs_rat(A) ).

% abs_minus
tff(fact_5476_abs__dvd__iff,axiom,
    ! [M: real,K: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(abs_abs_real,M)),K))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),K)) ) ).

% abs_dvd_iff
tff(fact_5477_abs__dvd__iff,axiom,
    ! [M: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(abs_abs_int,M)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),K)) ) ).

% abs_dvd_iff
tff(fact_5478_abs__dvd__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,abs_abs_Code_integer(M)),K))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,M),K)) ) ).

% abs_dvd_iff
tff(fact_5479_abs__dvd__iff,axiom,
    ! [M: rat,K: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,abs_abs_rat(M)),K))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,M),K)) ) ).

% abs_dvd_iff
tff(fact_5480_dvd__abs__iff,axiom,
    ! [M: real,K: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),aa_real_real(abs_abs_real,K)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),K)) ) ).

% dvd_abs_iff
tff(fact_5481_dvd__abs__iff,axiom,
    ! [M: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),aa_int_int(abs_abs_int,K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),K)) ) ).

% dvd_abs_iff
tff(fact_5482_dvd__abs__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,M),abs_abs_Code_integer(K)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,M),K)) ) ).

% dvd_abs_iff
tff(fact_5483_dvd__abs__iff,axiom,
    ! [M: rat,K: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,M),abs_abs_rat(K)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,M),K)) ) ).

% dvd_abs_iff
tff(fact_5484_abs__bool__eq,axiom,
    ! [P: bool] : aa_real_real(abs_abs_real,zero_n3304061248610475627l_real(P)) = zero_n3304061248610475627l_real(P) ).

% abs_bool_eq
tff(fact_5485_abs__bool__eq,axiom,
    ! [P: bool] : abs_abs_rat(zero_n2052037380579107095ol_rat(P)) = zero_n2052037380579107095ol_rat(P) ).

% abs_bool_eq
tff(fact_5486_abs__bool__eq,axiom,
    ! [P: bool] : aa_int_int(abs_abs_int,aa_bool_int(zero_n2684676970156552555ol_int,P)) = aa_bool_int(zero_n2684676970156552555ol_int,P) ).

% abs_bool_eq
tff(fact_5487_abs__bool__eq,axiom,
    ! [P: bool] : abs_abs_Code_integer(zero_n356916108424825756nteger(P)) = zero_n356916108424825756nteger(P) ).

% abs_bool_eq
tff(fact_5488_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_5489_tanh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% tanh_real_le_iff
tff(fact_5490_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(ord_less_num(aa_num_num(bit1,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(ord_less_num(M),N)) ) ).

% semiring_norm(80)
tff(fact_5491_semiring__norm_I73_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(ord_less_eq_num(aa_num_num(bit1,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(ord_less_eq_num(M),N)) ) ).

% semiring_norm(73)
tff(fact_5492_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_5493_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_5494_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(A)),zero_zero_rat))
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_5495_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_5496_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_5497_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_5498_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_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_5499_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_5500_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_5501_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_5502_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( abs_abs_rat(A) = A ) ) ).

% abs_of_nonneg
tff(fact_5503_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_5504_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_5505_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_5506_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),abs_abs_rat(A)))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_5507_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_5508_abs__neg__numeral,axiom,
    ! [N: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = numeral_numeral_real(N) ).

% abs_neg_numeral
tff(fact_5509_abs__neg__numeral,axiom,
    ! [N: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,N) ).

% abs_neg_numeral
tff(fact_5510_abs__neg__numeral,axiom,
    ! [N: num] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_num_Code_integer(numera6620942414471956472nteger,N) ).

% abs_neg_numeral
tff(fact_5511_abs__neg__numeral,axiom,
    ! [N: num] : abs_abs_rat(uminus_uminus_rat(numeral_numeral_rat(N))) = numeral_numeral_rat(N) ).

% abs_neg_numeral
tff(fact_5512_abs__neg__one,axiom,
    aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,one_one_real)) = one_one_real ).

% abs_neg_one
tff(fact_5513_abs__neg__one,axiom,
    aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,one_one_int)) = one_one_int ).

% abs_neg_one
tff(fact_5514_abs__neg__one,axiom,
    abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_5515_abs__neg__one,axiom,
    abs_abs_rat(uminus_uminus_rat(one_one_rat)) = one_one_rat ).

% abs_neg_one
tff(fact_5516_abs__power__minus,axiom,
    ! [A: real,N: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N)) = aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).

% abs_power_minus
tff(fact_5517_abs__power__minus,axiom,
    ! [A: int,N: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N)) = aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),N)) ).

% abs_power_minus
tff(fact_5518_abs__power__minus,axiom,
    ! [A: code_integer,N: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N)) = abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).

% abs_power_minus
tff(fact_5519_abs__power__minus,axiom,
    ! [A: rat,N: nat] : abs_abs_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N)) = abs_abs_rat(aa_nat_rat(power_power_rat(A),N)) ).

% abs_power_minus
tff(fact_5520_semiring__norm_I7_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% semiring_norm(7)
tff(fact_5521_semiring__norm_I9_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% semiring_norm(9)
tff(fact_5522_semiring__norm_I14_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M),aa_num_num(bit1,N))) ).

% semiring_norm(14)
tff(fact_5523_semiring__norm_I15_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),N)) ).

% semiring_norm(15)
tff(fact_5524_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(ord_less_num(aa_num_num(bit1,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(ord_less_num(M),N)) ) ).

% semiring_norm(81)
tff(fact_5525_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(ord_less_eq_num(aa_num_num(bit0,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(ord_less_eq_num(M),N)) ) ).

% semiring_norm(72)
tff(fact_5526_semiring__norm_I77_J,axiom,
    ! [N: num] : pp(aa_num_bool(ord_less_num(one),aa_num_num(bit1,N))) ).

% semiring_norm(77)
tff(fact_5527_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(ord_less_eq_num(aa_num_num(bit1,M)),one)) ).

% semiring_norm(70)
tff(fact_5528_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_5529_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_5530_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% tanh_real_nonneg_iff
tff(fact_5531_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tanh_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% tanh_real_nonpos_iff
tff(fact_5532_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_5533_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(A,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_5534_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_5535_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),divide_divide_rat(A,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_5536_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_5537_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_5538_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( abs_abs_rat(A) = uminus_uminus_rat(A) ) ) ).

% abs_of_nonpos
tff(fact_5539_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_5540_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit1
tff(fact_5541_semiring__norm_I10_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)),one)) ).

% semiring_norm(10)
tff(fact_5542_semiring__norm_I8_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),one) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one)) ).

% semiring_norm(8)
tff(fact_5543_semiring__norm_I5_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,M)),one) = aa_num_num(bit1,M) ).

% semiring_norm(5)
tff(fact_5544_semiring__norm_I4_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit1,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% semiring_norm(4)
tff(fact_5545_semiring__norm_I3_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit0,N)) = aa_num_num(bit1,N) ).

% semiring_norm(3)
tff(fact_5546_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_5547_semiring__norm_I16_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)),aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)))) ).

% semiring_norm(16)
tff(fact_5548_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(ord_less_eq_num(aa_num_num(bit1,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(ord_less_num(M),N)) ) ).

% semiring_norm(74)
tff(fact_5549_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(ord_less_num(aa_num_num(bit0,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(ord_less_eq_num(M),N)) ) ).

% semiring_norm(79)
tff(fact_5550_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_5551_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_5552_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(abs_abs_rat(A)),N)))
    <=> ( ( A != zero_zero_rat )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5553_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_5554_abs__power2,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_5555_abs__power2,axiom,
    ! [A: rat] : abs_abs_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_5556_abs__power2,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_5557_abs__power2,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_5558_power2__abs,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_5559_power2__abs,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(abs_abs_rat(A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_5560_power2__abs,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_5561_power2__abs,axiom,
    ! [A: int] : aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_5562_power__even__abs__numeral,axiom,
    ! [W: num,A: 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))),aa_num_nat(numeral_numeral_nat,W)))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_5563_power__even__abs__numeral,axiom,
    ! [W: num,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))),aa_num_nat(numeral_numeral_nat,W)))
     => ( aa_nat_rat(power_power_rat(abs_abs_rat(A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_5564_power__even__abs__numeral,axiom,
    ! [W: num,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))),aa_num_nat(numeral_numeral_nat,W)))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_5565_power__even__abs__numeral,axiom,
    ! [W: num,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))),aa_num_nat(numeral_numeral_nat,W)))
     => ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_5566_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N)) ).

% div_Suc_eq_div_add3
tff(fact_5567_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_5568_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N)) ).

% mod_Suc_eq_mod_add3
tff(fact_5569_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_5570_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)))),one_one_int) ).

% zmod_numeral_Bit1
tff(fact_5571_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_5572_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_5573_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_5574_abs__ge__self,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),abs_abs_rat(A))) ).

% abs_ge_self
tff(fact_5575_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_5576_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_5577_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_5578_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_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_5579_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_5580_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_5581_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_5582_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_5583_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
tff(fact_5584_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_5585_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] : 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_mult
tff(fact_5586_abs__mult,axiom,
    ! [A: real,B: 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_mult
tff(fact_5587_abs__mult,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(abs_abs_rat(A)),abs_abs_rat(B)) ).

% abs_mult
tff(fact_5588_abs__mult,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(times_times_int(A),B)) = aa_int_int(times_times_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_mult
tff(fact_5589_abs__one,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_5590_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_5591_abs__one,axiom,
    abs_abs_rat(one_one_rat) = one_one_rat ).

% abs_one
tff(fact_5592_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_5593_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_5594_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_5595_abs__minus__commute,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(minus_minus_rat(A,B)) = abs_abs_rat(minus_minus_rat(B,A)) ).

% abs_minus_commute
tff(fact_5596_abs__minus__commute,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(A),B)) = aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(B),A)) ).

% abs_minus_commute
tff(fact_5597_abs__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(abs_abs_real,X) = aa_real_real(abs_abs_real,Y) )
    <=> ( ( X = Y )
        | ( X = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% abs_eq_iff
tff(fact_5598_abs__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(abs_abs_int,X) = aa_int_int(abs_abs_int,Y) )
    <=> ( ( X = Y )
        | ( X = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% abs_eq_iff
tff(fact_5599_abs__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( abs_abs_Code_integer(X) = abs_abs_Code_integer(Y) )
    <=> ( ( X = Y )
        | ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% abs_eq_iff
tff(fact_5600_abs__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( abs_abs_rat(X) = abs_abs_rat(Y) )
    <=> ( ( X = Y )
        | ( X = uminus_uminus_rat(Y) ) ) ) ).

% abs_eq_iff
tff(fact_5601_power__abs,axiom,
    ! [A: code_integer,N: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),N) ).

% power_abs
tff(fact_5602_power__abs,axiom,
    ! [A: rat,N: nat] : abs_abs_rat(aa_nat_rat(power_power_rat(A),N)) = aa_nat_rat(power_power_rat(abs_abs_rat(A)),N) ).

% power_abs
tff(fact_5603_power__abs,axiom,
    ! [A: real,N: nat] : aa_real_real(abs_abs_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,aa_real_real(abs_abs_real,A)),N) ).

% power_abs
tff(fact_5604_power__abs,axiom,
    ! [A: int,N: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),N)) = aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N) ).

% power_abs
tff(fact_5605_dvd__if__abs__eq,axiom,
    ! [L: real,K: real] :
      ( ( aa_real_real(abs_abs_real,L) = aa_real_real(abs_abs_real,K) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,L),K)) ) ).

% dvd_if_abs_eq
tff(fact_5606_dvd__if__abs__eq,axiom,
    ! [L: int,K: int] :
      ( ( aa_int_int(abs_abs_int,L) = aa_int_int(abs_abs_int,K) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)) ) ).

% dvd_if_abs_eq
tff(fact_5607_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K: code_integer] :
      ( ( abs_abs_Code_integer(L) = abs_abs_Code_integer(K) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,L),K)) ) ).

% dvd_if_abs_eq
tff(fact_5608_dvd__if__abs__eq,axiom,
    ! [L: rat,K: rat] :
      ( ( abs_abs_rat(L) = abs_abs_rat(K) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,L),K)) ) ).

% dvd_if_abs_eq
tff(fact_5609_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] : aa_num_num(bit0,X22) != aa_num_num(bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_5610_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != aa_num_num(bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_5611_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_5612_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_5613_abs__ge__zero,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),abs_abs_rat(A))) ).

% abs_ge_zero
tff(fact_5614_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_5615_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_5616_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_5617_abs__not__less__zero,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(A)),zero_zero_rat)) ).

% abs_not_less_zero
tff(fact_5618_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_5619_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_5620_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_5621_abs__of__pos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( abs_abs_rat(A) = A ) ) ).

% abs_of_pos
tff(fact_5622_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_5623_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_5624_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_5625_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(plus_plus_rat(A,B))),plus_plus_rat(abs_abs_rat(A),abs_abs_rat(B)))) ).

% abs_triangle_ineq
tff(fact_5626_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(plus_plus_int(A),B))),aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))) ).

% abs_triangle_ineq
tff(fact_5627_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_5628_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_5629_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(A)),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(B)),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(abs_abs_rat(A)),abs_abs_rat(B))),aa_rat_rat(times_times_rat(C),D))) ) ) ).

% abs_mult_less
tff(fact_5630_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(times_times_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(times_times_int(C),D))) ) ) ).

% abs_mult_less
tff(fact_5631_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_5632_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_5633_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(abs_abs_rat(A),abs_abs_rat(B))),abs_abs_rat(minus_minus_rat(B,A)))) ).

% abs_triangle_ineq2_sym
tff(fact_5634_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(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(minus_minus_int(B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_5635_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_5636_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_5637_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(abs_abs_rat(A),abs_abs_rat(B)))),abs_abs_rat(minus_minus_rat(A,B)))) ).

% abs_triangle_ineq3
tff(fact_5638_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(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(minus_minus_int(A),B)))) ).

% abs_triangle_ineq3
tff(fact_5639_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_5640_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_5641_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(abs_abs_rat(A),abs_abs_rat(B))),abs_abs_rat(minus_minus_rat(A,B)))) ).

% abs_triangle_ineq2
tff(fact_5642_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_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(minus_minus_int(A),B)))) ).

% abs_triangle_ineq2
tff(fact_5643_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_5644_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( abs_abs_rat(divide_divide_rat(A,B)) = divide_divide_rat(abs_abs_rat(A),abs_abs_rat(B)) ) ) ).

% nonzero_abs_divide
tff(fact_5645_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_5646_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_5647_abs__ge__minus__self,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(A)),abs_abs_rat(A))) ).

% abs_ge_minus_self
tff(fact_5648_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_5649_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_5650_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_5651_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_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,uminus_uminus_rat(A)),B)) ) ) ).

% abs_le_iff
tff(fact_5652_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_5653_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_5654_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_5655_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(A)),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(A)),B)) ) ).

% abs_le_D2
tff(fact_5656_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_5657_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_5658_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_5659_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,uminus_uminus_rat(A)),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(A)),B)) ) ) ).

% abs_leI
tff(fact_5660_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_5661_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_5662_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_5663_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_5664_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_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,uminus_uminus_rat(A)),B)) ) ) ).

% abs_less_iff
tff(fact_5665_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_5666_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))
         => ( ! [M3: num] : X != product_Pair_num_num(aa_num_num(bit0,M3),one)
           => ( ! [M3: num,N2: num] : X != product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N2))
             => ( ! [M3: num,N2: num] : X != product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N2))
               => ( ! [M3: num] : X != product_Pair_num_num(aa_num_num(bit1,M3),one)
                 => ( ! [M3: num,N2: num] : X != product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N2))
                   => ~ ! [M3: num,N2: num] : X != product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N2)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_5667_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] : Y != aa_num_num(bit0,X23)
       => ~ ! [X33: num] : Y != aa_num_num(bit1,X33) ) ) ).

% num.exhaust
tff(fact_5668_sin__bound__lemma,axiom,
    ! [X: real,Y: real,U: real,V: real] :
      ( ( X = Y )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,U)),V))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),U)),Y))),V)) ) ) ).

% sin_bound_lemma
tff(fact_5669_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_5670_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_5671_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),E2)) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_5672_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(X)),E2)) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_5673_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_5674_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_5675_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(times_times_rat(abs_abs_rat(Y)),X) = abs_abs_rat(aa_rat_rat(times_times_rat(Y),X)) ) ) ).

% abs_mult_pos
tff(fact_5676_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(times_times_int(aa_int_int(abs_abs_int,Y)),X) = aa_int_int(abs_abs_int,aa_int_int(times_times_int(Y),X)) ) ) ).

% abs_mult_pos
tff(fact_5677_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_5678_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_5679_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)) ) )
     => ( abs_abs_rat(aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(abs_abs_rat(A)),abs_abs_rat(B)) ) ) ).

% abs_eq_mult
tff(fact_5680_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(times_times_int(A),B)) = aa_int_int(times_times_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).

% abs_eq_mult
tff(fact_5681_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_5682_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_5683_abs__eq__iff_H,axiom,
    ! [A: rat,B: 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 = uminus_uminus_rat(B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5684_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_5685_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_5686_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_5687_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A = abs_abs_rat(B) )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
        & ( ( B = A )
          | ( B = uminus_uminus_rat(A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5688_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_5689_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_5690_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_5691_abs__minus__le__zero,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(abs_abs_rat(A))),zero_zero_rat)) ).

% abs_minus_le_zero
tff(fact_5692_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_5693_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_5694_abs__div__pos,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( divide_divide_rat(abs_abs_rat(X),Y) = abs_abs_rat(divide_divide_rat(X,Y)) ) ) ).

% abs_div_pos
tff(fact_5695_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_5696_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_5697_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(abs_abs_rat(A)),N))) ).

% zero_le_power_abs
tff(fact_5698_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_5699_abs__if__raw,axiom,
    ! [X4: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X4) = aa_real_real(uminus_uminus_real,X4) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X4) = X4 ) ) ) ).

% abs_if_raw
tff(fact_5700_abs__if__raw,axiom,
    ! [X4: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X4) = aa_int_int(uminus_uminus_int,X4) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X4) = X4 ) ) ) ).

% abs_if_raw
tff(fact_5701_abs__if__raw,axiom,
    ! [X4: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X4),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(X4) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X4) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X4),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(X4) = X4 ) ) ) ).

% abs_if_raw
tff(fact_5702_abs__if__raw,axiom,
    ! [X4: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),zero_zero_rat))
       => ( abs_abs_rat(X4) = uminus_uminus_rat(X4) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),zero_zero_rat))
       => ( abs_abs_rat(X4) = X4 ) ) ) ).

% abs_if_raw
tff(fact_5703_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_5704_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_5705_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_5706_abs__if,axiom,
    ! [A: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( abs_abs_rat(A) = uminus_uminus_rat(A) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( abs_abs_rat(A) = A ) ) ) ).

% abs_if
tff(fact_5707_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_5708_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_5709_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_5710_abs__of__neg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( abs_abs_rat(A) = uminus_uminus_rat(A) ) ) ).

% abs_of_neg
tff(fact_5711_abs__diff__le__iff,axiom,
    ! [X: code_integer,A: code_integer,R: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,A))),R))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R)),X))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),plus_p5714425477246183910nteger(A,R))) ) ) ).

% abs_diff_le_iff
tff(fact_5712_abs__diff__le__iff,axiom,
    ! [X: real,A: real,R: 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))),R))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R))) ) ) ).

% abs_diff_le_iff
tff(fact_5713_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(X,A))),R))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(A,R)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),plus_plus_rat(A,R))) ) ) ).

% abs_diff_le_iff
tff(fact_5714_abs__diff__le__iff,axiom,
    ! [X: int,A: int,R: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(X),A))),R))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(minus_minus_int(A),R)),X))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(plus_plus_int(A),R))) ) ) ).

% abs_diff_le_iff
tff(fact_5715_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_5716_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_5717_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(A,B))),plus_plus_rat(abs_abs_rat(A),abs_abs_rat(B)))) ).

% abs_triangle_ineq4
tff(fact_5718_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(minus_minus_int(A),B))),aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))) ).

% abs_triangle_ineq4
tff(fact_5719_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_5720_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_5721_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,abs_abs_rat(minus_minus_rat(plus_plus_rat(A,B),plus_plus_rat(C,D)))),plus_plus_rat(abs_abs_rat(minus_minus_rat(A,C)),abs_abs_rat(minus_minus_rat(B,D))))) ).

% abs_diff_triangle_ineq
tff(fact_5722_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(minus_minus_int(aa_int_int(plus_plus_int(A),B)),aa_int_int(plus_plus_int(C),D)))),aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(A),C))),aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(B),D))))) ).

% abs_diff_triangle_ineq
tff(fact_5723_abs__diff__less__iff,axiom,
    ! [X: code_integer,A: code_integer,R: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,A))),R))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R)),X))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),plus_p5714425477246183910nteger(A,R))) ) ) ).

% abs_diff_less_iff
tff(fact_5724_abs__diff__less__iff,axiom,
    ! [X: real,A: real,R: 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))),R))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R))) ) ) ).

% abs_diff_less_iff
tff(fact_5725_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(minus_minus_rat(X,A))),R))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(A,R)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),plus_plus_rat(A,R))) ) ) ).

% abs_diff_less_iff
tff(fact_5726_abs__diff__less__iff,axiom,
    ! [X: int,A: int,R: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(X),A))),R))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(minus_minus_int(A),R)),X))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_int_int(plus_plus_int(A),R))) ) ) ).

% abs_diff_less_iff
tff(fact_5727_numeral__Bit1,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit1,N)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N))),one_one_complex) ).

% numeral_Bit1
tff(fact_5728_numeral__Bit1,axiom,
    ! [N: num] : numeral_numeral_real(aa_num_num(bit1,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(N)),numeral_numeral_real(N))),one_one_real) ).

% numeral_Bit1
tff(fact_5729_numeral__Bit1,axiom,
    ! [N: num] : numeral_numeral_rat(aa_num_num(bit1,N)) = plus_plus_rat(plus_plus_rat(numeral_numeral_rat(N),numeral_numeral_rat(N)),one_one_rat) ).

% numeral_Bit1
tff(fact_5730_numeral__Bit1,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N))),one_one_nat) ).

% numeral_Bit1
tff(fact_5731_numeral__Bit1,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N))),one_one_int) ).

% numeral_Bit1
tff(fact_5732_eval__nat__numeral_I3_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))) ).

% eval_nat_numeral(3)
tff(fact_5733_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = 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,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_5734_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: 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,Q2))) = 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,Q2))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_5735_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,M)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,N)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,N),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_5736_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != 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,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_5737_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: 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,Q2))) != 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,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_5738_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,M)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,N)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_5739_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != 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,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_5740_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != 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,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_5741_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,M)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,N)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_5742_power__minus__Bit1,axiom,
    ! [X: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_5743_power__minus__Bit1,axiom,
    ! [X: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_5744_power__minus__Bit1,axiom,
    ! [X: complex,K: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_5745_power__minus__Bit1,axiom,
    ! [X: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_5746_power__minus__Bit1,axiom,
    ! [X: rat,K: num] : aa_nat_rat(power_power_rat(uminus_uminus_rat(X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = uminus_uminus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_5747_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))
       => ? [D2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
            & ! [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))),D2))
               => ( 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_5748_numeral__code_I3_J,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit1,N)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N))),one_one_complex) ).

% numeral_code(3)
tff(fact_5749_numeral__code_I3_J,axiom,
    ! [N: num] : numeral_numeral_real(aa_num_num(bit1,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(N)),numeral_numeral_real(N))),one_one_real) ).

% numeral_code(3)
tff(fact_5750_numeral__code_I3_J,axiom,
    ! [N: num] : numeral_numeral_rat(aa_num_num(bit1,N)) = plus_plus_rat(plus_plus_rat(numeral_numeral_rat(N),numeral_numeral_rat(N)),one_one_rat) ).

% numeral_code(3)
tff(fact_5751_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N))),one_one_nat) ).

% numeral_code(3)
tff(fact_5752_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N))),one_one_int) ).

% numeral_code(3)
tff(fact_5753_power__numeral__odd,axiom,
    ! [Z: complex,W: num] : aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(Z),aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_odd
tff(fact_5754_power__numeral__odd,axiom,
    ! [Z: real,W: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_odd
tff(fact_5755_power__numeral__odd,axiom,
    ! [Z: rat,W: num] : aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Z),aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_odd
tff(fact_5756_power__numeral__odd,axiom,
    ! [Z: nat,W: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_odd
tff(fact_5757_power__numeral__odd,axiom,
    ! [Z: int,W: num] : aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_int_int(times_times_int(aa_int_int(times_times_int(Z),aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_odd
tff(fact_5758_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_5759_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_5760_abs__add__one__gt__zero,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),plus_plus_rat(one_one_rat,abs_abs_rat(X)))) ).

% abs_add_one_gt_zero
tff(fact_5761_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(plus_plus_int(one_one_int),aa_int_int(abs_abs_int,X)))) ).

% abs_add_one_gt_zero
tff(fact_5762_numeral__Bit1__div__2,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,N) ).

% numeral_Bit1_div_2
tff(fact_5763_numeral__Bit1__div__2,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,N) ).

% numeral_Bit1_div_2
tff(fact_5764_odd__numeral,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,N)))) ).

% odd_numeral
tff(fact_5765_odd__numeral,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)))) ).

% odd_numeral
tff(fact_5766_odd__numeral,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) ).

% odd_numeral
tff(fact_5767_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: 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,Q2))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_5768_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: 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,Q2))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_5769_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,N)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_5770_power3__eq__cube,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(A),A)),A) ).

% power3_eq_cube
tff(fact_5771_power3__eq__cube,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),A) ).

% power3_eq_cube
tff(fact_5772_power3__eq__cube,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),A)),A) ).

% power3_eq_cube
tff(fact_5773_power3__eq__cube,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A)),A) ).

% power3_eq_cube
tff(fact_5774_power3__eq__cube,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_int_int(times_times_int(aa_int_int(times_times_int(A),A)),A) ).

% power3_eq_cube
tff(fact_5775_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_5776_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))
       => ? [D2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
            & ! [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))),D2))
               => ( 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_5777_Suc3__eq__add__3,axiom,
    ! [N: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N) ).

% Suc3_eq_add_3
tff(fact_5778_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_5779_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_5780_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_5781_abs__le__square__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(X)),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_5782_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_5783_abs__square__eq__1,axiom,
    ! [X: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( abs_abs_Code_integer(X) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_5784_abs__square__eq__1,axiom,
    ! [X: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( abs_abs_rat(X) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_5785_abs__square__eq__1,axiom,
    ! [X: 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(abs_abs_real,X) = one_one_real ) ) ).

% abs_square_eq_1
tff(fact_5786_abs__square__eq__1,axiom,
    ! [X: int] :
      ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).

% abs_square_eq_1
tff(fact_5787_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_5788_power__even__abs,axiom,
    ! [N: nat,A: 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))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),N) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) ) ) ).

% power_even_abs
tff(fact_5789_power__even__abs,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))
     => ( aa_nat_rat(power_power_rat(abs_abs_rat(A)),N) = aa_nat_rat(power_power_rat(A),N) ) ) ).

% power_even_abs
tff(fact_5790_power__even__abs,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))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) ) ) ).

% power_even_abs
tff(fact_5791_power__even__abs,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))
     => ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N) = aa_nat_int(power_power_int(A),N) ) ) ).

% power_even_abs
tff(fact_5792_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: 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,Q2))) = 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,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5793_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: 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,Q2))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5794_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,M)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5795_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: 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,Q2))) = 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,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5796_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: 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,Q2))) = 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,Q2))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5797_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,N)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,N),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5798_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),N) ).

% Suc_div_eq_add3_div
tff(fact_5799_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),N) ).

% Suc_mod_eq_add3_mod
tff(fact_5800_abs__sqrt__wlog,axiom,
    ! [P: fun_Co8310303586393106575r_bool,X: code_integer] :
      ( ! [X2: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,X2),aa_nat_Code_integer(power_8256067586552552935nteger(X2),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_5801_abs__sqrt__wlog,axiom,
    ! [P: fun_re1255401033344146673l_bool,X: real] :
      ( ! [X2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,X2),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),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_5802_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,X: rat] :
      ( ! [X2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,X2),aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,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_5803_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_bool,X: int] :
      ( ! [X2: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,X2),aa_nat_int(power_power_int(X2),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_5804_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_5805_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_5806_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,abs_abs_rat(X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_5807_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_5808_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_5809_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_5810_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,abs_abs_rat(X)),one_one_rat)) ) ).

% abs_square_le_1
tff(fact_5811_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_5812_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_5813_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_5814_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,abs_abs_rat(X)),one_one_rat)) ) ).

% abs_square_less_1
tff(fact_5815_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_5816_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_5817_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_5818_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,abs_abs_rat(A)),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_5819_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_5820_eq__diff__eq_H,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( X = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z) )
    <=> ( Y = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Z) ) ) ).

% eq_diff_eq'
tff(fact_5821_abs__ln__one__plus__x__minus__x__bound__nonneg,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_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),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_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_5822_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_5823_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(pred_numeral,L)),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_5824_dbl__dec__simps_I4_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_5825_dbl__dec__simps_I4_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_5826_dbl__dec__simps_I4_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_5827_dbl__dec__simps_I4_J,axiom,
    neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_5828_dbl__dec__simps_I4_J,axiom,
    neg_nu3179335615603231917ec_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_5829_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_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(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_5830_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(ord_less_num(M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))) ) )
      & ( ~ pp(aa_num_bool(ord_less_num(M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N)) = unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_5831_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(ord_less_num(M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,M))) ) )
      & ( ~ pp(aa_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_5832_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_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(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_5833_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(ord_less_eq_num(M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))) ) )
      & ( ~ pp(aa_num_bool(ord_less_eq_num(M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,N)) = unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_5834_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(ord_less_eq_num(M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,M))) ) )
      & ( ~ pp(aa_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_5835_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_bit1
tff(fact_5836_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,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,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_5837_dbl__inc__simps_I3_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_5838_dbl__inc__simps_I3_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,one_one_real) = numeral_numeral_real(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_5839_dbl__inc__simps_I3_J,axiom,
    neg_nu5219082963157363817nc_rat(one_one_rat) = numeral_numeral_rat(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_5840_dbl__inc__simps_I3_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_5841_dbl__dec__simps_I3_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,one_one_complex) = one_one_complex ).

% dbl_dec_simps(3)
tff(fact_5842_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_5843_dbl__dec__simps_I3_J,axiom,
    neg_nu3179335615603231917ec_rat(one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_5844_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_5845_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_5846_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_5847_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( aa_nat_nat(suc,N) = aa_num_nat(numeral_numeral_nat,K) )
    <=> ( N = aa_num_nat(pred_numeral,K) ) ) ).

% Suc_eq_numeral
tff(fact_5848_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,N) )
    <=> ( aa_num_nat(pred_numeral,K) = N ) ) ).

% eq_numeral_Suc
tff(fact_5849_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_5850_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_5851_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% zero_le_arctan_iff
tff(fact_5852_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arctan,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% arctan_le_zero_iff
tff(fact_5853_dbl__inc__simps_I2_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).

% dbl_inc_simps(2)
tff(fact_5854_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_5855_dbl__inc__simps_I2_J,axiom,
    neg_nu5219082963157363817nc_rat(zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_5856_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_5857_dbl__inc__simps_I4_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% dbl_inc_simps(4)
tff(fact_5858_dbl__inc__simps_I4_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_inc_simps(4)
tff(fact_5859_dbl__inc__simps_I4_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% dbl_inc_simps(4)
tff(fact_5860_dbl__inc__simps_I4_J,axiom,
    neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_inc_simps(4)
tff(fact_5861_dbl__inc__simps_I4_J,axiom,
    neg_nu5219082963157363817nc_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(one_one_rat) ).

% dbl_inc_simps(4)
tff(fact_5862_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_5863_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,numeral_numeral_real(K)) = numeral_numeral_real(aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_5864_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : neg_nu5219082963157363817nc_rat(numeral_numeral_rat(K)) = numeral_numeral_rat(aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_5865_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_5866_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_5867_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_5868_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit1,K)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K)) ).

% pred_numeral_simps(3)
tff(fact_5869_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_5870_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_5871_diff__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(pred_numeral,K)) ).

% diff_Suc_numeral
tff(fact_5872_diff__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),N) ).

% diff_numeral_Suc
tff(fact_5873_max__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),aa_num_nat(pred_numeral,K))) ).

% max_Suc_numeral
tff(fact_5874_max__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),N)) ).

% max_numeral_Suc
tff(fact_5875_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_5876_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_5877_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_5878_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_5879_dbl__dec__simps_I2_J,axiom,
    neg_nu3179335615603231917ec_rat(zero_zero_rat) = uminus_uminus_rat(one_one_rat) ).

% dbl_dec_simps(2)
tff(fact_5880_numeral__div__minus__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M,N))) ).

% numeral_div_minus_numeral
tff(fact_5881_minus__numeral__div__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M,N))) ).

% minus_numeral_div_numeral
tff(fact_5882_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu6075765906172075777c_real,numeral_numeral_real(K))) ).

% dbl_inc_simps(1)
tff(fact_5883_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_inc_simps(1)
tff(fact_5884_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K))) ).

% dbl_inc_simps(1)
tff(fact_5885_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu7757733837767384882nteger(aa_num_Code_integer(numera6620942414471956472nteger,K))) ).

% dbl_inc_simps(1)
tff(fact_5886_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : neg_nu5219082963157363817nc_rat(uminus_uminus_rat(numeral_numeral_rat(K))) = uminus_uminus_rat(neg_nu3179335615603231917ec_rat(numeral_numeral_rat(K))) ).

% dbl_inc_simps(1)
tff(fact_5887_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu8295874005876285629c_real,numeral_numeral_real(K))) ).

% dbl_dec_simps(1)
tff(fact_5888_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_dec_simps(1)
tff(fact_5889_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K))) ).

% dbl_dec_simps(1)
tff(fact_5890_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu5831290666863070958nteger(aa_num_Code_integer(numera6620942414471956472nteger,K))) ).

% dbl_dec_simps(1)
tff(fact_5891_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : neg_nu3179335615603231917ec_rat(uminus_uminus_rat(numeral_numeral_rat(K))) = uminus_uminus_rat(neg_nu5219082963157363817nc_rat(numeral_numeral_rat(K))) ).

% dbl_dec_simps(1)
tff(fact_5892_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
    <=> unique6319869463603278526ux_int(unique5052692396658037445od_int(N,M)) ) ).

% dvd_numeral_simp
tff(fact_5893_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
    <=> unique6322359934112328802ux_nat(unique5055182867167087721od_nat(N,M)) ) ).

% dvd_numeral_simp
tff(fact_5894_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N)))
    <=> unique5706413561485394159nteger(unique3479559517661332726nteger(N,M)) ) ).

% dvd_numeral_simp
tff(fact_5895_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5052692396658037445od_int(M,one) = product_Pair_int_int(aa_num_int(numeral_numeral_int,M),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_5896_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_5897_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique3479559517661332726nteger(M,one) = produc1086072967326762835nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_5898_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_5899_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_5900_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% divmod_algorithm_code(3)
tff(fact_5901_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_5902_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_5903_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% divmod_algorithm_code(4)
tff(fact_5904_one__div__minus__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,N))) ).

% one_div_minus_numeral
tff(fact_5905_minus__one__div__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,N))) ).

% minus_one_div_numeral
tff(fact_5906_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(times_times_int(aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_bit0
tff(fact_5907_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(times_times_int(aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_5908_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_5909_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_5910_arctan__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% arctan_le_iff
tff(fact_5911_arctan__monotone_H,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_eq_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y))) ) ).

% arctan_monotone'
tff(fact_5912_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(times_times_int(M),N)) = one_one_int )
     => ( aa_int_int(abs_abs_int,M) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_5913_abs__div,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Y),X))
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,X)),aa_int_int(abs_abs_int,Y)) ) ) ).

% abs_div
tff(fact_5914_numeral__eq__Suc,axiom,
    ! [K: num] : aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,aa_num_nat(pred_numeral,K)) ).

% numeral_eq_Suc
tff(fact_5915_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_5916_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_5917_pred__numeral__def,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),one_one_nat) ).

% pred_numeral_def
tff(fact_5918_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(M),N)),M))
      <=> ( aa_int_int(abs_abs_int,N) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_5919_dbl__inc__def,axiom,
    ! [X: complex] : aa_complex_complex(neg_nu8557863876264182079omplex,X) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(X),X)),one_one_complex) ).

% dbl_inc_def
tff(fact_5920_dbl__inc__def,axiom,
    ! [X: real] : aa_real_real(neg_nu8295874005876285629c_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),X)),one_one_real) ).

% dbl_inc_def
tff(fact_5921_dbl__inc__def,axiom,
    ! [X: rat] : neg_nu5219082963157363817nc_rat(X) = plus_plus_rat(plus_plus_rat(X,X),one_one_rat) ).

% dbl_inc_def
tff(fact_5922_dbl__inc__def,axiom,
    ! [X: int] : aa_int_int(neg_nu5851722552734809277nc_int,X) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(X),X)),one_one_int) ).

% dbl_inc_def
tff(fact_5923_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,K)),L)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(K),L))) ) ).

% even_abs_add_iff
tff(fact_5924_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(K),aa_int_int(abs_abs_int,L))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(K),L))) ) ).

% even_add_abs_iff
tff(fact_5925_divmod__int__def,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(M,N) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ).

% divmod_int_def
tff(fact_5926_divmod__def,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(M,N) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ).

% divmod_def
tff(fact_5927_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_5928_divmod__def,axiom,
    ! [M: num,N: num] : unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_num_Code_integer(numera6620942414471956472nteger,N)),modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_num_Code_integer(numera6620942414471956472nteger,N))) ).

% divmod_def
tff(fact_5929_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_5930_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_nat_int(F,aa_nat_nat(suc,I3))),aa_nat_int(F,I3)))),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)))
           => ? [I3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I3))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
                & ( aa_nat_int(F,I3) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_5931_dbl__dec__def,axiom,
    ! [X: complex] : aa_complex_complex(neg_nu6511756317524482435omplex,X) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(X),X)),one_one_complex) ).

% dbl_dec_def
tff(fact_5932_dbl__dec__def,axiom,
    ! [X: real] : aa_real_real(neg_nu6075765906172075777c_real,X) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),X)),one_one_real) ).

% dbl_dec_def
tff(fact_5933_dbl__dec__def,axiom,
    ! [X: rat] : neg_nu3179335615603231917ec_rat(X) = minus_minus_rat(plus_plus_rat(X,X),one_one_rat) ).

% dbl_dec_def
tff(fact_5934_dbl__dec__def,axiom,
    ! [X: int] : aa_int_int(neg_nu3811975205180677377ec_int,X) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(X),X)),one_one_int) ).

% dbl_dec_def
tff(fact_5935_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(minus_minus_int(X),aa_int_int(times_times_int(aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(X),Z))),one_one_int)),D))),Z)) ) ).

% decr_lemma
tff(fact_5936_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(plus_plus_int(X),aa_int_int(times_times_int(aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(X),Z))),one_one_int)),D)))) ) ).

% incr_lemma
tff(fact_5937_nat__ivt__aux,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_nat_int(F,aa_nat_nat(suc,I3))),aa_nat_int(F,I3)))),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)))
         => ? [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
              & ( aa_nat_int(F,I3) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_5938_nat0__intermed__int__val,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),one_one_nat))),aa_nat_int(F,I3)))),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)))
         => ? [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
              & ( aa_nat_int(F,I3) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_5939_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_5940_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_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(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_5941_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(ord_less_num(M),N))
       => ( unique5052692396658037445od_int(M,N) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,M)) ) )
      & ( ~ pp(aa_num_bool(ord_less_num(M),N))
       => ( unique5052692396658037445od_int(M,N) = unique5024387138958732305ep_int(N,unique5052692396658037445od_int(M,aa_num_num(bit0,N))) ) ) ) ).

% divmod_divmod_step
tff(fact_5942_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(ord_less_num(M),N))
       => ( unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)) ) )
      & ( ~ pp(aa_num_bool(ord_less_num(M),N))
       => ( unique3479559517661332726nteger(M,N) = unique4921790084139445826nteger(N,unique3479559517661332726nteger(M,aa_num_num(bit0,N))) ) ) ) ).

% divmod_divmod_step
tff(fact_5943_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,numeral_numeral_real(aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_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,numeral_numeral_real(aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_real)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_5944_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(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(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(plus_plus_int(aa_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(aa_int_fun_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_5945_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(aa_int_fun_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(aa_int_fun_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_5946_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,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_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,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_Code_integer)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_5947_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) = 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) = 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(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_rat)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_5948_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,N)) = produc4245557441103728435nt_int(aTP_Lamm_gi,unique5052692396658037445od_int(M,N)) ).

% divmod_algorithm_code(6)
tff(fact_5949_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_gj),unique5055182867167087721od_nat(M,N)) ).

% divmod_algorithm_code(6)
tff(fact_5950_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] : unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,N)) = produc6916734918728496179nteger(aTP_Lamm_gk,unique3479559517661332726nteger(M,N)) ).

% divmod_algorithm_code(6)
tff(fact_5951_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),N))
     => ( groups4538972089207619220nt_int(aTP_Lamm_ao,set_or1266510415728281911st_int(M,N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(minus_minus_int(aa_int_int(times_times_int(N),aa_int_int(plus_plus_int(N),one_one_int))),aa_int_int(times_times_int(M),aa_int_int(minus_minus_int(M),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_5952_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_gl(L)),Qr) ).

% divmod_step_def
tff(fact_5953_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_gm(L),Qr) ).

% divmod_step_def
tff(fact_5954_divmod__step__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_gn(L),Qr) ).

% divmod_step_def
tff(fact_5955_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_5956_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_5957_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_5958_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_5959_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_5960_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_5961_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_5962_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_5963_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_5964_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_5965_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_5966_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_5967_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_5968_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_5969_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).

% of_int_le_iff
tff(fact_5970_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).

% of_int_le_iff
tff(fact_5971_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).

% of_int_le_iff
tff(fact_5972_of__int__numeral,axiom,
    ! [K: num] : aa_int_complex(ring_17405671764205052669omplex,aa_num_int(numeral_numeral_int,K)) = numera6690914467698888265omplex(K) ).

% of_int_numeral
tff(fact_5973_of__int__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K)) = numeral_numeral_real(K) ).

% of_int_numeral
tff(fact_5974_of__int__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K)) = numeral_numeral_rat(K) ).

% of_int_numeral
tff(fact_5975_of__int__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,K) ).

% of_int_numeral
tff(fact_5976_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = numera6690914467698888265omplex(N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5977_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = numeral_numeral_real(N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5978_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = numeral_numeral_rat(N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5979_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = aa_num_int(numeral_numeral_int,N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5980_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% of_int_less_iff
tff(fact_5981_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% of_int_less_iff
tff(fact_5982_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% of_int_less_iff
tff(fact_5983_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(times_times_int(W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_mult
tff(fact_5984_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(times_times_int(W),Z)) = aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_mult
tff(fact_5985_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(times_times_int(W),Z)) = aa_int_int(times_times_int(aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_mult
tff(fact_5986_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(plus_plus_int(W),Z)) = aa_int_int(plus_plus_int(aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_add
tff(fact_5987_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_add
tff(fact_5988_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(plus_plus_int(W),Z)) = plus_plus_rat(aa_int_rat(ring_1_of_int_rat,W),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_add
tff(fact_5989_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(minus_minus_int(W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_diff
tff(fact_5990_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(minus_minus_int(W),Z)) = minus_minus_rat(aa_int_rat(ring_1_of_int_rat,W),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_diff
tff(fact_5991_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(minus_minus_int(W),Z)) = aa_int_int(minus_minus_int(aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_diff
tff(fact_5992_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(power_power_int(Z),N)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z)),N) ).

% of_int_power
tff(fact_5993_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Z),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,Z)),N) ).

% of_int_power
tff(fact_5994_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(power_power_int(Z),N)) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,Z)),N) ).

% of_int_power
tff(fact_5995_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Z),N)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Z)),N) ).

% of_int_power
tff(fact_5996_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) = aa_int_rat(ring_1_of_int_rat,X) )
    <=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5997_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W) = aa_int_real(ring_1_of_int_real,X) )
    <=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5998_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W) = aa_int_complex(ring_17405671764205052669omplex,X) )
    <=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5999_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W) = aa_int_int(ring_1_of_int_int,X) )
    <=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_6000_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,X) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) )
    <=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_6001_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( 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)),W) )
    <=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_6002_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,X) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W) )
    <=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_6003_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( aa_int_int(ring_1_of_int_int,X) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W) )
    <=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_6004_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_complex] :
      ( finite_finite_real(S3)
     => ( ( member_real(A,S3)
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_go(A),B),S3) = aa_real_complex(B,A) ) )
        & ( ~ member_real(A,S3)
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_go(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_6005_sum_Odelta_H,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VE5569070535936029429omplex] :
      ( finite5795047828879050333T_VEBT(S3)
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S3))
         => ( groups1794756597179926696omplex(aa_fun8982142231764154876omplex(aTP_Lamm_gp(A),B),S3) = aa_VEBT_VEBT_complex(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S3))
         => ( groups1794756597179926696omplex(aa_fun8982142231764154876omplex(aTP_Lamm_gp(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_6006_sum_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_gq(A),B),S3) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_gq(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_6007_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_gr(A),B),S3) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_gr(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_6008_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( finite_finite_real(S3)
     => ( ( member_real(A,S3)
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_gs(A),B),S3) = aa_real_real(B,A) ) )
        & ( ~ member_real(A,S3)
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_gs(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_6009_sum_Odelta_H,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(S3)
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S3))
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_gt(A),B),S3) = aa_VEBT_VEBT_real(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S3))
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_gt(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_6010_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_gu(A),B),S3) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_gu(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_6011_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_gv(A),B),S3) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_gv(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_6012_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_rat] :
      ( finite_finite_real(S3)
     => ( ( member_real(A,S3)
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_gw(A),B),S3) = aa_real_rat(B,A) ) )
        & ( ~ member_real(A,S3)
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_gw(A),B),S3) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_6013_sum_Odelta_H,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(S3)
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S3))
         => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_gx(A),B),S3) = aa_VEBT_VEBT_rat(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S3))
         => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_gx(A),B),S3) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_6014_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_complex] :
      ( finite_finite_real(S3)
     => ( ( member_real(A,S3)
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_gy(A),B),S3) = aa_real_complex(B,A) ) )
        & ( ~ member_real(A,S3)
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_gy(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_6015_sum_Odelta,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VE5569070535936029429omplex] :
      ( finite5795047828879050333T_VEBT(S3)
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S3))
         => ( groups1794756597179926696omplex(aa_fun8982142231764154876omplex(aTP_Lamm_gz(A),B),S3) = aa_VEBT_VEBT_complex(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S3))
         => ( groups1794756597179926696omplex(aa_fun8982142231764154876omplex(aTP_Lamm_gz(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_6016_sum_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ha(A),B),S3) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ha(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_6017_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_hb(A),B),S3) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_hb(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_6018_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( finite_finite_real(S3)
     => ( ( member_real(A,S3)
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_hc(A),B),S3) = aa_real_real(B,A) ) )
        & ( ~ member_real(A,S3)
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_hc(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_6019_sum_Odelta,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(S3)
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S3))
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_hd(A),B),S3) = aa_VEBT_VEBT_real(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S3))
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_hd(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_6020_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_he(A),B),S3) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_he(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_6021_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_hf(A),B),S3) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_hf(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_6022_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_rat] :
      ( finite_finite_real(S3)
     => ( ( member_real(A,S3)
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_hg(A),B),S3) = aa_real_rat(B,A) ) )
        & ( ~ member_real(A,S3)
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_hg(A),B),S3) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_6023_sum_Odelta,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(S3)
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S3))
         => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_hh(A),B),S3) = aa_VEBT_VEBT_rat(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S3))
         => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_hh(A),B),S3) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_6024_sum__abs,axiom,
    ! [F: fun_int_int,A2: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(F,A2))),groups4538972089207619220nt_int(aTP_Lamm_hi(F),A2))) ).

% sum_abs
tff(fact_6025_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_hj(F),A2))) ).

% sum_abs
tff(fact_6026_set__encode__inverse,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_nat_set_nat(nat_set_decode,aa_set_nat_nat(nat_set_encode,A2)) = A2 ) ) ).

% set_encode_inverse
tff(fact_6027_sum__abs__ge__zero,axiom,
    ! [F: fun_int_int,A2: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_hi(F),A2))) ).

% sum_abs_ge_zero
tff(fact_6028_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_hj(F),A2))) ).

% sum_abs_ge_zero
tff(fact_6029_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_6030_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_6031_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_6032_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_6033_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_6034_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_6035_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_6036_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_6037_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_6038_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_6039_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_6040_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_6041_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)),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_6042_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)),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_6043_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_6044_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_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_6045_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_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_6046_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_6047_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)),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_6048_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)),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_6049_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_6050_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_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_6051_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_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_6052_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_6053_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_6054_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_6055_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_6056_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_6057_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_6058_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_6059_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_6060_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_6061_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_6062_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_6063_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_6064_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_6065_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: 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)),W)),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),W)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6066_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: 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)),W)),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),W)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6067_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: 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)),W)),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),W)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6068_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: 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)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_6069_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: 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)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_6070_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: 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)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_6071_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_6072_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_6073_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_6074_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_6075_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_6076_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_6077_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_6078_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_6079_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: 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)),W)),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),W)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_6080_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: 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)),W)),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),W)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_6081_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: 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)),W)),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),W)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_6082_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: 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)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_6083_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: 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)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_6084_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: 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)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_6085_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,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_6086_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(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_6087_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_6088_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,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_6089_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(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_6090_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_6091_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,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_6092_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(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_6093_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_6094_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,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_6095_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(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_6096_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_6097_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6098_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6099_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6100_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6101_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6102_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),N) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6103_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: 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,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6104_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X))),N) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6105_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X))),N) = aa_int_Code_integer(ring_18347121197199848620nteger,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6106_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(X))),N) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6107_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,N)) = produc4245557441103728435nt_int(aTP_Lamm_hk,unique5052692396658037445od_int(M,N)) ).

% divmod_algorithm_code(5)
tff(fact_6108_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_hl),unique5055182867167087721od_nat(M,N)) ).

% divmod_algorithm_code(5)
tff(fact_6109_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] : unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,N)) = produc6916734918728496179nteger(aTP_Lamm_hm,unique3479559517661332726nteger(M,N)) ).

% divmod_algorithm_code(5)
tff(fact_6110_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,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_6111_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,aa_num_Code_integer(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_6112_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(uminus_uminus_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_6113_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_6114_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,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_6115_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,aa_num_Code_integer(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_6116_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(uminus_uminus_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_6117_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_6118_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,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_6119_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_6120_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,aa_num_Code_integer(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_6121_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(uminus_uminus_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_6122_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,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_6123_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_6124_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,aa_num_Code_integer(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_6125_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(uminus_uminus_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_6126_mult__of__int__commute,axiom,
    ! [X: int,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_int_real(ring_1_of_int_real,X)) ).

% mult_of_int_commute
tff(fact_6127_mult__of__int__commute,axiom,
    ! [X: int,Y: rat] : aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,X)),Y) = aa_rat_rat(times_times_rat(Y),aa_int_rat(ring_1_of_int_rat,X)) ).

% mult_of_int_commute
tff(fact_6128_mult__of__int__commute,axiom,
    ! [X: int,Y: int] : aa_int_int(times_times_int(aa_int_int(ring_1_of_int_int,X)),Y) = aa_int_int(times_times_int(Y),aa_int_int(ring_1_of_int_int,X)) ).

% mult_of_int_commute
tff(fact_6129_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,K5)),groups2906978787729119204at_rat(G,K5))) ) ).

% sum_mono
tff(fact_6130_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I3: real] :
          ( member_real(I3,K5)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,K5)),groups1300246762558778688al_rat(G,K5))) ) ).

% sum_mono
tff(fact_6131_sum__mono,axiom,
    ! [K5: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( ! [I3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I3)),aa_VEBT_VEBT_rat(G,I3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups136491112297645522BT_rat(F,K5)),groups136491112297645522BT_rat(G,K5))) ) ).

% sum_mono
tff(fact_6132_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,K5)),groups3906332499630173760nt_rat(G,K5))) ) ).

% sum_mono
tff(fact_6133_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I3)),aa_complex_rat(G,I3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,K5)),groups5058264527183730370ex_rat(G,K5))) ) ).

% sum_mono
tff(fact_6134_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I3: real] :
          ( member_real(I3,K5)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I3)),aa_real_nat(G,I3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,K5)),groups1935376822645274424al_nat(G,K5))) ) ).

% sum_mono
tff(fact_6135_sum__mono,axiom,
    ! [K5: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [I3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,I3)),aa_VEBT_VEBT_nat(G,I3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups771621172384141258BT_nat(F,K5)),groups771621172384141258BT_nat(G,K5))) ) ).

% sum_mono
tff(fact_6136_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I3)),aa_int_nat(G,I3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,K5)),groups4541462559716669496nt_nat(G,K5))) ) ).

% sum_mono
tff(fact_6137_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I3)),aa_complex_nat(G,I3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups5693394587270226106ex_nat(F,K5)),groups5693394587270226106ex_nat(G,K5))) ) ).

% sum_mono
tff(fact_6138_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),K5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,I3)),aa_nat_int(G,I3))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,K5)),groups3539618377306564664at_int(G,K5))) ) ).

% sum_mono
tff(fact_6139_sum__product,axiom,
    ! [F: fun_int_int,A2: set_int,G: fun_int_int,B2: set_int] : aa_int_int(times_times_int(groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,B2)) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_ho(F),G),B2),A2) ).

% sum_product
tff(fact_6140_sum__product,axiom,
    ! [F: fun_complex_complex,A2: set_complex,G: fun_complex_complex,B2: set_complex] : aa_complex_complex(times_times_complex(groups7754918857620584856omplex(F,A2)),groups7754918857620584856omplex(G,B2)) = groups7754918857620584856omplex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_hq(F),G),B2),A2) ).

% sum_product
tff(fact_6141_sum__product,axiom,
    ! [F: fun_nat_nat,A2: set_nat,G: fun_nat_nat,B2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_hs(F),G),B2)),A2) ).

% sum_product
tff(fact_6142_sum__product,axiom,
    ! [F: fun_nat_real,A2: set_nat,G: fun_nat_real,B2: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(F,A2)),groups6591440286371151544t_real(G,B2)) = groups6591440286371151544t_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_hu(F),G),B2),A2) ).

% sum_product
tff(fact_6143_sum__distrib__right,axiom,
    ! [F: fun_int_int,A2: set_int,R: int] : aa_int_int(times_times_int(groups4538972089207619220nt_int(F,A2)),R) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_hv(F),R),A2) ).

% sum_distrib_right
tff(fact_6144_sum__distrib__right,axiom,
    ! [F: fun_complex_complex,A2: set_complex,R: complex] : aa_complex_complex(times_times_complex(groups7754918857620584856omplex(F,A2)),R) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_hw(F),R),A2) ).

% sum_distrib_right
tff(fact_6145_sum__distrib__right,axiom,
    ! [F: fun_nat_nat,A2: set_nat,R: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),R) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_hx(F),R)),A2) ).

% sum_distrib_right
tff(fact_6146_sum__distrib__right,axiom,
    ! [F: fun_nat_real,A2: set_nat,R: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(F,A2)),R) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_hy(F),R),A2) ).

% sum_distrib_right
tff(fact_6147_sum__distrib__left,axiom,
    ! [R: int,F: fun_int_int,A2: set_int] : aa_int_int(times_times_int(R),groups4538972089207619220nt_int(F,A2)) = groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_hz(R),F),A2) ).

% sum_distrib_left
tff(fact_6148_sum__distrib__left,axiom,
    ! [R: complex,F: fun_complex_complex,A2: set_complex] : aa_complex_complex(times_times_complex(R),groups7754918857620584856omplex(F,A2)) = groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_ia(R),F),A2) ).

% sum_distrib_left
tff(fact_6149_sum__distrib__left,axiom,
    ! [R: nat,F: fun_nat_nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ib(R),F)),A2) ).

% sum_distrib_left
tff(fact_6150_sum__distrib__left,axiom,
    ! [R: real,F: fun_nat_real,A2: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),groups6591440286371151544t_real(F,A2)) = groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ic(R),F),A2) ).

% sum_distrib_left
tff(fact_6151_sum_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A2: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_id(G),H),A2) = aa_int_int(plus_plus_int(groups4538972089207619220nt_int(G,A2)),groups4538972089207619220nt_int(H,A2)) ).

% sum.distrib
tff(fact_6152_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_ie(G),H),A2) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,A2)),groups7754918857620584856omplex(H,A2)) ).

% sum.distrib
tff(fact_6153_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_if(G),H)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(H),A2)) ).

% sum.distrib
tff(fact_6154_sum_Odistrib,axiom,
    ! [G: fun_nat_real,H: fun_nat_real,A2: set_nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ig(G),H),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,A2)),groups6591440286371151544t_real(H,A2)) ).

% sum.distrib
tff(fact_6155_sum__subtractf,axiom,
    ! [F: fun_int_int,G: fun_int_int,A2: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ih(F),G),A2) = aa_int_int(minus_minus_int(groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,A2)) ).

% sum_subtractf
tff(fact_6156_sum__subtractf,axiom,
    ! [F: fun_complex_complex,G: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_ii(F),G),A2) = aa_complex_complex(minus_minus_complex(groups7754918857620584856omplex(F,A2)),groups7754918857620584856omplex(G,A2)) ).

% sum_subtractf
tff(fact_6157_sum__subtractf,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,A2: set_nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ij(F),G),A2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(F,A2)),groups6591440286371151544t_real(G,A2)) ).

% sum_subtractf
tff(fact_6158_sum__divide__distrib,axiom,
    ! [F: fun_complex_complex,A2: set_complex,R: complex] : aa_complex_complex(divide1717551699836669952omplex(groups7754918857620584856omplex(F,A2)),R) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_ik(F),R),A2) ).

% sum_divide_distrib
tff(fact_6159_sum__divide__distrib,axiom,
    ! [F: fun_nat_real,A2: set_nat,R: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups6591440286371151544t_real(F,A2)),R) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_il(F),R),A2) ).

% sum_divide_distrib
tff(fact_6160_sum_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_int,G: fun_VE294566951569886236nt_int,R2: fun_VE3303673267288458579t_bool] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( groups769130701875090982BT_int(aa_fun4615758601980977314BT_int(aa_fun627869447348354137BT_int(aTP_Lamm_in(B2),G),R2),A2) = groups4538972089207619220nt_int(aa_fun5422038952152218214nt_int(aa_fun3638225402958239493nt_int(aTP_Lamm_ip(A2),G),R2),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_6161_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_int,G: fun_real_fun_int_int,R2: fun_re8661203372455305201t_bool] :
      ( finite_finite_real(A2)
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( groups1932886352136224148al_int(aa_fun529217645232493962al_int(aa_fun5832317237790237469al_int(aTP_Lamm_ir(B2),G),R2),A2) = groups4538972089207619220nt_int(aa_fun5532095235224157706nt_int(aa_fun4786959800906006429nt_int(aTP_Lamm_it(A2),G),R2),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_6162_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_int,G: fun_nat_fun_int_int,R2: fun_nat_fun_int_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( groups3539618377306564664at_int(aa_fun3504929462025808522at_int(aa_fun3325927984641084153at_int(aTP_Lamm_iv(B2),G),R2),A2) = groups4538972089207619220nt_int(aa_fun4504283173926863078nt_int(aa_fun5512849559334105941nt_int(aTP_Lamm_ix(A2),G),R2),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_6163_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B2: set_int,G: fun_co333780937270390912nt_int,R2: fun_co2381122952212249071t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( groups5690904116761175830ex_int(aa_fun3700423334941611530ex_int(aa_fun3520494530470439451ex_int(aTP_Lamm_iz(B2),G),R2),A2) = groups4538972089207619220nt_int(aa_fun855801025254967176nt_int(aa_fun315996177946036377nt_int(aTP_Lamm_jc(A2),G),R2),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_6164_sum_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_complex,G: fun_VE2006399705683203104omplex,R2: fun_VE2128610149277289041x_bool] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( groups1794756597179926696omplex(aa_fun6203732500129703078omplex(aa_fun7613540521737166425omplex(aTP_Lamm_je(B2),G),R2),A2) = groups7754918857620584856omplex(aa_fun1095524714264483688omplex(aa_fun8876071380179997707omplex(aTP_Lamm_jg(A2),G),R2),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_6165_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_complex,G: fun_re7208826350674946562omplex,R2: fun_re3010880487650781423x_bool] :
      ( finite_finite_real(A2)
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( groups5754745047067104278omplex(aa_fun4873726854752153866omplex(aa_fun883667864959905433omplex(aTP_Lamm_ji(B2),G),R2),A2) = groups7754918857620584856omplex(aa_fun8128191506076533516omplex(aa_fun840110154952133019omplex(aTP_Lamm_jk(A2),G),R2),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_6166_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_complex,G: fun_na1591161823130732966omplex,R2: 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_jm(B2),G),R2),A2) = groups7754918857620584856omplex(aa_fun1613615970879741416omplex(aa_fun7586814990830484307omplex(aTP_Lamm_jo(A2),G),R2),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_6167_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_complex,G: fun_in1307110748760623106omplex,R2: 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_jq(B2),G),R2),A2) = groups7754918857620584856omplex(aa_fun2686559102040213132omplex(aa_fun5422379709423809691omplex(aTP_Lamm_jt(A2),G),R2),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_6168_sum_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_nat,G: fun_VE597728521808591204at_nat,R2: fun_VE3763956218935105199t_bool] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups771621172384141258BT_nat(aa_fun1096289392377091562BT_nat(aa_fun7476968357888788569BT_nat(aTP_Lamm_jv(B2),G),R2),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5562694175296226570at_nat(aa_fun8318684123515291377at_nat(aTP_Lamm_jx(A2),G),R2)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_6169_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_nat,G: fun_real_fun_nat_nat,R2: fun_re9121486324101951821t_bool] :
      ( finite_finite_real(A2)
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups1935376822645274424al_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_jz(B2),G),R2),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_kb(A2),G),R2)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_6170_mod__sum__eq,axiom,
    ! [F: fun_int_int,A: int,A2: set_int] : aa_int_int(modulo_modulo_int(groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_kc(F),A),A2)),A) = aa_int_int(modulo_modulo_int(groups4538972089207619220nt_int(F,A2)),A) ).

% mod_sum_eq
tff(fact_6171_mod__sum__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kd(F),A)),A2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),A) ).

% mod_sum_eq
tff(fact_6172_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A2))) ) ).

% sum_nonneg
tff(fact_6173_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups2240296850493347238T_real(F,A2))) ) ).

% sum_nonneg
tff(fact_6174_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A2))) ) ).

% sum_nonneg
tff(fact_6175_sum__nonneg,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X2: complex] :
          ( pp(aa_set_complex_bool(member_complex(X2),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups5808333547571424918x_real(F,A2))) ) ).

% sum_nonneg
tff(fact_6176_sum__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_6177_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_6178_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups136491112297645522BT_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_6179_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_6180_sum__nonneg,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X2: complex] :
          ( pp(aa_set_complex_bool(member_complex(X2),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_6181_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A2))) ) ).

% sum_nonneg
tff(fact_6182_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X2)),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_6183_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,X2)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2240296850493347238T_real(F,A2)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_6184_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X2)),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_6185_sum__nonpos,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X2: complex] :
          ( pp(aa_set_complex_bool(member_complex(X2),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X2)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,A2)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_6186_sum__nonpos,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X2)),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_6187_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,X2)),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_6188_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,X2)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups136491112297645522BT_rat(F,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_6189_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X2)),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_6190_sum__nonpos,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X2: complex] :
          ( pp(aa_set_complex_bool(member_complex(X2),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X2)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_6191_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X2)),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_6192_sum__mono__inv,axiom,
    ! [F: fun_real_rat,I5: set_real,G: fun_real_rat,I: real] :
      ( ( groups1300246762558778688al_rat(F,I5) = groups1300246762558778688al_rat(G,I5) )
     => ( ! [I3: real] :
            ( member_real(I3,I5)
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3))) )
       => ( member_real(I,I5)
         => ( finite_finite_real(I5)
           => ( aa_real_rat(F,I) = aa_real_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6193_sum__mono__inv,axiom,
    ! [F: fun_VEBT_VEBT_rat,I5: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,I: vEBT_VEBT] :
      ( ( groups136491112297645522BT_rat(F,I5) = groups136491112297645522BT_rat(G,I5) )
     => ( ! [I3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I3)),aa_VEBT_VEBT_rat(G,I3))) )
       => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),I5))
         => ( finite5795047828879050333T_VEBT(I5)
           => ( aa_VEBT_VEBT_rat(F,I) = aa_VEBT_VEBT_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6194_sum__mono__inv,axiom,
    ! [F: fun_nat_rat,I5: set_nat,G: fun_nat_rat,I: nat] :
      ( ( groups2906978787729119204at_rat(F,I5) = groups2906978787729119204at_rat(G,I5) )
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3))) )
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( pp(aa_set_nat_bool(finite_finite_nat,I5))
           => ( aa_nat_rat(F,I) = aa_nat_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6195_sum__mono__inv,axiom,
    ! [F: fun_int_rat,I5: set_int,G: fun_int_rat,I: int] :
      ( ( groups3906332499630173760nt_rat(F,I5) = groups3906332499630173760nt_rat(G,I5) )
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3))) )
       => ( pp(aa_set_int_bool(member_int(I),I5))
         => ( pp(aa_set_int_bool(finite_finite_int,I5))
           => ( aa_int_rat(F,I) = aa_int_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6196_sum__mono__inv,axiom,
    ! [F: fun_complex_rat,I5: set_complex,G: fun_complex_rat,I: complex] :
      ( ( groups5058264527183730370ex_rat(F,I5) = groups5058264527183730370ex_rat(G,I5) )
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I3)),aa_complex_rat(G,I3))) )
       => ( pp(aa_set_complex_bool(member_complex(I),I5))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
           => ( aa_complex_rat(F,I) = aa_complex_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6197_sum__mono__inv,axiom,
    ! [F: fun_real_nat,I5: set_real,G: fun_real_nat,I: real] :
      ( ( groups1935376822645274424al_nat(F,I5) = groups1935376822645274424al_nat(G,I5) )
     => ( ! [I3: real] :
            ( member_real(I3,I5)
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I3)),aa_real_nat(G,I3))) )
       => ( member_real(I,I5)
         => ( finite_finite_real(I5)
           => ( aa_real_nat(F,I) = aa_real_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6198_sum__mono__inv,axiom,
    ! [F: fun_VEBT_VEBT_nat,I5: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,I: vEBT_VEBT] :
      ( ( groups771621172384141258BT_nat(F,I5) = groups771621172384141258BT_nat(G,I5) )
     => ( ! [I3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,I3)),aa_VEBT_VEBT_nat(G,I3))) )
       => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),I5))
         => ( finite5795047828879050333T_VEBT(I5)
           => ( aa_VEBT_VEBT_nat(F,I) = aa_VEBT_VEBT_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6199_sum__mono__inv,axiom,
    ! [F: fun_int_nat,I5: set_int,G: fun_int_nat,I: int] :
      ( ( groups4541462559716669496nt_nat(F,I5) = groups4541462559716669496nt_nat(G,I5) )
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I3)),aa_int_nat(G,I3))) )
       => ( pp(aa_set_int_bool(member_int(I),I5))
         => ( pp(aa_set_int_bool(finite_finite_int,I5))
           => ( aa_int_nat(F,I) = aa_int_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6200_sum__mono__inv,axiom,
    ! [F: fun_complex_nat,I5: set_complex,G: fun_complex_nat,I: complex] :
      ( ( groups5693394587270226106ex_nat(F,I5) = groups5693394587270226106ex_nat(G,I5) )
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I3)),aa_complex_nat(G,I3))) )
       => ( pp(aa_set_complex_bool(member_complex(I),I5))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
           => ( aa_complex_nat(F,I) = aa_complex_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6201_sum__mono__inv,axiom,
    ! [F: fun_real_int,I5: set_real,G: fun_real_int,I: real] :
      ( ( groups1932886352136224148al_int(F,I5) = groups1932886352136224148al_int(G,I5) )
     => ( ! [I3: real] :
            ( member_real(I3,I5)
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,I3)),aa_real_int(G,I3))) )
       => ( member_real(I,I5)
         => ( finite_finite_real(I5)
           => ( aa_real_int(F,I) = aa_real_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6202_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_6203_sum_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,P: fun_VEBT_VEBT_bool] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( groups1794756597179926696omplex(G,collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_ke(A2),P))) = groups1794756597179926696omplex(aa_fun6792494735270171281omplex(aTP_Lamm_kf(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6204_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_complex,P: fun_real_bool] :
      ( finite_finite_real(A2)
     => ( groups5754745047067104278omplex(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_kg(A2),P))) = groups5754745047067104278omplex(aa_fun6813637976755515871omplex(aTP_Lamm_kh(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6205_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_ki(A2),P))) = groups2073611262835488442omplex(aa_fun8588439219086927967omplex(aTP_Lamm_kj(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6206_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_kk(A2),P))) = groups3049146728041665814omplex(aa_fun3606824691168040799omplex(aTP_Lamm_kl(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6207_sum_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,P: fun_VEBT_VEBT_bool] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( groups2240296850493347238T_real(G,collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_ke(A2),P))) = groups2240296850493347238T_real(aa_fun2072160908118572431T_real(aTP_Lamm_km(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6208_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_real,P: fun_real_bool] :
      ( finite_finite_real(A2)
     => ( groups8097168146408367636l_real(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_kg(A2),P))) = groups8097168146408367636l_real(aa_fun6348708079608492509l_real(aTP_Lamm_kn(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6209_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_kk(A2),P))) = groups8778361861064173332t_real(aa_fun6974987029372589917t_real(aTP_Lamm_ko(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6210_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_6211_sum_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,P: fun_VEBT_VEBT_bool] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( groups136491112297645522BT_rat(G,collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_ke(A2),P))) = groups136491112297645522BT_rat(aa_fun3193099451372820155BT_rat(aTP_Lamm_kr(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6212_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_rat,P: fun_real_bool] :
      ( finite_finite_real(A2)
     => ( groups1300246762558778688al_rat(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_kg(A2),P))) = groups1300246762558778688al_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_ks(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6213_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( finite_finite_real(A2)
     => ( ! [X2: real] :
            ( member_real(X2,A2)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2))) )
       => ( ( groups8097168146408367636l_real(F,A2) = zero_zero_real )
        <=> ! [X3: real] :
              ( member_real(X3,A2)
             => ( aa_real_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6214_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( ! [X2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X2))) )
       => ( ( groups2240296850493347238T_real(F,A2) = zero_zero_real )
        <=> ! [X3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),A2))
             => ( aa_VEBT_VEBT_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6215_sum__nonneg__eq__0__iff,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))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X2))) )
       => ( ( 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_6216_sum__nonneg__eq__0__iff,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))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X2))) )
       => ( ( 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_6217_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( finite_finite_real(A2)
     => ( ! [X2: real] :
            ( member_real(X2,A2)
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X2))) )
       => ( ( groups1300246762558778688al_rat(F,A2) = zero_zero_rat )
        <=> ! [X3: real] :
              ( member_real(X3,A2)
             => ( aa_real_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6218_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( ! [X2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X2))) )
       => ( ( groups136491112297645522BT_rat(F,A2) = zero_zero_rat )
        <=> ! [X3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),A2))
             => ( aa_VEBT_VEBT_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6219_sum__nonneg__eq__0__iff,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))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X2))) )
       => ( ( 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_6220_sum__nonneg__eq__0__iff,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))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X2))) )
       => ( ( 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_6221_sum__nonneg__eq__0__iff,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))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X2))) )
       => ( ( 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_6222_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( finite_finite_real(A2)
     => ( ! [X2: real] :
            ( member_real(X2,A2)
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X2))) )
       => ( ( groups1935376822645274424al_nat(F,A2) = zero_zero_nat )
        <=> ! [X3: real] :
              ( member_real(X3,A2)
             => ( aa_real_nat(F,X3) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6223_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))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(G,X2))) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_int(I,Xa) = X2 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X2)),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_6224_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))
       => ( ! [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X2))) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_int(I,Xa) = X2 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X2)),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_6225_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))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(G,X2))) )
         => ( ! [X2: complex] :
                ( pp(aa_set_complex_bool(member_complex(X2),S))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_complex(I,Xa) = X2 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X2)),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_6226_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))
       => ( ! [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X2))) )
         => ( ! [X2: complex] :
                ( pp(aa_set_complex_bool(member_complex(X2),S))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_complex(I,Xa) = X2 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X2)),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_6227_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))
       => ( ! [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X2))) )
         => ( ! [X2: nat] :
                ( pp(aa_set_nat_bool(member_nat(X2),S))
               => ? [Xa: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Xa),T))
                    & ( aa_nat_nat(I,Xa) = X2 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X2)),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_6228_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))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X2))) )
         => ( ! [X2: nat] :
                ( pp(aa_set_nat_bool(member_nat(X2),S))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_nat(I,Xa) = X2 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X2)),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_6229_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))
       => ( ! [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X2))) )
         => ( ! [X2: nat] :
                ( pp(aa_set_nat_bool(member_nat(X2),S))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_nat(I,Xa) = X2 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X2)),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_6230_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))
       => ( ! [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X2))) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S))
               => ? [Xa: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Xa),T))
                    & ( aa_nat_int(I,Xa) = X2 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X2)),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_6231_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))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X2))) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_int(I,Xa) = X2 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X2)),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_6232_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))
       => ( ! [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X2))) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_int(I,Xa) = X2 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X2)),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_6233_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))
     => ( ! [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X2)),aa_int_real(G,X2))) )
       => ( ? [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_ex1
tff(fact_6234_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X2: complex] :
            ( pp(aa_set_complex_bool(member_complex(X2),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X2)),aa_complex_real(G,X2))) )
       => ( ? [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_ex1
tff(fact_6235_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))
     => ( ! [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(G,X2))) )
       => ( ? [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_ex1
tff(fact_6236_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))
     => ( ! [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X2)),aa_int_rat(G,X2))) )
       => ( ? [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_ex1
tff(fact_6237_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X2: complex] :
            ( pp(aa_set_complex_bool(member_complex(X2),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X2)),aa_complex_rat(G,X2))) )
       => ( ? [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_ex1
tff(fact_6238_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))
     => ( ! [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X2)),aa_int_nat(G,X2))) )
       => ( ? [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_ex1
tff(fact_6239_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X2: complex] :
            ( pp(aa_set_complex_bool(member_complex(X2),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,X2)),aa_complex_nat(G,X2))) )
       => ( ? [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_ex1
tff(fact_6240_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))
     => ( ! [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),A2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,X2)),aa_nat_int(G,X2))) )
       => ( ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A2))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,X4)),aa_nat_int(G,X4))) )
         => 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_6241_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X2: complex] :
            ( pp(aa_set_complex_bool(member_complex(X2),A2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_complex_int(F,X2)),aa_complex_int(G,X2))) )
       => ( ? [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A2))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_complex_int(F,X4)),aa_complex_int(G,X4))) )
         => 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_6242_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))
     => ( ! [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,X2)),aa_int_int(G,X2))) )
       => ( ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A2))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(F,X4)),aa_int_int(G,X4))) )
         => 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_6243_sum_Orelated,axiom,
    ! [R2: fun_co1378794544173401837x_bool,S3: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,zero_zero_complex),zero_zero_complex))
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,X1),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,Y1),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_complex_complex(plus_plus_complex(X1),Y1)),aa_complex_complex(plus_plus_complex(X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S3))
         => ( ! [X2: nat] :
                ( pp(aa_set_nat_bool(member_nat(X2),S3))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_nat_complex(H,X2)),aa_nat_complex(G,X2))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,groups2073611262835488442omplex(H,S3)),groups2073611262835488442omplex(G,S3))) ) ) ) ) ).

% sum.related
tff(fact_6244_sum_Orelated,axiom,
    ! [R2: fun_co1378794544173401837x_bool,S3: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,zero_zero_complex),zero_zero_complex))
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,X1),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,Y1),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_complex_complex(plus_plus_complex(X1),Y1)),aa_complex_complex(plus_plus_complex(X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S3))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_int_complex(H,X2)),aa_int_complex(G,X2))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,groups3049146728041665814omplex(H,S3)),groups3049146728041665814omplex(G,S3))) ) ) ) ) ).

% sum.related
tff(fact_6245_sum_Orelated,axiom,
    ! [R2: fun_re1255401033344146673l_bool,S3: set_int,H: fun_int_real,G: fun_int_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,zero_zero_real),zero_zero_real))
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_int_real(H,X2)),aa_int_real(G,X2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,groups8778361861064173332t_real(H,S3)),groups8778361861064173332t_real(G,S3))) ) ) ) ) ).

% sum.related
tff(fact_6246_sum_Orelated,axiom,
    ! [R2: fun_re1255401033344146673l_bool,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,zero_zero_real),zero_zero_real))
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
         => ( ! [X2: complex] :
                ( pp(aa_set_complex_bool(member_complex(X2),S3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_complex_real(H,X2)),aa_complex_real(G,X2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,groups5808333547571424918x_real(H,S3)),groups5808333547571424918x_real(G,S3))) ) ) ) ) ).

% sum.related
tff(fact_6247_sum_Orelated,axiom,
    ! [R2: fun_rat_fun_rat_bool,S3: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,zero_zero_rat),zero_zero_rat))
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,X1),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,plus_plus_rat(X1,Y1)),plus_plus_rat(X23,Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S3))
         => ( ! [X2: nat] :
                ( pp(aa_set_nat_bool(member_nat(X2),S3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,aa_nat_rat(H,X2)),aa_nat_rat(G,X2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,groups2906978787729119204at_rat(H,S3)),groups2906978787729119204at_rat(G,S3))) ) ) ) ) ).

% sum.related
tff(fact_6248_sum_Orelated,axiom,
    ! [R2: fun_rat_fun_rat_bool,S3: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,zero_zero_rat),zero_zero_rat))
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,X1),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,plus_plus_rat(X1,Y1)),plus_plus_rat(X23,Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,aa_int_rat(H,X2)),aa_int_rat(G,X2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,groups3906332499630173760nt_rat(H,S3)),groups3906332499630173760nt_rat(G,S3))) ) ) ) ) ).

% sum.related
tff(fact_6249_sum_Orelated,axiom,
    ! [R2: fun_rat_fun_rat_bool,S3: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,zero_zero_rat),zero_zero_rat))
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,X1),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,plus_plus_rat(X1,Y1)),plus_plus_rat(X23,Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
         => ( ! [X2: complex] :
                ( pp(aa_set_complex_bool(member_complex(X2),S3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,aa_complex_rat(H,X2)),aa_complex_rat(G,X2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,groups5058264527183730370ex_rat(H,S3)),groups5058264527183730370ex_rat(G,S3))) ) ) ) ) ).

% sum.related
tff(fact_6250_sum_Orelated,axiom,
    ! [R2: fun_nat_fun_nat_bool,S3: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,zero_zero_nat),zero_zero_nat))
     => ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X1),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,Y1),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,aa_int_nat(H,X2)),aa_int_nat(G,X2))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,groups4541462559716669496nt_nat(H,S3)),groups4541462559716669496nt_nat(G,S3))) ) ) ) ) ).

% sum.related
tff(fact_6251_sum_Orelated,axiom,
    ! [R2: fun_nat_fun_nat_bool,S3: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,zero_zero_nat),zero_zero_nat))
     => ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X1),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,Y1),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
         => ( ! [X2: complex] :
                ( pp(aa_set_complex_bool(member_complex(X2),S3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,aa_complex_nat(H,X2)),aa_complex_nat(G,X2))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,groups5693394587270226106ex_nat(H,S3)),groups5693394587270226106ex_nat(G,S3))) ) ) ) ) ).

% sum.related
tff(fact_6252_sum_Orelated,axiom,
    ! [R2: fun_int_fun_int_bool,S3: set_nat,H: fun_nat_int,G: fun_nat_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(R2,zero_zero_int),zero_zero_int))
     => ( ! [X1: int,Y1: int,X23: int,Y23: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(R2,X1),X23))
              & pp(aa_int_bool(aa_int_fun_int_bool(R2,Y1),Y23)) )
           => pp(aa_int_bool(aa_int_fun_int_bool(R2,aa_int_int(plus_plus_int(X1),Y1)),aa_int_int(plus_plus_int(X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S3))
         => ( ! [X2: nat] :
                ( pp(aa_set_nat_bool(member_nat(X2),S3))
               => pp(aa_int_bool(aa_int_fun_int_bool(R2,aa_nat_int(H,X2)),aa_nat_int(G,X2))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(R2,groups3539618377306564664at_int(H,S3)),groups3539618377306564664at_int(G,S3))) ) ) ) ) ).

% sum.related
tff(fact_6253_sum__strict__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( ( A2 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_VEBT_VEBT_real(F,X2)),aa_VEBT_VEBT_real(G,X2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups2240296850493347238T_real(F,A2)),groups2240296850493347238T_real(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6254_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 )
       => ( ! [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
tff(fact_6255_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 )
       => ( ! [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
tff(fact_6256_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( finite_finite_real(A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X2: real] :
              ( member_real(X2,A2)
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X2)),aa_real_real(G,X2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6257_sum__strict__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( ( A2 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_VEBT_VEBT_rat(F,X2)),aa_VEBT_VEBT_rat(G,X2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups136491112297645522BT_rat(F,A2)),groups136491112297645522BT_rat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6258_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 )
       => ( ! [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
tff(fact_6259_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 )
       => ( ! [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
tff(fact_6260_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 )
       => ( ! [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
tff(fact_6261_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( finite_finite_real(A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X2: real] :
              ( member_real(X2,A2)
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(G,X2))) )
         => 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_6262_sum__strict__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( ( A2 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_VEBT_VEBT_nat(F,X2)),aa_VEBT_VEBT_nat(G,X2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups771621172384141258BT_nat(F,A2)),groups771621172384141258BT_nat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6263_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_real,S3: set_real,I: fun_real_real,J: fun_real_real,T2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(S4)
     => ( finite_finite_real(T3)
       => ( ! [A4: real] :
              ( member_real(A4,minus_minus_set_real(S3,S4))
             => ( aa_real_real(I,aa_real_real(J,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( member_real(A4,minus_minus_set_real(S3,S4))
               => member_real(aa_real_real(J,A4),minus_minus_set_real(T2,T3)) )
           => ( ! [B4: real] :
                  ( member_real(B4,minus_minus_set_real(T2,T3))
                 => ( aa_real_real(J,aa_real_real(I,B4)) = B4 ) )
             => ( ! [B4: real] :
                    ( member_real(B4,minus_minus_set_real(T2,T3))
                   => member_real(aa_real_real(I,B4),minus_minus_set_real(S3,S4)) )
               => ( ! [A4: real] :
                      ( member_real(A4,S4)
                     => ( aa_real_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: real] :
                        ( member_real(B4,T3)
                       => ( aa_real_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: real] :
                          ( member_real(A4,S3)
                         => ( aa_real_complex(H,aa_real_real(J,A4)) = aa_real_complex(G,A4) ) )
                     => ( groups5754745047067104278omplex(G,S3) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6264_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_VEBT_VEBT,S3: set_real,I: fun_VEBT_VEBT_real,J: fun_real_VEBT_VEBT,T2: set_VEBT_VEBT,G: fun_real_complex,H: fun_VE5569070535936029429omplex] :
      ( finite_finite_real(S4)
     => ( finite5795047828879050333T_VEBT(T3)
       => ( ! [A4: real] :
              ( member_real(A4,minus_minus_set_real(S3,S4))
             => ( aa_VEBT_VEBT_real(I,aa_real_VEBT_VEBT(J,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( member_real(A4,minus_minus_set_real(S3,S4))
               => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_real_VEBT_VEBT(J,A4)),minus_5127226145743854075T_VEBT(T2,T3))) )
           => ( ! [B4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(T2,T3)))
                 => ( aa_real_VEBT_VEBT(J,aa_VEBT_VEBT_real(I,B4)) = B4 ) )
             => ( ! [B4: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(T2,T3)))
                   => member_real(aa_VEBT_VEBT_real(I,B4),minus_minus_set_real(S3,S4)) )
               => ( ! [A4: real] :
                      ( member_real(A4,S4)
                     => ( aa_real_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),T3))
                       => ( aa_VEBT_VEBT_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: real] :
                          ( member_real(A4,S3)
                         => ( aa_VEBT_VEBT_complex(H,aa_real_VEBT_VEBT(J,A4)) = aa_real_complex(G,A4) ) )
                     => ( groups5754745047067104278omplex(G,S3) = groups1794756597179926696omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6265_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T3: set_real,S3: set_VEBT_VEBT,I: fun_real_VEBT_VEBT,J: fun_VEBT_VEBT_real,T2: set_real,G: fun_VE5569070535936029429omplex,H: fun_real_complex] :
      ( finite5795047828879050333T_VEBT(S4)
     => ( finite_finite_real(T3)
       => ( ! [A4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(S3,S4)))
             => ( aa_real_VEBT_VEBT(I,aa_VEBT_VEBT_real(J,A4)) = A4 ) )
         => ( ! [A4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(S3,S4)))
               => member_real(aa_VEBT_VEBT_real(J,A4),minus_minus_set_real(T2,T3)) )
           => ( ! [B4: real] :
                  ( member_real(B4,minus_minus_set_real(T2,T3))
                 => ( aa_VEBT_VEBT_real(J,aa_real_VEBT_VEBT(I,B4)) = B4 ) )
             => ( ! [B4: real] :
                    ( member_real(B4,minus_minus_set_real(T2,T3))
                   => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_real_VEBT_VEBT(I,B4)),minus_5127226145743854075T_VEBT(S3,S4))) )
               => ( ! [A4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),S4))
                     => ( aa_VEBT_VEBT_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: real] :
                        ( member_real(B4,T3)
                       => ( aa_real_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),S3))
                         => ( aa_real_complex(H,aa_VEBT_VEBT_real(J,A4)) = aa_VEBT_VEBT_complex(G,A4) ) )
                     => ( groups1794756597179926696omplex(G,S3) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6266_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T3: set_VEBT_VEBT,S3: set_VEBT_VEBT,I: fun_VE3165209603783583043T_VEBT,J: fun_VE3165209603783583043T_VEBT,T2: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( finite5795047828879050333T_VEBT(S4)
     => ( finite5795047828879050333T_VEBT(T3)
       => ( ! [A4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(S3,S4)))
             => ( aa_VEB5832953974749077948T_VEBT(I,aa_VEB5832953974749077948T_VEBT(J,A4)) = A4 ) )
         => ( ! [A4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(S3,S4)))
               => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_VEB5832953974749077948T_VEBT(J,A4)),minus_5127226145743854075T_VEBT(T2,T3))) )
           => ( ! [B4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(T2,T3)))
                 => ( aa_VEB5832953974749077948T_VEBT(J,aa_VEB5832953974749077948T_VEBT(I,B4)) = B4 ) )
             => ( ! [B4: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(T2,T3)))
                   => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_VEB5832953974749077948T_VEBT(I,B4)),minus_5127226145743854075T_VEBT(S3,S4))) )
               => ( ! [A4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),S4))
                     => ( aa_VEBT_VEBT_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),T3))
                       => ( aa_VEBT_VEBT_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),S3))
                         => ( aa_VEBT_VEBT_complex(H,aa_VEB5832953974749077948T_VEBT(J,A4)) = aa_VEBT_VEBT_complex(G,A4) ) )
                     => ( groups1794756597179926696omplex(G,S3) = groups1794756597179926696omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6267_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_int,S3: set_real,I: fun_int_real,J: fun_real_int,T2: set_int,G: fun_real_complex,H: fun_int_complex] :
      ( finite_finite_real(S4)
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A4: real] :
              ( member_real(A4,minus_minus_set_real(S3,S4))
             => ( aa_int_real(I,aa_real_int(J,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( member_real(A4,minus_minus_set_real(S3,S4))
               => pp(aa_set_int_bool(member_int(aa_real_int(J,A4)),minus_minus_set_int(T2,T3))) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T2,T3)))
                 => ( aa_real_int(J,aa_int_real(I,B4)) = B4 ) )
             => ( ! [B4: int] :
                    ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T2,T3)))
                   => member_real(aa_int_real(I,B4),minus_minus_set_real(S3,S4)) )
               => ( ! [A4: real] :
                      ( member_real(A4,S4)
                     => ( aa_real_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: int] :
                        ( pp(aa_set_int_bool(member_int(B4),T3))
                       => ( aa_int_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: real] :
                          ( member_real(A4,S3)
                         => ( aa_int_complex(H,aa_real_int(J,A4)) = aa_real_complex(G,A4) ) )
                     => ( groups5754745047067104278omplex(G,S3) = groups3049146728041665814omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6268_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T3: set_int,S3: set_VEBT_VEBT,I: fun_int_VEBT_VEBT,J: fun_VEBT_VEBT_int,T2: set_int,G: fun_VE5569070535936029429omplex,H: fun_int_complex] :
      ( finite5795047828879050333T_VEBT(S4)
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(S3,S4)))
             => ( aa_int_VEBT_VEBT(I,aa_VEBT_VEBT_int(J,A4)) = A4 ) )
         => ( ! [A4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(S3,S4)))
               => pp(aa_set_int_bool(member_int(aa_VEBT_VEBT_int(J,A4)),minus_minus_set_int(T2,T3))) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T2,T3)))
                 => ( aa_VEBT_VEBT_int(J,aa_int_VEBT_VEBT(I,B4)) = B4 ) )
             => ( ! [B4: int] :
                    ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T2,T3)))
                   => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_int_VEBT_VEBT(I,B4)),minus_5127226145743854075T_VEBT(S3,S4))) )
               => ( ! [A4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),S4))
                     => ( aa_VEBT_VEBT_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: int] :
                        ( pp(aa_set_int_bool(member_int(B4),T3))
                       => ( aa_int_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),S3))
                         => ( aa_int_complex(H,aa_VEBT_VEBT_int(J,A4)) = aa_VEBT_VEBT_complex(G,A4) ) )
                     => ( groups1794756597179926696omplex(G,S3) = groups3049146728041665814omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6269_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_real,S3: 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,S4))
     => ( finite_finite_real(T3)
       => ( ! [A4: int] :
              ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S3,S4)))
             => ( aa_real_int(I,aa_int_real(J,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S3,S4)))
               => member_real(aa_int_real(J,A4),minus_minus_set_real(T2,T3)) )
           => ( ! [B4: real] :
                  ( member_real(B4,minus_minus_set_real(T2,T3))
                 => ( aa_int_real(J,aa_real_int(I,B4)) = B4 ) )
             => ( ! [B4: real] :
                    ( member_real(B4,minus_minus_set_real(T2,T3))
                   => pp(aa_set_int_bool(member_int(aa_real_int(I,B4)),minus_minus_set_int(S3,S4))) )
               => ( ! [A4: int] :
                      ( pp(aa_set_int_bool(member_int(A4),S4))
                     => ( aa_int_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: real] :
                        ( member_real(B4,T3)
                       => ( aa_real_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: int] :
                          ( pp(aa_set_int_bool(member_int(A4),S3))
                         => ( aa_real_complex(H,aa_int_real(J,A4)) = aa_int_complex(G,A4) ) )
                     => ( groups3049146728041665814omplex(G,S3) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6270_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_VEBT_VEBT,S3: set_int,I: fun_VEBT_VEBT_int,J: fun_int_VEBT_VEBT,T2: set_VEBT_VEBT,G: fun_int_complex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( finite5795047828879050333T_VEBT(T3)
       => ( ! [A4: int] :
              ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S3,S4)))
             => ( aa_VEBT_VEBT_int(I,aa_int_VEBT_VEBT(J,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S3,S4)))
               => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_int_VEBT_VEBT(J,A4)),minus_5127226145743854075T_VEBT(T2,T3))) )
           => ( ! [B4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(T2,T3)))
                 => ( aa_int_VEBT_VEBT(J,aa_VEBT_VEBT_int(I,B4)) = B4 ) )
             => ( ! [B4: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(T2,T3)))
                   => pp(aa_set_int_bool(member_int(aa_VEBT_VEBT_int(I,B4)),minus_minus_set_int(S3,S4))) )
               => ( ! [A4: int] :
                      ( pp(aa_set_int_bool(member_int(A4),S4))
                     => ( aa_int_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),T3))
                       => ( aa_VEBT_VEBT_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: int] :
                          ( pp(aa_set_int_bool(member_int(A4),S3))
                         => ( aa_VEBT_VEBT_complex(H,aa_int_VEBT_VEBT(J,A4)) = aa_int_complex(G,A4) ) )
                     => ( groups3049146728041665814omplex(G,S3) = groups1794756597179926696omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6271_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_int,S3: 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,S4))
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A4: int] :
              ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S3,S4)))
             => ( aa_int_int(I,aa_int_int(J,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S3,S4)))
               => pp(aa_set_int_bool(member_int(aa_int_int(J,A4)),minus_minus_set_int(T2,T3))) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T2,T3)))
                 => ( aa_int_int(J,aa_int_int(I,B4)) = B4 ) )
             => ( ! [B4: int] :
                    ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T2,T3)))
                   => pp(aa_set_int_bool(member_int(aa_int_int(I,B4)),minus_minus_set_int(S3,S4))) )
               => ( ! [A4: int] :
                      ( pp(aa_set_int_bool(member_int(A4),S4))
                     => ( aa_int_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: int] :
                        ( pp(aa_set_int_bool(member_int(B4),T3))
                       => ( aa_int_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: int] :
                          ( pp(aa_set_int_bool(member_int(A4),S3))
                         => ( aa_int_complex(H,aa_int_int(J,A4)) = aa_int_complex(G,A4) ) )
                     => ( groups3049146728041665814omplex(G,S3) = groups3049146728041665814omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6272_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_real,S3: set_real,I: fun_real_real,J: fun_real_real,T2: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(S4)
     => ( finite_finite_real(T3)
       => ( ! [A4: real] :
              ( member_real(A4,minus_minus_set_real(S3,S4))
             => ( aa_real_real(I,aa_real_real(J,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( member_real(A4,minus_minus_set_real(S3,S4))
               => member_real(aa_real_real(J,A4),minus_minus_set_real(T2,T3)) )
           => ( ! [B4: real] :
                  ( member_real(B4,minus_minus_set_real(T2,T3))
                 => ( aa_real_real(J,aa_real_real(I,B4)) = B4 ) )
             => ( ! [B4: real] :
                    ( member_real(B4,minus_minus_set_real(T2,T3))
                   => member_real(aa_real_real(I,B4),minus_minus_set_real(S3,S4)) )
               => ( ! [A4: real] :
                      ( member_real(A4,S4)
                     => ( aa_real_real(G,A4) = zero_zero_real ) )
                 => ( ! [B4: real] :
                        ( member_real(B4,T3)
                       => ( aa_real_real(H,B4) = zero_zero_real ) )
                   => ( ! [A4: real] :
                          ( member_real(A4,S3)
                         => ( aa_real_real(H,aa_real_real(J,A4)) = aa_real_real(G,A4) ) )
                     => ( groups8097168146408367636l_real(G,S3) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6273_real__of__int__div4,axiom,
    ! [N: int,X: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),X))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,X)))) ).

% real_of_int_div4
tff(fact_6274_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_real,I: real] :
      ( finite_finite_real(S)
     => ( ! [I3: real] :
            ( member_real(I3,S)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3))) )
       => ( ( groups8097168146408367636l_real(F,S) = zero_zero_real )
         => ( member_real(I,S)
           => ( aa_real_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6275_sum__nonneg__0,axiom,
    ! [S: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,I: vEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(S)
     => ( ! [I3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I3))) )
       => ( ( groups2240296850493347238T_real(F,S) = zero_zero_real )
         => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),S))
           => ( aa_VEBT_VEBT_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6276_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_real,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3))) )
       => ( ( 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_6277_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_real,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3))) )
       => ( ( 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_6278_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_rat,I: real] :
      ( finite_finite_real(S)
     => ( ! [I3: real] :
            ( member_real(I3,S)
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3))) )
       => ( ( groups1300246762558778688al_rat(F,S) = zero_zero_rat )
         => ( member_real(I,S)
           => ( aa_real_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6279_sum__nonneg__0,axiom,
    ! [S: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,I: vEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(S)
     => ( ! [I3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3))) )
       => ( ( groups136491112297645522BT_rat(F,S) = zero_zero_rat )
         => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),S))
           => ( aa_VEBT_VEBT_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6280_sum__nonneg__0,axiom,
    ! [S: set_nat,F: fun_nat_rat,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3))) )
       => ( ( 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_6281_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_rat,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3))) )
       => ( ( 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_6282_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_rat,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3))) )
       => ( ( 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_6283_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_nat,I: real] :
      ( finite_finite_real(S)
     => ( ! [I3: real] :
            ( member_real(I3,S)
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3))) )
       => ( ( groups1935376822645274424al_nat(F,S) = zero_zero_nat )
         => ( member_real(I,S)
           => ( aa_real_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6284_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_real,B2: real,I: real] :
      ( finite_finite_real(S)
     => ( ! [I3: real] :
            ( member_real(I3,S)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3))) )
       => ( ( groups8097168146408367636l_real(F,S) = B2 )
         => ( 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_6285_sum__nonneg__leq__bound,axiom,
    ! [S: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,B2: real,I: vEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(S)
     => ( ! [I3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I3))) )
       => ( ( groups2240296850493347238T_real(F,S) = B2 )
         => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6286_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))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3))) )
       => ( ( 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_6287_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_real,B2: real,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3))) )
       => ( ( 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_6288_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_rat,B2: rat,I: real] :
      ( finite_finite_real(S)
     => ( ! [I3: real] :
            ( member_real(I3,S)
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3))) )
       => ( ( groups1300246762558778688al_rat(F,S) = B2 )
         => ( 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_6289_sum__nonneg__leq__bound,axiom,
    ! [S: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,B2: rat,I: vEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(S)
     => ( ! [I3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3))) )
       => ( ( groups136491112297645522BT_rat(F,S) = B2 )
         => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6290_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))
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3))) )
       => ( ( 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_6291_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))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3))) )
       => ( ( 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_6292_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_rat,B2: rat,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3))) )
       => ( ( 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_6293_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_nat,B2: nat,I: real] :
      ( finite_finite_real(S)
     => ( ! [I3: real] :
            ( member_real(I3,S)
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3))) )
       => ( ( groups1935376822645274424al_nat(F,S) = B2 )
         => ( 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_6294_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_complex] :
      ( finite_finite_real(A2)
     => ( groups5754745047067104278omplex(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_kt(G)))) = groups5754745047067104278omplex(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6295_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_ku(G)))) = groups3049146728041665814omplex(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6296_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_real] :
      ( finite_finite_real(A2)
     => ( groups8097168146408367636l_real(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_kv(G)))) = groups8097168146408367636l_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6297_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_kw(G)))) = groups8778361861064173332t_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6298_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_kx(G)))) = groups5808333547571424918x_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6299_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_rat] :
      ( finite_finite_real(A2)
     => ( groups1300246762558778688al_rat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_ky(G)))) = groups1300246762558778688al_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6300_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_kz(G)))) = groups3906332499630173760nt_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6301_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_la(G)))) = groups5058264527183730370ex_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6302_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_nat] :
      ( finite_finite_real(A2)
     => ( groups1935376822645274424al_nat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_lb(G)))) = groups1935376822645274424al_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6303_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_lc(G)))) = groups4541462559716669496nt_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6304_real__of__int__div,axiom,
    ! [D: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),N))
     => ( aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),D)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,D)) ) ) ).

% real_of_int_div
tff(fact_6305_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_real] :
      ( finite_finite_real(I5)
     => ( member_real(I,I5)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,I)))
         => ( ! [I3: real] :
                ( member_real(I3,I5)
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_6306_sum__pos2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(I5)
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(F,I)))
         => ( ! [I3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups2240296850493347238T_real(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_6307_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( pp(aa_set_int_bool(member_int(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,I)))
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_6308_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( pp(aa_set_complex_bool(member_complex(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I)))
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_6309_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_rat] :
      ( finite_finite_real(I5)
     => ( member_real(I,I5)
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,I)))
         => ( ! [I3: real] :
                ( member_real(I3,I5)
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_6310_sum__pos2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(I5)
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I)))
         => ( ! [I3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups136491112297645522BT_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_6311_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( pp(aa_set_nat_bool(member_nat(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I)))
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_6312_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( pp(aa_set_int_bool(member_int(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(F,I)))
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_6313_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( pp(aa_set_complex_bool(member_complex(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I)))
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_6314_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_nat] :
      ( finite_finite_real(I5)
     => ( member_real(I,I5)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F,I)))
         => ( ! [I3: real] :
                ( member_real(I3,I5)
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_6315_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(I5)
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups2240296850493347238T_real(F,I5))) ) ) ) ).

% sum_pos
tff(fact_6316_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I5))) ) ) ) ).

% sum_pos
tff(fact_6317_sum__pos,axiom,
    ! [I5: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I5))) ) ) ) ).

% sum_pos
tff(fact_6318_sum__pos,axiom,
    ! [I5: set_real,F: fun_real_real] :
      ( finite_finite_real(I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( member_real(I3,I5)
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I5))) ) ) ) ).

% sum_pos
tff(fact_6319_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(I5)
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups136491112297645522BT_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_6320_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_6321_sum__pos,axiom,
    ! [I5: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( pp(aa_set_nat_bool(member_nat(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_6322_sum__pos,axiom,
    ! [I5: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_6323_sum__pos,axiom,
    ! [I5: set_real,F: fun_real_rat] :
      ( finite_finite_real(I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( member_real(I3,I5)
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_6324_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( finite5795047828879050333T_VEBT(I5)
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I5))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,I3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups771621172384141258BT_nat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_6325_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_6326_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S3: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(T2)
     => ( ord_less_eq_set_real(S3,T2)
       => ( ! [X2: real] :
              ( member_real(X2,minus_minus_set_real(T2,S3))
             => ( aa_real_complex(G,X2) = zero_zero_complex ) )
         => ( ! [X2: real] :
                ( member_real(X2,S3)
               => ( aa_real_complex(G,X2) = aa_real_complex(H,X2) ) )
           => ( groups5754745047067104278omplex(G,T2) = groups5754745047067104278omplex(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6327_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_VEBT_VEBT,S3: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( finite5795047828879050333T_VEBT(T2)
     => ( ord_le4337996190870823476T_VEBT(S3,T2)
       => ( ! [X2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),minus_5127226145743854075T_VEBT(T2,S3)))
             => ( aa_VEBT_VEBT_complex(G,X2) = zero_zero_complex ) )
         => ( ! [X2: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),S3))
               => ( aa_VEBT_VEBT_complex(G,X2) = aa_VEBT_VEBT_complex(H,X2) ) )
           => ( groups1794756597179926696omplex(G,T2) = groups1794756597179926696omplex(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6328_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T2))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T2,S3)))
             => ( aa_int_complex(G,X2) = zero_zero_complex ) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S3))
               => ( aa_int_complex(G,X2) = aa_int_complex(H,X2) ) )
           => ( groups3049146728041665814omplex(G,T2) = groups3049146728041665814omplex(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6329_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S3: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(T2)
     => ( ord_less_eq_set_real(S3,T2)
       => ( ! [X2: real] :
              ( member_real(X2,minus_minus_set_real(T2,S3))
             => ( aa_real_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: real] :
                ( member_real(X2,S3)
               => ( aa_real_real(G,X2) = aa_real_real(H,X2) ) )
           => ( groups8097168146408367636l_real(G,T2) = groups8097168146408367636l_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6330_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_VEBT_VEBT,S3: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(T2)
     => ( ord_le4337996190870823476T_VEBT(S3,T2)
       => ( ! [X2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),minus_5127226145743854075T_VEBT(T2,S3)))
             => ( aa_VEBT_VEBT_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),S3))
               => ( aa_VEBT_VEBT_real(G,X2) = aa_VEBT_VEBT_real(H,X2) ) )
           => ( groups2240296850493347238T_real(G,T2) = groups2240296850493347238T_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6331_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T2))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T2,S3)))
             => ( aa_int_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S3))
               => ( aa_int_real(G,X2) = aa_int_real(H,X2) ) )
           => ( groups8778361861064173332t_real(G,T2) = groups8778361861064173332t_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6332_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T2))
       => ( ! [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),minus_811609699411566653omplex(T2,S3)))
             => ( aa_complex_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: complex] :
                ( pp(aa_set_complex_bool(member_complex(X2),S3))
               => ( aa_complex_real(G,X2) = aa_complex_real(H,X2) ) )
           => ( groups5808333547571424918x_real(G,T2) = groups5808333547571424918x_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6333_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S3: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(T2)
     => ( ord_less_eq_set_real(S3,T2)
       => ( ! [X2: real] :
              ( member_real(X2,minus_minus_set_real(T2,S3))
             => ( aa_real_rat(G,X2) = zero_zero_rat ) )
         => ( ! [X2: real] :
                ( member_real(X2,S3)
               => ( aa_real_rat(G,X2) = aa_real_rat(H,X2) ) )
           => ( groups1300246762558778688al_rat(G,T2) = groups1300246762558778688al_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6334_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_VEBT_VEBT,S3: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(T2)
     => ( ord_le4337996190870823476T_VEBT(S3,T2)
       => ( ! [X2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),minus_5127226145743854075T_VEBT(T2,S3)))
             => ( aa_VEBT_VEBT_rat(G,X2) = zero_zero_rat ) )
         => ( ! [X2: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),S3))
               => ( aa_VEBT_VEBT_rat(G,X2) = aa_VEBT_VEBT_rat(H,X2) ) )
           => ( groups136491112297645522BT_rat(G,T2) = groups136491112297645522BT_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6335_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T2))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T2,S3)))
             => ( aa_int_rat(G,X2) = zero_zero_rat ) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S3))
               => ( aa_int_rat(G,X2) = aa_int_rat(H,X2) ) )
           => ( groups3906332499630173760nt_rat(G,T2) = groups3906332499630173760nt_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6336_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S3: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( finite_finite_real(T2)
     => ( ord_less_eq_set_real(S3,T2)
       => ( ! [X2: real] :
              ( member_real(X2,minus_minus_set_real(T2,S3))
             => ( aa_real_complex(H,X2) = zero_zero_complex ) )
         => ( ! [X2: real] :
                ( member_real(X2,S3)
               => ( aa_real_complex(G,X2) = aa_real_complex(H,X2) ) )
           => ( groups5754745047067104278omplex(G,S3) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6337_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_VEBT_VEBT,S3: set_VEBT_VEBT,H: fun_VE5569070535936029429omplex,G: fun_VE5569070535936029429omplex] :
      ( finite5795047828879050333T_VEBT(T2)
     => ( ord_le4337996190870823476T_VEBT(S3,T2)
       => ( ! [X2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),minus_5127226145743854075T_VEBT(T2,S3)))
             => ( aa_VEBT_VEBT_complex(H,X2) = zero_zero_complex ) )
         => ( ! [X2: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),S3))
               => ( aa_VEBT_VEBT_complex(G,X2) = aa_VEBT_VEBT_complex(H,X2) ) )
           => ( groups1794756597179926696omplex(G,S3) = groups1794756597179926696omplex(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6338_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_int,S3: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T2))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T2,S3)))
             => ( aa_int_complex(H,X2) = zero_zero_complex ) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S3))
               => ( aa_int_complex(G,X2) = aa_int_complex(H,X2) ) )
           => ( groups3049146728041665814omplex(G,S3) = groups3049146728041665814omplex(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6339_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S3: set_real,H: fun_real_real,G: fun_real_real] :
      ( finite_finite_real(T2)
     => ( ord_less_eq_set_real(S3,T2)
       => ( ! [X2: real] :
              ( member_real(X2,minus_minus_set_real(T2,S3))
             => ( aa_real_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: real] :
                ( member_real(X2,S3)
               => ( aa_real_real(G,X2) = aa_real_real(H,X2) ) )
           => ( groups8097168146408367636l_real(G,S3) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6340_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_VEBT_VEBT,S3: set_VEBT_VEBT,H: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(T2)
     => ( ord_le4337996190870823476T_VEBT(S3,T2)
       => ( ! [X2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),minus_5127226145743854075T_VEBT(T2,S3)))
             => ( aa_VEBT_VEBT_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),S3))
               => ( aa_VEBT_VEBT_real(G,X2) = aa_VEBT_VEBT_real(H,X2) ) )
           => ( groups2240296850493347238T_real(G,S3) = groups2240296850493347238T_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6341_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_int,S3: set_int,H: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T2))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T2,S3)))
             => ( aa_int_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S3))
               => ( aa_int_real(G,X2) = aa_int_real(H,X2) ) )
           => ( groups8778361861064173332t_real(G,S3) = groups8778361861064173332t_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6342_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T2))
       => ( ! [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),minus_811609699411566653omplex(T2,S3)))
             => ( aa_complex_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: complex] :
                ( pp(aa_set_complex_bool(member_complex(X2),S3))
               => ( aa_complex_real(G,X2) = aa_complex_real(H,X2) ) )
           => ( groups5808333547571424918x_real(G,S3) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6343_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S3: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( finite_finite_real(T2)
     => ( ord_less_eq_set_real(S3,T2)
       => ( ! [X2: real] :
              ( member_real(X2,minus_minus_set_real(T2,S3))
             => ( aa_real_rat(H,X2) = zero_zero_rat ) )
         => ( ! [X2: real] :
                ( member_real(X2,S3)
               => ( aa_real_rat(G,X2) = aa_real_rat(H,X2) ) )
           => ( groups1300246762558778688al_rat(G,S3) = groups1300246762558778688al_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6344_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_VEBT_VEBT,S3: set_VEBT_VEBT,H: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(T2)
     => ( ord_le4337996190870823476T_VEBT(S3,T2)
       => ( ! [X2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),minus_5127226145743854075T_VEBT(T2,S3)))
             => ( aa_VEBT_VEBT_rat(H,X2) = zero_zero_rat ) )
         => ( ! [X2: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),S3))
               => ( aa_VEBT_VEBT_rat(G,X2) = aa_VEBT_VEBT_rat(H,X2) ) )
           => ( groups136491112297645522BT_rat(G,S3) = groups136491112297645522BT_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6345_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_int,S3: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T2))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T2,S3)))
             => ( aa_int_rat(H,X2) = zero_zero_rat ) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S3))
               => ( aa_int_rat(G,X2) = aa_int_rat(H,X2) ) )
           => ( groups3906332499630173760nt_rat(G,S3) = groups3906332499630173760nt_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6346_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S3: 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(S3),T2))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T2,S3)))
             => ( aa_int_complex(G,X2) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,T2) = groups3049146728041665814omplex(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6347_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S3: 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(S3),T2))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T2,S3)))
             => ( aa_int_real(G,X2) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,T2) = groups8778361861064173332t_real(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6348_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T2))
       => ( ! [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),minus_811609699411566653omplex(T2,S3)))
             => ( aa_complex_real(G,X2) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T2) = groups5808333547571424918x_real(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6349_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S3: 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(S3),T2))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T2,S3)))
             => ( aa_int_rat(G,X2) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,T2) = groups3906332499630173760nt_rat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6350_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T2))
       => ( ! [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),minus_811609699411566653omplex(T2,S3)))
             => ( aa_complex_rat(G,X2) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,T2) = groups5058264527183730370ex_rat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6351_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T2))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T2,S3)))
             => ( aa_int_nat(G,X2) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,T2) = groups4541462559716669496nt_nat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6352_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T2))
       => ( ! [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),minus_811609699411566653omplex(T2,S3)))
             => ( aa_complex_nat(G,X2) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T2) = groups5693394587270226106ex_nat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6353_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T2))
       => ( ! [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),minus_811609699411566653omplex(T2,S3)))
             => ( aa_complex_int(G,X2) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T2) = groups5690904116761175830ex_int(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6354_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S3: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(S3),T2))
       => ( ! [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),minus_minus_set_nat(T2,S3)))
             => ( aa_nat_complex(G,X2) = zero_zero_complex ) )
         => ( groups2073611262835488442omplex(G,T2) = groups2073611262835488442omplex(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6355_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S3: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(S3),T2))
       => ( ! [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),minus_minus_set_nat(T2,S3)))
             => ( aa_nat_rat(G,X2) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,T2) = groups2906978787729119204at_rat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6356_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S3: 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(S3),T2))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T2,S3)))
             => ( aa_int_complex(G,X2) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,S3) = groups3049146728041665814omplex(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6357_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S3: 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(S3),T2))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T2,S3)))
             => ( aa_int_real(G,X2) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,S3) = groups8778361861064173332t_real(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6358_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T2))
       => ( ! [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),minus_811609699411566653omplex(T2,S3)))
             => ( aa_complex_real(G,X2) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S3) = groups5808333547571424918x_real(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6359_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S3: 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(S3),T2))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T2,S3)))
             => ( aa_int_rat(G,X2) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,S3) = groups3906332499630173760nt_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6360_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T2))
       => ( ! [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),minus_811609699411566653omplex(T2,S3)))
             => ( aa_complex_rat(G,X2) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,S3) = groups5058264527183730370ex_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6361_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T2))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T2,S3)))
             => ( aa_int_nat(G,X2) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,S3) = groups4541462559716669496nt_nat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6362_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T2))
       => ( ! [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),minus_811609699411566653omplex(T2,S3)))
             => ( aa_complex_nat(G,X2) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S3) = groups5693394587270226106ex_nat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6363_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T2))
       => ( ! [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),minus_811609699411566653omplex(T2,S3)))
             => ( aa_complex_int(G,X2) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S3) = groups5690904116761175830ex_int(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6364_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S3: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(S3),T2))
       => ( ! [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),minus_minus_set_nat(T2,S3)))
             => ( aa_nat_complex(G,X2) = zero_zero_complex ) )
         => ( groups2073611262835488442omplex(G,S3) = groups2073611262835488442omplex(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6365_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S3: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(S3),T2))
       => ( ! [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),minus_minus_set_nat(T2,S3)))
             => ( aa_nat_rat(G,X2) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,S3) = groups2906978787729119204at_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6366_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A2,C4)
       => ( ord_less_eq_set_real(B2,C4)
         => ( ! [A4: real] :
                ( member_real(A4,minus_minus_set_real(C4,A2))
               => ( aa_real_complex(G,A4) = zero_zero_complex ) )
           => ( ! [B4: real] :
                  ( member_real(B4,minus_minus_set_real(C4,B2))
                 => ( aa_real_complex(H,B4) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,C4) = groups5754745047067104278omplex(H,C4) )
               => ( groups5754745047067104278omplex(G,A2) = groups5754745047067104278omplex(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6367_sum_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( finite5795047828879050333T_VEBT(C4)
     => ( ord_le4337996190870823476T_VEBT(A2,C4)
       => ( ord_le4337996190870823476T_VEBT(B2,C4)
         => ( ! [A4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_complex(G,A4) = zero_zero_complex ) )
           => ( ! [B4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_complex(H,B4) = zero_zero_complex ) )
             => ( ( groups1794756597179926696omplex(G,C4) = groups1794756597179926696omplex(H,C4) )
               => ( groups1794756597179926696omplex(G,A2) = groups1794756597179926696omplex(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6368_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A2)))
               => ( aa_int_complex(G,A4) = zero_zero_complex ) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C4,B2)))
                 => ( aa_int_complex(H,B4) = zero_zero_complex ) )
             => ( ( groups3049146728041665814omplex(G,C4) = groups3049146728041665814omplex(H,C4) )
               => ( groups3049146728041665814omplex(G,A2) = groups3049146728041665814omplex(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6369_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A2,C4)
       => ( ord_less_eq_set_real(B2,C4)
         => ( ! [A4: real] :
                ( member_real(A4,minus_minus_set_real(C4,A2))
               => ( aa_real_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: real] :
                  ( member_real(B4,minus_minus_set_real(C4,B2))
                 => ( aa_real_real(H,B4) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) )
               => ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6370_sum_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(C4)
     => ( ord_le4337996190870823476T_VEBT(A2,C4)
       => ( ord_le4337996190870823476T_VEBT(B2,C4)
         => ( ! [A4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_real(H,B4) = zero_zero_real ) )
             => ( ( groups2240296850493347238T_real(G,C4) = groups2240296850493347238T_real(H,C4) )
               => ( groups2240296850493347238T_real(G,A2) = groups2240296850493347238T_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6371_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A2)))
               => ( aa_int_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C4,B2)))
                 => ( aa_int_real(H,B4) = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real(G,C4) = groups8778361861064173332t_real(H,C4) )
               => ( groups8778361861064173332t_real(G,A2) = groups8778361861064173332t_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6372_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A4: complex] :
                ( pp(aa_set_complex_bool(member_complex(A4),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_real(H,B4) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) )
               => ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6373_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A2,C4)
       => ( ord_less_eq_set_real(B2,C4)
         => ( ! [A4: real] :
                ( member_real(A4,minus_minus_set_real(C4,A2))
               => ( aa_real_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: real] :
                  ( member_real(B4,minus_minus_set_real(C4,B2))
                 => ( aa_real_rat(H,B4) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) )
               => ( groups1300246762558778688al_rat(G,A2) = groups1300246762558778688al_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6374_sum_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(C4)
     => ( ord_le4337996190870823476T_VEBT(A2,C4)
       => ( ord_le4337996190870823476T_VEBT(B2,C4)
         => ( ! [A4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_rat(H,B4) = zero_zero_rat ) )
             => ( ( groups136491112297645522BT_rat(G,C4) = groups136491112297645522BT_rat(H,C4) )
               => ( groups136491112297645522BT_rat(G,A2) = groups136491112297645522BT_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6375_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A2)))
               => ( aa_int_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C4,B2)))
                 => ( aa_int_rat(H,B4) = zero_zero_rat ) )
             => ( ( groups3906332499630173760nt_rat(G,C4) = groups3906332499630173760nt_rat(H,C4) )
               => ( groups3906332499630173760nt_rat(G,A2) = groups3906332499630173760nt_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6376_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A2,C4)
       => ( ord_less_eq_set_real(B2,C4)
         => ( ! [A4: real] :
                ( member_real(A4,minus_minus_set_real(C4,A2))
               => ( aa_real_complex(G,A4) = zero_zero_complex ) )
           => ( ! [B4: real] :
                  ( member_real(B4,minus_minus_set_real(C4,B2))
                 => ( aa_real_complex(H,B4) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,A2) = groups5754745047067104278omplex(H,B2) )
              <=> ( groups5754745047067104278omplex(G,C4) = groups5754745047067104278omplex(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6377_sum_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( finite5795047828879050333T_VEBT(C4)
     => ( ord_le4337996190870823476T_VEBT(A2,C4)
       => ( ord_le4337996190870823476T_VEBT(B2,C4)
         => ( ! [A4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_complex(G,A4) = zero_zero_complex ) )
           => ( ! [B4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_complex(H,B4) = zero_zero_complex ) )
             => ( ( groups1794756597179926696omplex(G,A2) = groups1794756597179926696omplex(H,B2) )
              <=> ( groups1794756597179926696omplex(G,C4) = groups1794756597179926696omplex(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6378_sum_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A2)))
               => ( aa_int_complex(G,A4) = zero_zero_complex ) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C4,B2)))
                 => ( aa_int_complex(H,B4) = zero_zero_complex ) )
             => ( ( groups3049146728041665814omplex(G,A2) = groups3049146728041665814omplex(H,B2) )
              <=> ( groups3049146728041665814omplex(G,C4) = groups3049146728041665814omplex(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6379_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A2,C4)
       => ( ord_less_eq_set_real(B2,C4)
         => ( ! [A4: real] :
                ( member_real(A4,minus_minus_set_real(C4,A2))
               => ( aa_real_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: real] :
                  ( member_real(B4,minus_minus_set_real(C4,B2))
                 => ( aa_real_real(H,B4) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B2) )
              <=> ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6380_sum_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(C4)
     => ( ord_le4337996190870823476T_VEBT(A2,C4)
       => ( ord_le4337996190870823476T_VEBT(B2,C4)
         => ( ! [A4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_real(H,B4) = zero_zero_real ) )
             => ( ( groups2240296850493347238T_real(G,A2) = groups2240296850493347238T_real(H,B2) )
              <=> ( groups2240296850493347238T_real(G,C4) = groups2240296850493347238T_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6381_sum_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A2)))
               => ( aa_int_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C4,B2)))
                 => ( aa_int_real(H,B4) = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real(G,A2) = groups8778361861064173332t_real(H,B2) )
              <=> ( groups8778361861064173332t_real(G,C4) = groups8778361861064173332t_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6382_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A4: complex] :
                ( pp(aa_set_complex_bool(member_complex(A4),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_real(H,B4) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B2) )
              <=> ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6383_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A2,C4)
       => ( ord_less_eq_set_real(B2,C4)
         => ( ! [A4: real] :
                ( member_real(A4,minus_minus_set_real(C4,A2))
               => ( aa_real_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: real] :
                  ( member_real(B4,minus_minus_set_real(C4,B2))
                 => ( aa_real_rat(H,B4) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,A2) = groups1300246762558778688al_rat(H,B2) )
              <=> ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6384_sum_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(C4)
     => ( ord_le4337996190870823476T_VEBT(A2,C4)
       => ( ord_le4337996190870823476T_VEBT(B2,C4)
         => ( ! [A4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_rat(H,B4) = zero_zero_rat ) )
             => ( ( groups136491112297645522BT_rat(G,A2) = groups136491112297645522BT_rat(H,B2) )
              <=> ( groups136491112297645522BT_rat(G,C4) = groups136491112297645522BT_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6385_sum_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A2)))
               => ( aa_int_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C4,B2)))
                 => ( aa_int_rat(H,B4) = zero_zero_rat ) )
             => ( ( groups3906332499630173760nt_rat(G,A2) = groups3906332499630173760nt_rat(H,B2) )
              <=> ( groups3906332499630173760nt_rat(G,C4) = groups3906332499630173760nt_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6386_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_6387_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_6388_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) = plus_plus_rat(groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,B2)),groups3906332499630173760nt_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6389_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) = plus_plus_rat(groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,B2)),groups5058264527183730370ex_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6390_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_6391_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_6392_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(plus_plus_int(groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,B2))),groups5690904116761175830ex_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6393_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( groups2906978787729119204at_rat(G,A2) = plus_plus_rat(groups2906978787729119204at_rat(G,minus_minus_set_nat(A2,B2)),groups2906978787729119204at_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6394_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( groups3539618377306564664at_int(G,A2) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,minus_minus_set_nat(A2,B2))),groups3539618377306564664at_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6395_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(plus_plus_int(groups4538972089207619220nt_int(G,minus_minus_set_int(A2,B2))),groups4538972089207619220nt_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6396_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_6397_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_6398_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)) = minus_minus_rat(groups3906332499630173760nt_rat(F,A2),groups3906332499630173760nt_rat(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6399_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)) = minus_minus_rat(groups5058264527183730370ex_rat(F,A2),groups5058264527183730370ex_rat(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6400_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(minus_minus_int(groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6401_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(ord_less_eq_set_nat(B2),A2))
       => ( groups2906978787729119204at_rat(F,minus_minus_set_nat(A2,B2)) = minus_minus_rat(groups2906978787729119204at_rat(F,A2),groups2906978787729119204at_rat(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6402_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(ord_less_eq_set_nat(B2),A2))
       => ( groups3539618377306564664at_int(F,minus_minus_set_nat(A2,B2)) = aa_int_int(minus_minus_int(groups3539618377306564664at_int(F,A2)),groups3539618377306564664at_int(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6403_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(minus_minus_int(groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6404_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_6405_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(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_6406_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_6407_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_6408_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_6409_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_6410_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_6411_of__int__leD,axiom,
    ! [N: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_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_6412_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_6413_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_6414_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_6415_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_6416_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_6417_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_6418_of__int__lessD,axiom,
    ! [N: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_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_6419_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_6420_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(K)) ).

% of_int_neg_numeral
tff(fact_6421_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)) ).

% of_int_neg_numeral
tff(fact_6422_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K)) ).

% of_int_neg_numeral
tff(fact_6423_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K)) ).

% of_int_neg_numeral
tff(fact_6424_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = uminus_uminus_rat(numeral_numeral_rat(K)) ).

% of_int_neg_numeral
tff(fact_6425_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_6426_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_6427_real__of__int__div__aux,axiom,
    ! [X: int,D: int] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,X)),aa_int_real(ring_1_of_int_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(modulo_modulo_int(X),D))),aa_int_real(ring_1_of_int_real,D))) ).

% real_of_int_div_aux
tff(fact_6428_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_real] :
      ( finite_finite_real(B2)
     => ( ord_less_eq_set_real(A2,B2)
       => ( ! [B4: real] :
              ( member_real(B4,minus_minus_set_real(B2,A2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,B4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6429_sum__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(B2)
     => ( ord_le4337996190870823476T_VEBT(A2,B2)
       => ( ! [B4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(B2,A2)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,B4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2240296850493347238T_real(F,A2)),groups2240296850493347238T_real(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6430_sum__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( ! [B4: int] :
              ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(B2,A2)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,B4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6431_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))
       => ( ! [B4: complex] :
              ( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(B2,A2)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,B4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6432_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_rat] :
      ( finite_finite_real(B2)
     => ( ord_less_eq_set_real(A2,B2)
       => ( ! [B4: real] :
              ( member_real(B4,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,B4))) )
         => 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_6433_sum__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(B2)
     => ( ord_le4337996190870823476T_VEBT(A2,B2)
       => ( ! [B4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(B2,A2)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,B4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups136491112297645522BT_rat(F,A2)),groups136491112297645522BT_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6434_sum__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( ! [B4: int] :
              ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(B2,A2)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,B4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6435_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))
       => ( ! [B4: complex] :
              ( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(B2,A2)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,B4))) )
         => 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_6436_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_nat] :
      ( finite_finite_real(B2)
     => ( ord_less_eq_set_real(A2,B2)
       => ( ! [B4: real] :
              ( member_real(B4,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,B4))) )
         => 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_6437_sum__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( finite5795047828879050333T_VEBT(B2)
     => ( ord_le4337996190870823476T_VEBT(A2,B2)
       => ( ! [B4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(B2,A2)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,B4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups771621172384141258BT_nat(F,A2)),groups771621172384141258BT_nat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6438_real__of__int__div2,axiom,
    ! [N: int,X: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,X))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),X))))) ).

% real_of_int_div2
tff(fact_6439_real__of__int__div3,axiom,
    ! [N: int,X: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,X))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),X)))),one_one_real)) ).

% real_of_int_div3
tff(fact_6440_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_real] :
      ( finite_finite_real(B2)
     => ( ord_less_eq_set_real(A2,B2)
       => ( 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)))
           => ( ! [X2: real] :
                  ( member_real(X2,B2)
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6441_sum__strict__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(B2)
     => ( ord_le4337996190870823476T_VEBT(A2,B2)
       => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B),minus_5127226145743854075T_VEBT(B2,A2)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(F,B)))
           => ( ! [X2: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),B2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X2))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups2240296850493347238T_real(F,A2)),groups2240296850493347238T_real(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6442_sum__strict__mono2,axiom,
    ! [B2: set_int,A2: set_int,B: int,F: fun_int_real] :
      ( 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(member_int(B),minus_minus_set_int(B2,A2)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,B)))
           => ( ! [X2: int] :
                  ( pp(aa_set_int_bool(member_int(X2),B2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X2))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6443_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)))
           => ( ! [X2: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X2),B2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X2))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6444_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_rat] :
      ( finite_finite_real(B2)
     => ( ord_less_eq_set_real(A2,B2)
       => ( 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)))
           => ( ! [X2: real] :
                  ( member_real(X2,B2)
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X2))) )
             => 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_6445_sum__strict__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(B2)
     => ( ord_le4337996190870823476T_VEBT(A2,B2)
       => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B),minus_5127226145743854075T_VEBT(B2,A2)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,B)))
           => ( ! [X2: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),B2))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X2))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups136491112297645522BT_rat(F,A2)),groups136491112297645522BT_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6446_sum__strict__mono2,axiom,
    ! [B2: set_int,A2: set_int,B: int,F: fun_int_rat] :
      ( 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(member_int(B),minus_minus_set_int(B2,A2)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(F,B)))
           => ( ! [X2: int] :
                  ( pp(aa_set_int_bool(member_int(X2),B2))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X2))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6447_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)))
           => ( ! [X2: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X2),B2))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X2))) )
             => 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_6448_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_nat] :
      ( finite_finite_real(B2)
     => ( ord_less_eq_set_real(A2,B2)
       => ( 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)))
           => ( ! [X2: real] :
                  ( member_real(X2,B2)
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X2))) )
             => 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_6449_sum__strict__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( finite5795047828879050333T_VEBT(B2)
     => ( ord_le4337996190870823476T_VEBT(A2,B2)
       => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B),minus_5127226145743854075T_VEBT(B2,A2)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,B)))
           => ( ! [X2: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),B2))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,X2))) )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups771621172384141258BT_nat(F,A2)),groups771621172384141258BT_nat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6450_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_Code_integer,A: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),I5))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(X,I3))) )
     => ( ( groups7501900531339628137nteger(X,I5) = one_one_Code_integer )
       => ( ! [I3: nat] :
              ( pp(aa_set_nat_bool(member_nat(I3),I5))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_nat_Code_integer(A,I3),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_ld(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_6451_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_re6978948838062957830nteger,A: fun_re6978948838062957830nteger,B: code_integer,Delta: code_integer] :
      ( ! [I3: real] :
          ( member_real(I3,I5)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(X,I3))) )
     => ( ( groups7713935264441627589nteger(X,I5) = one_one_Code_integer )
       => ( ! [I3: real] :
              ( member_real(I3,I5)
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_real_Code_integer(A,I3),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7713935264441627589nteger(aa_fun7923231501384339402nteger(aTP_Lamm_le(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_6452_convex__sum__bound__le,axiom,
    ! [I5: set_VEBT_VEBT,X: fun_VE9219535225822561060nteger,A: fun_VE9219535225822561060nteger,B: code_integer,Delta: code_integer] :
      ( ! [I3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I5))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_VEB5935388676523804373nteger(X,I3))) )
     => ( ( groups5748017345553531991nteger(X,I5) = one_one_Code_integer )
       => ( ! [I3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I5))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_VEB5935388676523804373nteger(A,I3),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups5748017345553531991nteger(aa_fun2400605388621188028nteger(aTP_Lamm_lf(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_6453_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_Code_integer,A: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),I5))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(X,I3))) )
     => ( ( groups7873554091576472773nteger(X,I5) = one_one_Code_integer )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I5))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_int_Code_integer(A,I3),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_lg(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_6454_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: fun_co8153928105989009288nteger,A: fun_co8153928105989009288nteger,B: code_integer,Delta: code_integer] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),I5))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(X,I3))) )
     => ( ( groups6621422865394947399nteger(X,I5) = one_one_Code_integer )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I5))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_com4217951221296830281nteger(A,I3),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups6621422865394947399nteger(aa_fun5267417455630836558nteger(aTP_Lamm_lh(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_6455_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_real_real,A: fun_real_real,B: real,Delta: real] :
      ( ! [I3: real] :
          ( member_real(I3,I5)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(X,I3))) )
     => ( ( groups8097168146408367636l_real(X,I5) = one_one_real )
       => ( ! [I3: real] :
              ( member_real(I3,I5)
             => 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,I3)),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_li(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_6456_convex__sum__bound__le,axiom,
    ! [I5: set_VEBT_VEBT,X: fun_VEBT_VEBT_real,A: fun_VEBT_VEBT_real,B: real,Delta: real] :
      ( ! [I3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(X,I3))) )
     => ( ( groups2240296850493347238T_real(X,I5) = one_one_real )
       => ( ! [I3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I5))
             => 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_VEBT_VEBT_real(A,I3)),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,groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_lj(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_6457_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_real,A: fun_int_real,B: real,Delta: real] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),I5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(X,I3))) )
     => ( ( groups8778361861064173332t_real(X,I5) = one_one_real )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I5))
             => 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,I3)),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_lk(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_6458_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: fun_complex_real,A: fun_complex_real,B: real,Delta: real] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),I5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(X,I3))) )
     => ( ( groups5808333547571424918x_real(X,I5) = one_one_real )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I5))
             => 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_complex_real(A,I3)),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,groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_ll(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_6459_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_rat,A: fun_nat_rat,B: rat,Delta: rat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),I5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(X,I3))) )
     => ( ( groups2906978787729119204at_rat(X,I5) = one_one_rat )
       => ( ! [I3: nat] :
              ( pp(aa_set_nat_bool(member_nat(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_nat_rat(A,I3),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_lm(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_6460_even__of__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)) ) ).

% even_of_int_iff
tff(fact_6461_even__of__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)) ) ).

% even_of_int_iff
tff(fact_6462_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_gl(L)),Qr) ).

% divmod_step_nat_def
tff(fact_6463_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_gm(L),Qr) ).

% divmod_step_int_def
tff(fact_6464_floor__exists,axiom,
    ! [X: real] :
    ? [Z4: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z4)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(Z4),one_one_int)))) ) ).

% floor_exists
tff(fact_6465_floor__exists,axiom,
    ! [X: rat] :
    ? [Z4: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z4)),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(plus_plus_int(Z4),one_one_int)))) ) ).

% floor_exists
tff(fact_6466_floor__exists1,axiom,
    ! [X: real] :
    ? [X2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X2)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(X2),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(plus_plus_int(Y3),one_one_int)))) )
         => ( Y3 = X2 ) ) ) ).

% floor_exists1
tff(fact_6467_floor__exists1,axiom,
    ! [X: rat] :
    ? [X2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X2)),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(plus_plus_int(X2),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(plus_plus_int(Y3),one_one_int)))) )
         => ( Y3 = X2 ) ) ) ).

% floor_exists1
tff(fact_6468_infinite__int__iff__unbounded,axiom,
    ! [S3: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S3))
    <=> ! [M7: int] :
        ? [N5: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M7),aa_int_int(abs_abs_int,N5)))
          & pp(aa_set_int_bool(member_int(N5),S3)) ) ) ).

% infinite_int_iff_unbounded
tff(fact_6469_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_ln),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_6470_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique5052692396658037445od_int(bitM(M),aa_num_num(bit0,one)) = product_Pair_int_int(aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,M)),one_one_int),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_6471_mask__numeral,axiom,
    ! [N: num] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(numeral_numeral_nat,N)) = 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(bit_se2002935070580805687sk_nat,aa_num_nat(pred_numeral,N)))) ).

% mask_numeral
tff(fact_6472_mask__numeral,axiom,
    ! [N: num] : aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(numeral_numeral_nat,N)) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(pred_numeral,N)))) ).

% mask_numeral
tff(fact_6473_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_6474_mask__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(bit_se2002935070580805687sk_nat,N) = zero_zero_nat )
    <=> ( N = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_6475_mask__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_int(bit_se2000444600071755411sk_int,N) = zero_zero_int )
    <=> ( N = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_6476_mask__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,zero_zero_nat) = zero_zero_nat ).

% mask_0
tff(fact_6477_mask__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,zero_zero_nat) = zero_zero_int ).

% mask_0
tff(fact_6478_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(bitM(K)) ).

% dbl_dec_simps(5)
tff(fact_6479_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,numeral_numeral_real(K)) = numeral_numeral_real(bitM(K)) ).

% dbl_dec_simps(5)
tff(fact_6480_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : neg_nu3179335615603231917ec_rat(numeral_numeral_rat(K)) = numeral_numeral_rat(bitM(K)) ).

% dbl_dec_simps(5)
tff(fact_6481_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,bitM(K)) ).

% dbl_dec_simps(5)
tff(fact_6482_mask__Suc__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% mask_Suc_0
tff(fact_6483_mask__Suc__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% mask_Suc_0
tff(fact_6484_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit0,K)) = aa_num_nat(numeral_numeral_nat,bitM(K)) ).

% pred_numeral_simps(2)
tff(fact_6485_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,aa_nat_set_nat(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,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_complex_complex(plus_plus_complex(groups2073611262835488442omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_complex(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_6486_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,aa_nat_set_nat(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,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)),aa_nat_rat(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_6487_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,aa_nat_set_nat(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,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_6488_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),aa_nat_set_nat(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),aa_nat_set_nat(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),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_6489_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,aa_nat_set_nat(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,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_6490_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_lo(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_lo(C),A2) = zero_zero_complex ) ) ) ).

% sum_zero_power
tff(fact_6491_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_lp(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_lp(C),A2) = zero_zero_rat ) ) ) ).

% sum_zero_power
tff(fact_6492_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_lq(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_lq(C),A2) = zero_zero_real ) ) ) ).

% sum_zero_power
tff(fact_6493_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_lr(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_lr(C),D),A2) = zero_zero_complex ) ) ) ).

% sum_zero_power'
tff(fact_6494_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_ls(C),D),A2) = 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_ls(C),D),A2) = zero_zero_rat ) ) ) ).

% sum_zero_power'
tff(fact_6495_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_lt(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_lt(C),D),A2) = zero_zero_real ) ) ) ).

% sum_zero_power'
tff(fact_6496_of__int__mask__eq,axiom,
    ! [N: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(bit_se2000444600071755411sk_int,N)) = aa_nat_int(bit_se2000444600071755411sk_int,N) ).

% of_int_mask_eq
tff(fact_6497_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_6498_semiring__norm_I26_J,axiom,
    bitM(one) = one ).

% semiring_norm(26)
tff(fact_6499_sum__subtractf__nat,axiom,
    ! [A2: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(G,X2)),aa_real_nat(F,X2))) )
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_lu(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_6500_sum__subtractf__nat,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,F: fun_VEBT_VEBT_nat] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(G,X2)),aa_VEBT_VEBT_nat(F,X2))) )
     => ( groups771621172384141258BT_nat(aa_fun8284231930576791164BT_nat(aTP_Lamm_lv(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups771621172384141258BT_nat(F,A2)),groups771621172384141258BT_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_6501_sum__subtractf__nat,axiom,
    ! [A2: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(G,X2)),aa_int_nat(F,X2))) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_lw(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_6502_sum__subtractf__nat,axiom,
    ! [A2: set_complex,G: fun_complex_nat,F: fun_complex_nat] :
      ( ! [X2: complex] :
          ( pp(aa_set_complex_bool(member_complex(X2),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(G,X2)),aa_complex_nat(F,X2))) )
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_lx(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_6503_sum__subtractf__nat,axiom,
    ! [A2: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(G,X2)),aa_nat_nat(F,X2))) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ly(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_6504_mask__nonnegative__int,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,N))) ).

% mask_nonnegative_int
tff(fact_6505_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_6506_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lz(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_6507_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,M: nat,K: nat,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ma(G),K),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_6508_semiring__norm_I27_J,axiom,
    ! [N: num] : bitM(aa_num_num(bit0,N)) = aa_num_num(bit1,bitM(N)) ).

% semiring_norm(27)
tff(fact_6509_semiring__norm_I28_J,axiom,
    ! [N: num] : bitM(aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(bit0,N)) ).

% semiring_norm(28)
tff(fact_6510_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) )
            & ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_int_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6511_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) )
            & ! [Xa4: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_complex_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6512_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) )
            & ! [Xa4: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_nat_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6513_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) )
     => ? [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),A2))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X2))) ) ) ).

% sum_SucD
tff(fact_6514_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 )
            & ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_int_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6515_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 )
            & ! [Xa4: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_complex_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6516_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 )
            & ! [Xa4: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_nat_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6517_sum__power__add,axiom,
    ! [X: complex,M: nat,I5: set_nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_mb(X),M),I5) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),groups2073611262835488442omplex(power_power_complex(X),I5)) ).

% sum_power_add
tff(fact_6518_sum__power__add,axiom,
    ! [X: rat,M: nat,I5: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_mc(X),M),I5) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M)),groups2906978787729119204at_rat(power_power_rat(X),I5)) ).

% sum_power_add
tff(fact_6519_sum__power__add,axiom,
    ! [X: int,M: nat,I5: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_md(X),M),I5) = aa_int_int(times_times_int(aa_nat_int(power_power_int(X),M)),groups3539618377306564664at_int(power_power_int(X),I5)) ).

% sum_power_add
tff(fact_6520_sum__power__add,axiom,
    ! [X: real,M: nat,I5: set_nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_me(X),M),I5) = 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),I5)) ).

% sum_power_add
tff(fact_6521_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_mf(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ).

% sum.atLeastAtMost_rev
tff(fact_6522_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,N: nat,M: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_mg(G),N),M),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ).

% sum.atLeastAtMost_rev
tff(fact_6523_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_6524_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_ak,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_ga(N),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_6525_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_ak,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_cg,N))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_6526_eval__nat__numeral_I2_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,bitM(N))) ).

% eval_nat_numeral(2)
tff(fact_6527_one__plus__BitM,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),bitM(N)) = aa_num_num(bit0,N) ).

% one_plus_BitM
tff(fact_6528_BitM__plus__one,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,bitM(N)),one) = aa_num_num(bit0,N) ).

% BitM_plus_one
tff(fact_6529_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_6530_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_6531_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(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_6532_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_6533_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_6534_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(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),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_6535_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_6536_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,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = plus_plus_rat(aa_nat_rat(G,aa_nat_nat(suc,N)),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6537_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,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_int_int(plus_plus_int(aa_nat_int(G,aa_nat_nat(suc,N))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6538_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),aa_nat_set_nat(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),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6539_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,aa_nat_set_nat(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,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6540_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,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = plus_plus_rat(aa_nat_rat(G,M),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6541_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,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(plus_plus_int(aa_nat_int(G,M)),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6542_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),aa_nat_set_nat(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),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6543_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,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6544_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))
     => ( plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)),aa_nat_rat(G,aa_nat_nat(suc,N))) = plus_plus_rat(aa_nat_rat(G,M),groups2906978787729119204at_rat(aTP_Lamm_mh(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6545_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(plus_plus_int(groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) = aa_int_int(plus_plus_int(aa_nat_int(G,M)),groups3539618377306564664at_int(aTP_Lamm_mi(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6546_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),aa_nat_set_nat(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_mj(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6547_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,aa_nat_set_nat(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_mk(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6548_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_ml(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,N)),aa_nat_rat(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_6549_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_mm(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(minus_minus_int(aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_6550_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_mn(F),aa_nat_set_nat(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_6551_numeral__BitM,axiom,
    ! [N: num] : numera6690914467698888265omplex(bitM(N)) = aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(aa_num_num(bit0,N))),one_one_complex) ).

% numeral_BitM
tff(fact_6552_numeral__BitM,axiom,
    ! [N: num] : numeral_numeral_real(bitM(N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(aa_num_num(bit0,N))),one_one_real) ).

% numeral_BitM
tff(fact_6553_numeral__BitM,axiom,
    ! [N: num] : numeral_numeral_rat(bitM(N)) = minus_minus_rat(numeral_numeral_rat(aa_num_num(bit0,N)),one_one_rat) ).

% numeral_BitM
tff(fact_6554_numeral__BitM,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,bitM(N)) = aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),one_one_int) ).

% numeral_BitM
tff(fact_6555_odd__numeral__BitM,axiom,
    ! [W: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_num_Code_integer(numera6620942414471956472nteger,bitM(W)))) ).

% odd_numeral_BitM
tff(fact_6556_odd__numeral__BitM,axiom,
    ! [W: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,bitM(W)))) ).

% odd_numeral_BitM
tff(fact_6557_odd__numeral__BitM,axiom,
    ! [W: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,bitM(W)))) ).

% odd_numeral_BitM
tff(fact_6558_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,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2))) = plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)),groups2906978787729119204at_rat(G,aa_nat_set_nat(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_6559_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,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2))) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),groups3539618377306564664at_int(G,aa_nat_set_nat(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_6560_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),aa_nat_set_nat(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),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(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_6561_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,aa_nat_set_nat(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,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),groups6591440286371151544t_real(G,aa_nat_set_nat(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_6562_set__encode__def,axiom,
    nat_set_encode = groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% set_encode_def
tff(fact_6563_Suc__mask__eq__exp,axiom,
    ! [N: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% Suc_mask_eq_exp
tff(fact_6564_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_6565_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_mo(F),aa_nat_set_nat(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_mo(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_complex ) ) ) ).

% sum_natinterval_diff
tff(fact_6566_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_mp(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = 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_mp(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
tff(fact_6567_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_mq(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_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_mq(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_int ) ) ) ).

% sum_natinterval_diff
tff(fact_6568_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_mr(F),aa_nat_set_nat(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_mr(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_real ) ) ) ).

% sum_natinterval_diff
tff(fact_6569_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_ms(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = minus_minus_rat(aa_nat_rat(F,N),aa_nat_rat(F,M)) ) ) ).

% sum_telescope''
tff(fact_6570_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_mt(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_int_int(minus_minus_int(aa_nat_int(F,N)),aa_nat_int(F,M)) ) ) ).

% sum_telescope''
tff(fact_6571_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_mu(F),aa_nat_set_nat(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_6572_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(bit_se2119862282449309892nteger,N)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_6573_semiring__bit__operations__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(bit_se2002935070580805687sk_nat,N)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_6574_semiring__bit__operations__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_nat_int(bit_se2000444600071755411sk_int,N)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_6575_mask__nat__def,axiom,
    ! [N: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat) ).

% mask_nat_def
tff(fact_6576_mask__half__int,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(bit_se2000444600071755411sk_int,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ).

% mask_half_int
tff(fact_6577_mask__eq__sum__exp,axiom,
    ! [N: nat] : aa_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_bh,N))) ).

% mask_eq_sum_exp
tff(fact_6578_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_bh,N))) ).

% mask_eq_sum_exp
tff(fact_6579_finite__transitivity__chain,axiom,
    ! [A2: set_VEBT_VEBT,R2: fun_VE5877677516038742509T_bool] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( ! [X2: vEBT_VEBT] : ~ pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(R2,X2),X2))
       => ( ! [X2: vEBT_VEBT,Y2: vEBT_VEBT,Z4: vEBT_VEBT] :
              ( pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(R2,X2),Y2))
             => ( pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(R2,Y2),Z4))
               => pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(R2,X2),Z4)) ) )
         => ( ! [X2: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
               => ? [Y3: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y3),A2))
                    & pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(R2,X2),Y3)) ) )
           => ( A2 = bot_bo8194388402131092736T_VEBT ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_6580_finite__transitivity__chain,axiom,
    ! [A2: set_complex,R2: fun_co1378794544173401837x_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X2: complex] : ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,X2),X2))
       => ( ! [X2: complex,Y2: complex,Z4: complex] :
              ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,X2),Y2))
             => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,Y2),Z4))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,X2),Z4)) ) )
         => ( ! [X2: complex] :
                ( pp(aa_set_complex_bool(member_complex(X2),A2))
               => ? [Y3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y3),A2))
                    & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,X2),Y3)) ) )
           => ( A2 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_6581_finite__transitivity__chain,axiom,
    ! [A2: set_nat,R2: fun_nat_fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [X2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X2),X2))
       => ( ! [X2: nat,Y2: nat,Z4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X2),Y2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,Y2),Z4))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X2),Z4)) ) )
         => ( ! [X2: nat] :
                ( pp(aa_set_nat_bool(member_nat(X2),A2))
               => ? [Y3: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y3),A2))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X2),Y3)) ) )
           => ( A2 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_6582_finite__transitivity__chain,axiom,
    ! [A2: set_int,R2: fun_int_fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X2: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(R2,X2),X2))
       => ( ! [X2: int,Y2: int,Z4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(R2,X2),Y2))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(R2,Y2),Z4))
               => pp(aa_int_bool(aa_int_fun_int_bool(R2,X2),Z4)) ) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),A2))
               => ? [Y3: int] :
                    ( pp(aa_set_int_bool(member_int(Y3),A2))
                    & pp(aa_int_bool(aa_int_fun_int_bool(R2,X2),Y3)) ) )
           => ( A2 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_6583_finite__transitivity__chain,axiom,
    ! [A2: set_real,R2: fun_re1255401033344146673l_bool] :
      ( finite_finite_real(A2)
     => ( ! [X2: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,X2),X2))
       => ( ! [X2: real,Y2: real,Z4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,X2),Y2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,Y2),Z4))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,X2),Z4)) ) )
         => ( ! [X2: real] :
                ( member_real(X2,A2)
               => ? [Y3: real] :
                    ( member_real(Y3,A2)
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,X2),Y3)) ) )
           => ( A2 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_6584_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_6585_mask__int__def,axiom,
    ! [N: nat] : aa_nat_int(bit_se2000444600071755411sk_int,N) = aa_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) ).

% mask_int_def
tff(fact_6586_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z4: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z4))) ).

% ex_le_of_int
tff(fact_6587_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z4: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z4))) ).

% ex_le_of_int
tff(fact_6588_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z4: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z4)),X)) ).

% ex_of_int_less
tff(fact_6589_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z4: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z4)),X)) ).

% ex_of_int_less
tff(fact_6590_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z4: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z4))) ).

% ex_less_of_int
tff(fact_6591_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z4: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z4))) ).

% ex_less_of_int
tff(fact_6592_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),aa_nat_set_nat(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_6593_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(times_times_rat(minus_minus_rat(one_one_rat,X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = 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_6594_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(times_times_int(aa_int_int(minus_minus_int(one_one_int),X)),groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = aa_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_6595_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),aa_nat_set_nat(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_6596_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups2906978787729119204at_rat(aTP_Lamm_mv(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_6597_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups3539618377306564664at_int(aTP_Lamm_mw(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_6598_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_mx(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_6599_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups6591440286371151544t_real(aTP_Lamm_my(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_6600_mask__eq__exp__minus__1,axiom,
    ! [N: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat) ).

% mask_eq_exp_minus_1
tff(fact_6601_mask__eq__exp__minus__1,axiom,
    ! [N: nat] : aa_nat_int(bit_se2000444600071755411sk_int,N) = aa_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) ).

% mask_eq_exp_minus_1
tff(fact_6602_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S3))
    <=> ! [M7: nat] :
        ? [N5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M7),N5))
          & pp(aa_set_nat_bool(member_nat(N5),S3)) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_6603_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set_nat] :
      ( ! [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),M3))
         => ? [N6: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N6))
              & pp(aa_set_nat_bool(member_nat(N6),S3)) ) )
     => ~ pp(aa_set_nat_bool(finite_finite_nat,S3)) ) ).

% unbounded_k_infinite
tff(fact_6604_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S3))
    <=> ! [M7: nat] :
        ? [N5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
          & pp(aa_set_nat_bool(member_nat(N5),S3)) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_6605_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_bh,N))) ).

% mask_eq_sum_exp_nat
tff(fact_6606_gauss__sum__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_an),aa_nat_set_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,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_6607_arith__series__nat,axiom,
    ! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mz(A),D)),aa_nat_set_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(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_6608_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_an),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_6609_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),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),numeral_numeral_real(aa_num_num(bit0,one))))))
       => ( aa_real_int(archim8280529875227126926d_real,X) = Y ) ) ) ).

% round_unique
tff(fact_6610_round__unique,axiom,
    ! [X: rat,Y: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(X,divide_divide_rat(one_one_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)),plus_plus_rat(X,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))))
       => ( archim7778729529865785530nd_rat(X) = Y ) ) ) ).

% round_unique
tff(fact_6611_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),numeral_numeral_real(aa_num_num(bit0,one)))))
     => ( aa_real_int(archim8280529875227126926d_real,X) = N ) ) ).

% round_unique'
tff(fact_6612_round__unique_H,axiom,
    ! [X: rat,N: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(minus_minus_rat(X,aa_int_rat(ring_1_of_int_rat,N)))),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))))
     => ( archim7778729529865785530nd_rat(X) = N ) ) ).

% round_unique'
tff(fact_6613_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),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% of_int_round_abs_le
tff(fact_6614_of__int__round__abs__le,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(X)),X))),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))) ).

% of_int_round_abs_le
tff(fact_6615_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),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_6616_of__int__round__gt,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(X,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(X)))) ).

% of_int_round_gt
tff(fact_6617_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),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_6618_of__int__round__ge,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(X,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(X)))) ).

% of_int_round_ge
tff(fact_6619_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),numeral_numeral_real(aa_num_num(bit0,one)))))) ).

% of_int_round_le
tff(fact_6620_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,archim7778729529865785530nd_rat(X))),plus_plus_rat(X,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))))) ).

% of_int_round_le
tff(fact_6621_round__numeral,axiom,
    ! [N: num] : aa_real_int(archim8280529875227126926d_real,numeral_numeral_real(N)) = aa_num_int(numeral_numeral_int,N) ).

% round_numeral
tff(fact_6622_round__numeral,axiom,
    ! [N: num] : archim7778729529865785530nd_rat(numeral_numeral_rat(N)) = aa_num_int(numeral_numeral_int,N) ).

% round_numeral
tff(fact_6623_round__neg__numeral,axiom,
    ! [N: num] : aa_real_int(archim8280529875227126926d_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% round_neg_numeral
tff(fact_6624_round__neg__numeral,axiom,
    ! [N: num] : archim7778729529865785530nd_rat(uminus_uminus_rat(numeral_numeral_rat(N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% round_neg_numeral
tff(fact_6625_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,archim7778729529865785530nd_rat(X)),archim7778729529865785530nd_rat(Y))) ) ).

% round_mono
tff(fact_6626_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_6627_round__diff__minimal,axiom,
    ! [Z: rat,M: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(Z,aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Z))))),abs_abs_rat(minus_minus_rat(Z,aa_int_rat(ring_1_of_int_rat,M))))) ).

% round_diff_minimal
tff(fact_6628_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),aa_nat_set_nat(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),aa_nat_set_nat(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),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),M),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))),minus_minus_rat(one_one_rat,X)) ) ) ) ) ) ).

% sum_gp
tff(fact_6629_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),aa_nat_set_nat(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),aa_nat_set_nat(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),aa_nat_set_nat(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_6630_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),aa_nat_set_nat(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),aa_nat_set_nat(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),aa_nat_set_nat(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_6631_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,N)),aa_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_6632_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_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_6633_sum__gp__offset,axiom,
    ! [X: rat,M: nat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N),one_one_rat) ) )
      & ( ( X != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M)),minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N)))),minus_minus_rat(one_one_rat,X)) ) ) ) ).

% sum_gp_offset
tff(fact_6634_sum__gp__offset,axiom,
    ! [X: complex,M: nat,N: nat] :
      ( ( ( X = one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex) ) )
      & ( ( X != one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).

% sum_gp_offset
tff(fact_6635_sum__gp__offset,axiom,
    ! [X: real,M: nat,N: nat] :
      ( ( ( X = one_one_real )
       => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real) ) )
      & ( ( X != one_one_real )
       => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M)),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_nat_nat(suc,N))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ).

% sum_gp_offset
tff(fact_6636_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6637_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6638_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,aa_nat_set_nat(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_6639_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),aa_nat_set_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_6640_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),groups6591440286371151544t_real(semiri5074537144036343181t_real,aa_nat_set_nat(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_6641_gauss__sum,axiom,
    ! [N: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,N)),aa_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_6642_gauss__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_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_6643_arith__series,axiom,
    ! [A: int,D: int,N: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_na(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_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_6644_arith__series,axiom,
    ! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nb(A),D)),aa_nat_set_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(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_6645_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_6646_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_6647_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_6648_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,M) = aa_nat_complex(semiri8010041392384452111omplex,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_6649_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_num_int(numeral_numeral_int,V) )
    <=> ( M = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_6650_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_6651_abs__of__nat,axiom,
    ! [N: nat] : abs_abs_rat(aa_nat_rat(semiri681578069525770553at_rat,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ).

% abs_of_nat
tff(fact_6652_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_6653_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_6654_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_6655_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_6656_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_6657_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_6658_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_6659_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_6660_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_6661_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_6662_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_6663_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_6664_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_6665_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_6666_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_6667_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_6668_of__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% of_nat_0
tff(fact_6669_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_6670_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_6671_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_6672_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_6673_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_rat(semiri681578069525770553at_rat,aa_num_nat(numeral_numeral_nat,N)) = numeral_numeral_rat(N) ).

% of_nat_numeral
tff(fact_6674_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N)) = aa_num_int(numeral_numeral_int,N) ).

% of_nat_numeral
tff(fact_6675_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,N)) = numeral_numeral_real(N) ).

% of_nat_numeral
tff(fact_6676_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,N) ).

% of_nat_numeral
tff(fact_6677_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_complex(semiri8010041392384452111omplex,aa_num_nat(numeral_numeral_nat,N)) = numera6690914467698888265omplex(N) ).

% of_nat_numeral
tff(fact_6678_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_6679_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_6680_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_6681_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_6682_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)) = plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_add
tff(fact_6683_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(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_add
tff(fact_6684_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_6685_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_6686_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,N)) ).

% of_nat_add
tff(fact_6687_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_mult
tff(fact_6688_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(times_times_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_mult
tff(fact_6689_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_6690_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_6691_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_6692_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_6693_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_6694_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_6695_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_6696_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_6697_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_6698_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_6699_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_6700_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_6701_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_6702_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_6703_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_6704_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_6705_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_6706_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_6707_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,X) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_6708_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,X) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_6709_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,X) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_6710_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,X) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_6711_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) = aa_nat_int(semiri1314217659103216013at_int,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_6712_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) = aa_nat_real(semiri5074537144036343181t_real,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_6713_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) = aa_nat_nat(semiri1316708129612266289at_nat,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_6714_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) = aa_nat_complex(semiri8010041392384452111omplex,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_6715_of__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,M)),N) ).

% of_nat_power
tff(fact_6716_of__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,M)),N) ).

% of_nat_power
tff(fact_6717_of__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),N) ).

% of_nat_power
tff(fact_6718_of__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),N) ).

% of_nat_power
tff(fact_6719_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_6720_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_6721_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_complex(semiri8010041392384452111omplex,zero_n2687167440665602831ol_nat(P)) = zero_n1201886186963655149omplex(P) ).

% of_nat_of_bool
tff(fact_6722_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_6723_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_6724_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_Code_integer(semiri4939895301339042750nteger,zero_n2687167440665602831ol_nat(P)) = zero_n356916108424825756nteger(P) ).

% of_nat_of_bool
tff(fact_6725_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_6726_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_6727_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_6728_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_6729_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = plus_plus_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,M)) ).

% of_nat_Suc
tff(fact_6730_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M)) = aa_int_int(plus_plus_int(one_one_int),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% of_nat_Suc
tff(fact_6731_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_6732_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_6733_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_6734_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),numeral_numeral_real(W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,W))) ) ).

% real_of_nat_less_numeral_iff
tff(fact_6735_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(W)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,W)),N)) ) ).

% numeral_less_real_of_nat_iff
tff(fact_6736_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_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_6737_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_6738_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_6739_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_6740_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_6741_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: 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)),W)))
    <=> 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),W))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6742_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: 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)),W)))
    <=> 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),W))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6743_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: 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)),W)))
    <=> 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),W))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6744_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: 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)),W)))
    <=> 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),W))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6745_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: 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)),W)),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),W)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6746_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: 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)),W)),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),W)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6747_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: 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)),W)),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),W)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6748_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: 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)),W)),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),W)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6749_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Y) = aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6750_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6751_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6752_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6753_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6754_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N) = aa_nat_rat(semiri681578069525770553at_rat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_6755_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = aa_nat_int(semiri1314217659103216013at_int,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_6756_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N) = aa_nat_real(semiri5074537144036343181t_real,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_6757_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = aa_nat_nat(semiri1316708129612266289at_nat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_6758_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N) = aa_nat_complex(semiri8010041392384452111omplex,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_6759_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: 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)),W)))
    <=> 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),W))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6760_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: 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)),W)))
    <=> 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),W))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6761_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: 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)),W)))
    <=> 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),W))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6762_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: 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)),W)))
    <=> 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),W))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6763_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: 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)),W)),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),W)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6764_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: 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)),W)),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),W)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6765_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: 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)),W)),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),W)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6766_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: 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)),W)),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),W)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6767_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_6768_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_6769_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_6770_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_6771_even__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% even_of_nat
tff(fact_6772_even__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% even_of_nat
tff(fact_6773_even__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% even_of_nat
tff(fact_6774_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(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_6775_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_6776_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,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_6777_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_6778_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(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_6779_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_6780_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,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_6781_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_6782_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,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_6783_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(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_6784_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_6785_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_6786_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,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_6787_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(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_6788_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_6789_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_6790_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_6791_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_6792_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_6793_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_6794_mult__of__nat__commute,axiom,
    ! [X: nat,Y: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(semiri681578069525770553at_rat,X)) ).

% mult_of_nat_commute
tff(fact_6795_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] : aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,X)),Y) = aa_int_int(times_times_int(Y),aa_nat_int(semiri1314217659103216013at_int,X)) ).

% mult_of_nat_commute
tff(fact_6796_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_6797_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_6798_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_6799_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M3: nat,N2: nat] : Z != aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,M3)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% int_diff_cases
tff(fact_6800_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_6801_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_6802_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_6803_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_6804_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_6805_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_6806_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_6807_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_6808_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_6809_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_6810_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_6811_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_6812_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_6813_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_6814_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_6815_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_6816_div__mult2__eq_H,axiom,
    ! [A: int,M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% div_mult2_eq'
tff(fact_6817_div__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% div_mult2_eq'
tff(fact_6818_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_6819_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_6820_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_6821_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_6822_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_6823_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_6824_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_6825_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_6826_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_6827_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_6828_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_6829_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_6830_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_6831_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_6832_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).

% of_nat_dvd_iff
tff(fact_6833_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).

% of_nat_dvd_iff
tff(fact_6834_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).

% of_nat_dvd_iff
tff(fact_6835_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_6836_int__ops_I3_J,axiom,
    ! [N: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N)) = aa_num_int(numeral_numeral_int,N) ).

% int_ops(3)
tff(fact_6837_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_6838_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_6839_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_6840_of__nat__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = modulo364778990260209775nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).

% of_nat_mod
tff(fact_6841_of__nat__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = aa_int_int(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_mod
tff(fact_6842_of__nat__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_mod
tff(fact_6843_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z: int] : aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N)),Z)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))),Z) ).

% zadd_int_left
tff(fact_6844_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(5)
tff(fact_6845_int__plus,axiom,
    ! [N: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% int_plus
tff(fact_6846_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_6847_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(7)
tff(fact_6848_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z))
    <=> ? [N5: nat] : Z = aa_int_int(plus_plus_int(W),aa_nat_int(semiri1314217659103216013at_int,N5)) ) ).

% zle_iff_zadd
tff(fact_6849_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_6850_zdiv__int,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zdiv_int
tff(fact_6851_of__nat__max,axiom,
    ! [X: nat,Y: nat] : semiri4216267220026989637d_enat(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,semiri4216267220026989637d_enat(X)),semiri4216267220026989637d_enat(Y)) ).

% of_nat_max
tff(fact_6852_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_6853_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_6854_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_6855_zmod__int,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_int_int(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zmod_int
tff(fact_6856_nat__less__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).

% nat_less_as_int
tff(fact_6857_of__nat__mask__eq,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,N)) = aa_nat_nat(bit_se2002935070580805687sk_nat,N) ).

% of_nat_mask_eq
tff(fact_6858_of__nat__mask__eq,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(bit_se2002935070580805687sk_nat,N)) = aa_nat_int(bit_se2000444600071755411sk_int,N) ).

% of_nat_mask_eq
tff(fact_6859_nat__leq__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).

% nat_leq_as_int
tff(fact_6860_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(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N2)),X))) ) ).

% ex_less_of_nat_mult
tff(fact_6861_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_6862_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)) = minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,N)) ) ) ).

% of_nat_diff
tff(fact_6863_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(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ).

% of_nat_diff
tff(fact_6864_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_6865_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_6866_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_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,N)) ) ) ).

% of_nat_diff
tff(fact_6867_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_6868_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_6869_real__of__nat__div4,axiom,
    ! [N: nat,X: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X)))) ).

% real_of_nat_div4
tff(fact_6870_int__ops_I4_J,axiom,
    ! [A: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,A)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A)),one_one_int) ).

% int_ops(4)
tff(fact_6871_int__Suc,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ).

% int_Suc
tff(fact_6872_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z))
    <=> ? [N5: nat] : Z = aa_int_int(plus_plus_int(W),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N5))) ) ).

% zless_iff_Suc_zadd
tff(fact_6873_real__of__nat__div,axiom,
    ! [D: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),N))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),D)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,D)) ) ) ).

% real_of_nat_div
tff(fact_6874_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N: nat] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))),modulo364778990260209775nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,M))) ).

% mod_mult2_eq'
tff(fact_6875_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N: nat] : aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,N)))),aa_int_int(modulo_modulo_int(A),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% mod_mult2_eq'
tff(fact_6876_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N: 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_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,N)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))) ).

% mod_mult2_eq'
tff(fact_6877_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = divide_divide_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% field_char_0_class.of_nat_div
tff(fact_6878_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% field_char_0_class.of_nat_div
tff(fact_6879_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))),aa_nat_complex(semiri8010041392384452111omplex,N)) ).

% field_char_0_class.of_nat_div
tff(fact_6880_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_6881_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_6882_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_6883_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_6884_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_6885_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(times_times_int(aa_nat_int(semiri1314217659103216013at_int,K)),I)),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,K)),J))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_6886_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_6887_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_6888_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(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ) ) ).

% int_ops(6)
tff(fact_6889_real__of__nat__div__aux,axiom,
    ! [X: nat,D: nat] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(semiri5074537144036343181t_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),D))),aa_nat_real(semiri5074537144036343181t_real,D))) ).

% real_of_nat_div_aux
tff(fact_6890_nat__approx__posE,axiom,
    ! [E: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
     => ~ ! [N2: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)))),E)) ) ).

% nat_approx_posE
tff(fact_6891_nat__approx__posE,axiom,
    ! [E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
     => ~ ! [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)))),E)) ) ).

% nat_approx_posE
tff(fact_6892_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(numeral_numeral_rat(aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_6893_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_6894_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,numeral_numeral_real(aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_6895_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_6896_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,divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,M))),divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,N)))) ) ) ).

% inverse_of_nat_le
tff(fact_6897_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))
       => ( ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M3))
             => 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,M3)),X)),C)) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_6898_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_6899_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(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_6900_real__of__nat__div2,axiom,
    ! [N: nat,X: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X))))) ).

% real_of_nat_div2
tff(fact_6901_real__of__nat__div3,axiom,
    ! [N: nat,X: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X)))),one_one_real)) ).

% real_of_nat_div3
tff(fact_6902_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_6903_linear__plus__1__le__power,axiom,
    ! [X: real,N: nat] :
      ( 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(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)),N))) ) ).

% linear_plus_1_le_power
tff(fact_6904_Bernoulli__inequality,axiom,
    ! [X: real,N: nat] :
      ( 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,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),N))) ) ).

% Bernoulli_inequality
tff(fact_6905_Bernoulli__inequality__even,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))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),N))) ) ).

% Bernoulli_inequality_even
tff(fact_6906_double__arith__series,axiom,
    ! [A: rat,D: rat,N: nat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_nc(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_rat_rat(times_times_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N),one_one_rat)),plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),A),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),D))) ).

% double_arith_series
tff(fact_6907_double__arith__series,axiom,
    ! [A: int,D: int,N: nat] : aa_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_na(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_int_int(times_times_int(aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,N)),D))) ).

% double_arith_series
tff(fact_6908_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_nd(A),D),aa_nat_set_nat(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_6909_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_nb(A),D)),aa_nat_set_nat(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_6910_double__arith__series,axiom,
    ! [A: real,D: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ne(A),D),aa_nat_set_nat(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,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_6911_double__gauss__sum,axiom,
    ! [N: nat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N),one_one_rat)) ).

% double_gauss_sum
tff(fact_6912_double__gauss__sum,axiom,
    ! [N: nat] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ).

% double_gauss_sum
tff(fact_6913_double__gauss__sum,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,aa_nat_set_nat(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_6914_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),aa_nat_set_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_6915_double__gauss__sum,axiom,
    ! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),groups6591440286371151544t_real(semiri5074537144036343181t_real,aa_nat_set_nat(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_6916_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_nf,divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_6917_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_ng,divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_6918_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) = produc1703576794950452218t_real(aTP_Lamm_nh,divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_6919_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_ni),divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_6920_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_nj,divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_6921_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_nk,X)) ) ).

% monoseq_arctan_series
tff(fact_6922_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K5: 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)),K5))
       => ( 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))),K5))
         => 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,K5),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_6923_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K5: real,N: nat] :
      ( ( H != zero_zero_complex )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Z),H))),K5))
         => 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,K5),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_6924_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),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_real(ln_ln_real,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_nl,X)) ) ) ) ).

% ln_series
tff(fact_6925_lemma__termdiff2,axiom,
    ! [H: rat,Z: rat,N: nat] :
      ( ( H != zero_zero_rat )
     => ( minus_minus_rat(divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(plus_plus_rat(Z,H)),N),aa_nat_rat(power_power_rat(Z),N)),H),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(power_power_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) = aa_rat_rat(times_times_rat(H),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nn(H),Z),N),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_6926_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_np(H),Z),N),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_6927_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_nr(H),Z),N),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_6928_arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(arctan,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ns,X)) ) ) ).

% arctan_series
tff(fact_6929_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( 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_6930_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( member_num(I,set_ord_lessThan_num(K))
    <=> pp(aa_num_bool(ord_less_num(I),K)) ) ).

% lessThan_iff
tff(fact_6931_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_6932_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_ord_lessThan_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K)) ) ).

% lessThan_iff
tff(fact_6933_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( member_real(I,set_or5984915006950818249n_real(K))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),K)) ) ).

% lessThan_iff
tff(fact_6934_finite__lessThan,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_ord_lessThan_nat,K))) ).

% finite_lessThan
tff(fact_6935_lessThan__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( 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_6936_lessThan__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ord_less_eq_set_num(set_ord_lessThan_num(X),set_ord_lessThan_num(Y))
    <=> pp(aa_num_bool(ord_less_eq_num(X),Y)) ) ).

% lessThan_subset_iff
tff(fact_6937_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_6938_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,X)),aa_nat_set_nat(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_6939_lessThan__subset__iff,axiom,
    ! [X: real,Y: real] :
      ( 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_6940_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,N)),aa_nat_rat(G,N)) ).

% sum.lessThan_Suc
tff(fact_6941_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_int(G,N)) ).

% sum.lessThan_Suc
tff(fact_6942_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_nat(G,N)) ).

% sum.lessThan_Suc
tff(fact_6943_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_real(G,N)) ).

% sum.lessThan_Suc
tff(fact_6944_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_lo(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_6945_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_lq(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_6946_int__if,axiom,
    ! [P: bool,A: nat,B: nat] :
      ( ( pp(P)
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(P),A),B)) = aa_nat_int(semiri1314217659103216013at_int,A) ) )
      & ( ~ pp(P)
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(P),A),B)) = aa_nat_int(semiri1314217659103216013at_int,B) ) ) ) ).

% int_if
tff(fact_6947_nat__int__comparison_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_int(semiri1314217659103216013at_int,A) = aa_nat_int(semiri1314217659103216013at_int,B) ) ) ).

% nat_int_comparison(1)
tff(fact_6948_infinite__Iio,axiom,
    ! [A: int] : ~ pp(aa_set_int_bool(finite_finite_int,set_ord_lessThan_int(A))) ).

% infinite_Iio
tff(fact_6949_infinite__Iio,axiom,
    ! [A: real] : ~ finite_finite_real(set_or5984915006950818249n_real(A)) ).

% infinite_Iio
tff(fact_6950_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aTP_Lamm_nt(U)) ).

% lessThan_def
tff(fact_6951_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_nu,U)) ).

% lessThan_def
tff(fact_6952_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aTP_Lamm_nv(U)) ).

% lessThan_def
tff(fact_6953_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_nw,U)) ).

% lessThan_def
tff(fact_6954_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bh,U)) ).

% lessThan_def
tff(fact_6955_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_nx,U)) ).

% lessThan_def
tff(fact_6956_lessThan__strict__subset__iff,axiom,
    ! [M: rat,N: rat] :
      ( 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_6957_lessThan__strict__subset__iff,axiom,
    ! [M: num,N: num] :
      ( ord_less_set_num(set_ord_lessThan_num(M),set_ord_lessThan_num(N))
    <=> pp(aa_num_bool(ord_less_num(M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_6958_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_6959_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,M)),aa_nat_set_nat(set_ord_lessThan_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_6960_lessThan__strict__subset__iff,axiom,
    ! [M: real,N: real] :
      ( 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_6961_complex__mod__minus__le__complex__mod,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,X))),aa_complex_real(real_V1022390504157884413omplex,X))) ).

% complex_mod_minus_le_complex_mod
tff(fact_6962_complex__mod__triangle__ineq2,axiom,
    ! [B: complex,A: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(B),A))),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,A))) ).

% complex_mod_triangle_ineq2
tff(fact_6963_finite__nat__bounded,axiom,
    ! [S3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ? [K3: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(S3),aa_nat_set_nat(set_ord_lessThan_nat,K3))) ) ).

% finite_nat_bounded
tff(fact_6964_finite__nat__iff__bounded,axiom,
    ! [S3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
    <=> ? [K2: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(S3),aa_nat_set_nat(set_ord_lessThan_nat,K2))) ) ).

% finite_nat_iff_bounded
tff(fact_6965_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ny(G),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nat_diff_reindex
tff(fact_6966_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_nz(G),N),aa_nat_set_nat(set_ord_lessThan_nat,N)) = groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nat_diff_reindex
tff(fact_6967_sum__diff__distrib,axiom,
    ! [Q: fun_real_nat,P: fun_real_nat,N: real] :
      ( ! [X2: real] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(Q,X2)),aa_real_nat(P,X2)))
     => ( 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_lu(Q),P),set_or5984915006950818249n_real(N)) ) ) ).

% sum_diff_distrib
tff(fact_6968_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,P: fun_nat_nat,N: nat] :
      ( ! [X2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Q,X2)),aa_nat_nat(P,X2)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(P),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),aa_nat_set_nat(set_ord_lessThan_nat,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ly(Q),P)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ) ).

% sum_diff_distrib
tff(fact_6969_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = plus_plus_rat(aa_nat_rat(G,zero_zero_nat),groups2906978787729119204at_rat(aTP_Lamm_mh(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_6970_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(plus_plus_int(aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_mi(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_6971_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_mj(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_6972_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),groups6591440286371151544t_real(aTP_Lamm_mk(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_6973_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_rat,N: nat,R: rat] : minus_minus_rat(groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,N)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),R)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_oa(F),R),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sumr_diff_mult_const2
tff(fact_6974_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_int,N: nat,R: int] : aa_int_int(minus_minus_int(groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,N)),R)) = groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ob(F),R),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sumr_diff_mult_const2
tff(fact_6975_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_complex,N: nat,R: complex] : aa_complex_complex(minus_minus_complex(groups2073611262835488442omplex(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),R)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_oc(F),R),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sumr_diff_mult_const2
tff(fact_6976_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_real,N: nat,R: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),R)) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_od(F),R),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sumr_diff_mult_const2
tff(fact_6977_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_ml(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = minus_minus_rat(aa_nat_rat(F,M),aa_nat_rat(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6978_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_mm(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(minus_minus_int(aa_nat_int(F,M)),aa_nat_int(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6979_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,M: nat] : groups6591440286371151544t_real(aTP_Lamm_mn(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,M)),aa_nat_real(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6980_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_oe(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = minus_minus_rat(aa_nat_rat(F,zero_zero_nat),aa_nat_rat(F,M)) ).

% sum_lessThan_telescope'
tff(fact_6981_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_of(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(minus_minus_int(aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,M)) ).

% sum_lessThan_telescope'
tff(fact_6982_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,M: nat] : groups6591440286371151544t_real(aTP_Lamm_og(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,M)) ).

% sum_lessThan_telescope'
tff(fact_6983_power__diff__1__eq,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),one_one_complex)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_6984_power__diff__1__eq,axiom,
    ! [X: rat,N: nat] : minus_minus_rat(aa_nat_rat(power_power_rat(X),N),one_one_rat) = aa_rat_rat(times_times_rat(minus_minus_rat(X,one_one_rat)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_6985_power__diff__1__eq,axiom,
    ! [X: int,N: nat] : aa_int_int(minus_minus_int(aa_nat_int(power_power_int(X),N)),one_one_int) = aa_int_int(times_times_int(aa_int_int(minus_minus_int(X),one_one_int)),groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_6986_power__diff__1__eq,axiom,
    ! [X: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_6987_one__diff__power__eq,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_6988_one__diff__power__eq,axiom,
    ! [X: rat,N: nat] : minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(X),N)) = aa_rat_rat(times_times_rat(minus_minus_rat(one_one_rat,X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_6989_one__diff__power__eq,axiom,
    ! [X: int,N: nat] : aa_int_int(minus_minus_int(one_one_int),aa_nat_int(power_power_int(X),N)) = aa_int_int(times_times_int(aa_int_int(minus_minus_int(one_one_int),X)),groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_6990_one__diff__power__eq,axiom,
    ! [X: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),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),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_6991_geometric__sum,axiom,
    ! [X: complex,N: nat] :
      ( ( X != one_one_complex )
     => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),one_one_complex)),aa_complex_complex(minus_minus_complex(X),one_one_complex)) ) ) ).

% geometric_sum
tff(fact_6992_geometric__sum,axiom,
    ! [X: rat,N: nat] :
      ( ( X != one_one_rat )
     => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),N),one_one_rat),minus_minus_rat(X,one_one_rat)) ) ) ).

% geometric_sum
tff(fact_6993_geometric__sum,axiom,
    ! [X: real,N: nat] :
      ( ( X != one_one_real )
     => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)) ) ) ).

% geometric_sum
tff(fact_6994_monoseq__realpow,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_eq_real,X),one_one_real))
       => topolo6980174941875973593q_real(aa_real_fun_nat_real(power_power_real,X)) ) ) ).

% monoseq_realpow
tff(fact_6995_sum__gp__strict,axiom,
    ! [X: rat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ) )
      & ( ( X != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = divide_divide_rat(minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(X),N)),minus_minus_rat(one_one_rat,X)) ) ) ) ).

% sum_gp_strict
tff(fact_6996_sum__gp__strict,axiom,
    ! [X: complex,N: nat] :
      ( ( ( X = one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_complex(semiri8010041392384452111omplex,N) ) )
      & ( ( X != one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).

% sum_gp_strict
tff(fact_6997_sum__gp__strict,axiom,
    ! [X: real,N: nat] :
      ( ( ( X = one_one_real )
       => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ) )
      & ( ( X != one_one_real )
       => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ).

% sum_gp_strict
tff(fact_6998_lemma__termdiff1,axiom,
    ! [Z: complex,H: complex,M: nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_oh(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_oi(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_6999_lemma__termdiff1,axiom,
    ! [Z: rat,H: rat,M: nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_oj(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_ok(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_7000_lemma__termdiff1,axiom,
    ! [Z: int,H: int,M: nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ol(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_om(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_7001_lemma__termdiff1,axiom,
    ! [Z: real,H: real,M: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_on(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_oo(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_7002_diff__power__eq__sum,axiom,
    ! [X: complex,N: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))),aa_nat_complex(power_power_complex(Y),aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_op(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_7003_diff__power__eq__sum,axiom,
    ! [X: rat,N: nat,Y: rat] : minus_minus_rat(aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N)),aa_nat_rat(power_power_rat(Y),aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(minus_minus_rat(X,Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_oq(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_7004_diff__power__eq__sum,axiom,
    ! [X: int,N: nat,Y: int] : aa_int_int(minus_minus_int(aa_nat_int(power_power_int(X),aa_nat_nat(suc,N))),aa_nat_int(power_power_int(Y),aa_nat_nat(suc,N))) = aa_int_int(times_times_int(aa_int_int(minus_minus_int(X),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_or(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_7005_diff__power__eq__sum,axiom,
    ! [X: real,N: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_nat_nat(suc,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_nat1308781354268537904t_real(aTP_Lamm_os(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_7006_power__diff__sumr2,axiom,
    ! [X: complex,N: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(Y),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ot(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_7007_power__diff__sumr2,axiom,
    ! [X: rat,N: nat,Y: rat] : minus_minus_rat(aa_nat_rat(power_power_rat(X),N),aa_nat_rat(power_power_rat(Y),N)) = aa_rat_rat(times_times_rat(minus_minus_rat(X,Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_ou(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_7008_power__diff__sumr2,axiom,
    ! [X: int,N: nat,Y: int] : aa_int_int(minus_minus_int(aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),N)) = aa_int_int(times_times_int(aa_int_int(minus_minus_int(X),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_ov(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_7009_power__diff__sumr2,axiom,
    ! [X: real,N: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_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),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_nat1308781354268537904t_real(aTP_Lamm_ow(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_7010_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_rat,K5: 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)),K5)) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),K5))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7011_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_int,K5: 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)),K5)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K5))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,N)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7012_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_nat,K5: 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)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),K5))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7013_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_real,K5: 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)),K5)) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),K5))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7014_one__diff__power__eq_H,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_ox(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_7015_one__diff__power__eq_H,axiom,
    ! [X: rat,N: nat] : minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(X),N)) = aa_rat_rat(times_times_rat(minus_minus_rat(one_one_rat,X)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_oy(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_7016_one__diff__power__eq_H,axiom,
    ! [X: int,N: nat] : aa_int_int(minus_minus_int(one_one_int),aa_nat_int(power_power_int(X),N)) = aa_int_int(times_times_int(aa_int_int(minus_minus_int(one_one_int),X)),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_oz(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_7017_one__diff__power__eq_H,axiom,
    ! [X: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),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_nat_fun_nat_real(aTP_Lamm_pa(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_7018_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,N: nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_pb(F),G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_pc(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),groups6591440286371151544t_real(aTP_Lamm_pd(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum_split_even_odd
tff(fact_7019_norm__divide__numeral,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),numeral_numeral_real(W)) ).

% norm_divide_numeral
tff(fact_7020_norm__divide__numeral,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),numeral_numeral_real(W)) ).

% norm_divide_numeral
tff(fact_7021_norm__mult__numeral2,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,A)),numeral_numeral_real(W)) ).

% norm_mult_numeral2
tff(fact_7022_norm__mult__numeral2,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,A)),numeral_numeral_real(W)) ).

% norm_mult_numeral2
tff(fact_7023_norm__mult__numeral1,axiom,
    ! [W: num,A: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),aa_real_real(real_V7735802525324610683m_real,A)) ).

% norm_mult_numeral1
tff(fact_7024_norm__mult__numeral1,axiom,
    ! [W: num,A: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% norm_mult_numeral1
tff(fact_7025_norm__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = numeral_numeral_real(W) ).

% norm_neg_numeral
tff(fact_7026_norm__neg__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = numeral_numeral_real(W) ).

% norm_neg_numeral
tff(fact_7027_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_7028_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_7029_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_7030_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_7031_norm__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,numeral_numeral_real(W)) = numeral_numeral_real(W) ).

% norm_numeral
tff(fact_7032_norm__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,numera6690914467698888265omplex(W)) = numeral_numeral_real(W) ).

% norm_numeral
tff(fact_7033_norm__minus__commute,axiom,
    ! [A: real,B: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ).

% norm_minus_commute
tff(fact_7034_norm__minus__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B),A)) ).

% norm_minus_commute
tff(fact_7035_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_7036_norm__ge__zero,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,X))) ).

% norm_ge_zero
tff(fact_7037_norm__mult,axiom,
    ! [X: real,Y: 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,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y)) ).

% norm_mult
tff(fact_7038_norm__mult,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)) ).

% norm_mult
tff(fact_7039_norm__divide,axiom,
    ! [A: real,B: 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)) ).

% norm_divide
tff(fact_7040_norm__divide,axiom,
    ! [A: complex,B: 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)) ).

% norm_divide
tff(fact_7041_sum__norm__le,axiom,
    ! [S3: set_real,F: fun_real_complex,G: fun_real_real] :
      ( ! [X2: real] :
          ( member_real(X2,S3)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(F,X2))),aa_real_real(G,X2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups5754745047067104278omplex(F,S3))),groups8097168146408367636l_real(G,S3))) ) ).

% sum_norm_le
tff(fact_7042_sum__norm__le,axiom,
    ! [S3: set_VEBT_VEBT,F: fun_VE5569070535936029429omplex,G: fun_VEBT_VEBT_real] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),S3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_VEBT_VEBT_complex(F,X2))),aa_VEBT_VEBT_real(G,X2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups1794756597179926696omplex(F,S3))),groups2240296850493347238T_real(G,S3))) ) ).

% sum_norm_le
tff(fact_7043_sum__norm__le,axiom,
    ! [S3: set_int,F: fun_int_complex,G: fun_int_real] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),S3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(F,X2))),aa_int_real(G,X2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups3049146728041665814omplex(F,S3))),groups8778361861064173332t_real(G,S3))) ) ).

% sum_norm_le
tff(fact_7044_sum__norm__le,axiom,
    ! [S3: set_nat,F: fun_nat_complex,G: fun_nat_real] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),S3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,X2))),aa_nat_real(G,X2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,S3))),groups6591440286371151544t_real(G,S3))) ) ).

% sum_norm_le
tff(fact_7045_sum__norm__le,axiom,
    ! [S3: set_complex,F: fun_complex_complex,G: fun_complex_real] :
      ( ! [X2: complex] :
          ( pp(aa_set_complex_bool(member_complex(X2),S3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(F,X2))),aa_complex_real(G,X2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,S3))),groups5808333547571424918x_real(G,S3))) ) ).

% sum_norm_le
tff(fact_7046_sum__norm__le,axiom,
    ! [S3: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),S3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,X2))),aa_nat_real(G,X2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,groups6591440286371151544t_real(F,S3))),groups6591440286371151544t_real(G,S3))) ) ).

% sum_norm_le
tff(fact_7047_norm__power,axiom,
    ! [X: real,N: nat] : aa_real_real(real_V7735802525324610683m_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(real_V7735802525324610683m_real,X)),N) ).

% norm_power
tff(fact_7048_norm__power,axiom,
    ! [X: complex,N: nat] : aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(X),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,X)),N) ).

% norm_power
tff(fact_7049_norm__sum,axiom,
    ! [F: fun_nat_complex,A2: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,A2))),groups6591440286371151544t_real(aTP_Lamm_pe(F),A2))) ).

% norm_sum
tff(fact_7050_norm__sum,axiom,
    ! [F: fun_complex_complex,A2: set_complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,A2))),groups5808333547571424918x_real(aTP_Lamm_pf(F),A2))) ).

% norm_sum
tff(fact_7051_norm__sum,axiom,
    ! [F: fun_nat_real,A2: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,groups6591440286371151544t_real(F,A2))),groups6591440286371151544t_real(aTP_Lamm_pg(F),A2))) ).

% norm_sum
tff(fact_7052_norm__uminus__minus,axiom,
    ! [X: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X)),Y)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) ).

% norm_uminus_minus
tff(fact_7053_norm__uminus__minus,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),Y)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y)) ).

% norm_uminus_minus
tff(fact_7054_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_7055_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_7056_power__eq__imp__eq__norm,axiom,
    ! [W: real,N: nat,Z: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_real_real(real_V7735802525324610683m_real,W) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_7057_power__eq__imp__eq__norm,axiom,
    ! [W: complex,N: nat,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W),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,W) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_7058_norm__mult__less,axiom,
    ! [X: real,R: real,Y: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R))
     => ( 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,R),S))) ) ) ).

% norm_mult_less
tff(fact_7059_norm__mult__less,axiom,
    ! [X: complex,R: real,Y: complex,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R))
     => ( 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,R),S))) ) ) ).

% norm_mult_less
tff(fact_7060_norm__mult__ineq,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y)))) ).

% norm_mult_ineq
tff(fact_7061_norm__mult__ineq,axiom,
    ! [X: complex,Y: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))) ).

% norm_mult_ineq
tff(fact_7062_norm__triangle__lt,axiom,
    ! [X: real,Y: real,E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),E)) ) ).

% norm_triangle_lt
tff(fact_7063_norm__triangle__lt,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E)) ) ).

% norm_triangle_lt
tff(fact_7064_norm__add__less,axiom,
    ! [X: real,R: real,Y: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R))
     => ( 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,R),S))) ) ) ).

% norm_add_less
tff(fact_7065_norm__add__less,axiom,
    ! [X: complex,R: real,Y: complex,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R))
     => ( 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,R),S))) ) ) ).

% norm_add_less
tff(fact_7066_norm__power__ineq,axiom,
    ! [X: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,X)),N))) ).

% norm_power_ineq
tff(fact_7067_norm__power__ineq,axiom,
    ! [X: complex,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(X),N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,X)),N))) ).

% norm_power_ineq
tff(fact_7068_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),C))) ) ).

% norm_add_leD
tff(fact_7069_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),C))) ) ).

% norm_add_leD
tff(fact_7070_norm__triangle__le,axiom,
    ! [X: real,Y: real,E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),E)) ) ).

% norm_triangle_le
tff(fact_7071_norm__triangle__le,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E)) ) ).

% norm_triangle_le
tff(fact_7072_norm__triangle__ineq,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y)))) ).

% norm_triangle_ineq
tff(fact_7073_norm__triangle__ineq,axiom,
    ! [X: complex,Y: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))) ).

% norm_triangle_ineq
tff(fact_7074_norm__triangle__mono,axiom,
    ! [A: real,R: real,B: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A)),R))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),S))
       => 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,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R),S))) ) ) ).

% norm_triangle_mono
tff(fact_7075_norm__triangle__mono,axiom,
    ! [A: complex,R: real,B: complex,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,A)),R))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R),S))) ) ) ).

% norm_triangle_mono
tff(fact_7076_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_7077_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_7078_norm__triangle__sub,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Y)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y))))) ).

% norm_triangle_sub
tff(fact_7079_norm__triangle__sub,axiom,
    ! [X: complex,Y: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Y)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))))) ).

% norm_triangle_sub
tff(fact_7080_norm__triangle__ineq4,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)))) ).

% norm_triangle_ineq4
tff(fact_7081_norm__triangle__ineq4,axiom,
    ! [A: complex,B: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)))) ).

% norm_triangle_ineq4
tff(fact_7082_norm__diff__triangle__le,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_le
tff(fact_7083_norm__diff__triangle__le,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_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_le
tff(fact_7084_norm__triangle__le__diff,axiom,
    ! [X: real,Y: real,E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y))),E)) ) ).

% norm_triangle_le_diff
tff(fact_7085_norm__triangle__le__diff,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E)) ) ).

% norm_triangle_le_diff
tff(fact_7086_norm__diff__ineq,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(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))) ).

% norm_diff_ineq
tff(fact_7087_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B)))) ).

% norm_diff_ineq
tff(fact_7088_norm__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(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))) ).

% norm_triangle_ineq2
tff(fact_7089_norm__triangle__ineq2,axiom,
    ! [A: complex,B: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B)))) ).

% norm_triangle_ineq2
tff(fact_7090_power__eq__1__iff,axiom,
    ! [W: real,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),N) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W) = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_7091_power__eq__1__iff,axiom,
    ! [W: complex,N: nat] :
      ( ( aa_nat_complex(power_power_complex(W),N) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W) = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_7092_norm__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(real_V7735802525324610683m_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(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D))))) ).

% norm_diff_triangle_ineq
tff(fact_7093_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : 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(plus_plus_complex(A),B)),aa_complex_complex(plus_plus_complex(C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),C))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B),D))))) ).

% norm_diff_triangle_ineq
tff(fact_7094_norm__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(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))) ).

% norm_triangle_ineq3
tff(fact_7095_norm__triangle__ineq3,axiom,
    ! [A: complex,B: complex] : 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_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B)))) ).

% norm_triangle_ineq3
tff(fact_7096_square__norm__one,axiom,
    ! [X: 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(real_V7735802525324610683m_real,X) = one_one_real ) ) ).

% square_norm_one
tff(fact_7097_square__norm__one,axiom,
    ! [X: complex] :
      ( ( aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,X) = one_one_real ) ) ).

% square_norm_one
tff(fact_7098_norm__power__diff,axiom,
    ! [Z: real,W: real,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W)),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,W),M)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),W))))) ) ) ).

% norm_power_diff
tff(fact_7099_norm__power__diff,axiom,
    ! [Z: complex,W: complex,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,W)),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Z),M)),aa_nat_complex(power_power_complex(W),M)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Z),W))))) ) ) ).

% norm_power_diff
tff(fact_7100_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_7101_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_7102_sumr__cos__zero__one,axiom,
    ! [N: nat] : groups6591440286371151544t_real(aTP_Lamm_ph,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_7103_suminf__finite,axiom,
    ! [N3: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N3))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N3))
           => ( aa_nat_complex(F,N2) = zero_zero_complex ) )
       => ( suminf_complex(F) = groups2073611262835488442omplex(F,N3) ) ) ) ).

% suminf_finite
tff(fact_7104_suminf__finite,axiom,
    ! [N3: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N3))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N3))
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => ( suminf_int(F) = groups3539618377306564664at_int(F,N3) ) ) ) ).

% suminf_finite
tff(fact_7105_suminf__finite,axiom,
    ! [N3: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N3))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N3))
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => ( suminf_nat(F) = aa_set_nat_nat(groups3542108847815614940at_nat(F),N3) ) ) ) ).

% suminf_finite
tff(fact_7106_suminf__finite,axiom,
    ! [N3: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N3))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N3))
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => ( suminf_real(F) = groups6591440286371151544t_real(F,N3) ) ) ) ).

% suminf_finite
tff(fact_7107_pi__series,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_pi) ).

% pi_series
tff(fact_7108_summable__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_ns,X)) ) ).

% summable_arctan_series
tff(fact_7109_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_pj,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_7110_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_pk(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_7111_summable__iff__shift,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(aa_nat_fun_nat_real(aTP_Lamm_ma(F),K))
    <=> summable_real(F) ) ).

% summable_iff_shift
tff(fact_7112_summable__cmult__iff,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_pl(C),F))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_cmult_iff
tff(fact_7113_summable__cmult__iff,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ic(C),F))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_7114_summable__divide__iff,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_pm(F),C))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_divide_iff
tff(fact_7115_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_il(F),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_7116_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_pn(P),F)) ) ).

% summable_If_finite
tff(fact_7117_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_po(P),F)) ) ).

% summable_If_finite
tff(fact_7118_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_pp(P),F)) ) ).

% summable_If_finite
tff(fact_7119_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_pq(P),F)) ) ).

% summable_If_finite
tff(fact_7120_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_pr(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7121_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_ps(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7122_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_pt(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7123_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_pu(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7124_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_complex,X: complex] :
      ( pp(aa_complex_bool(sums_complex(aTP_Lamm_lo(A)),X))
    <=> ( aa_nat_complex(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_7125_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_real,X: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_lq(A)),X))
    <=> ( aa_nat_real(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_7126_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_7127_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_7128_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N3: nat,F: fun_nat_real] :
      ( summable_real(G)
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N3),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_7129_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N3: nat,F: fun_nat_complex] :
      ( summable_real(G)
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N3),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_7130_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_7131_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_7132_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => summable_real(aa_nat_fun_nat_real(aTP_Lamm_ma(F),K)) ) ).

% summable_ignore_initial_segment
tff(fact_7133_summable__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ig(F),G)) ) ) ).

% summable_add
tff(fact_7134_summable__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_if(F),G)) ) ) ).

% summable_add
tff(fact_7135_summable__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_pv(F),G)) ) ) ).

% summable_add
tff(fact_7136_sums__add,axiom,
    ! [F: fun_nat_real,A: real,G: fun_nat_real,B: real] :
      ( pp(aa_real_bool(sums_real(F),A))
     => ( pp(aa_real_bool(sums_real(G),B))
       => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ig(F),G)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% sums_add
tff(fact_7137_sums__add,axiom,
    ! [F: fun_nat_nat,A: nat,G: fun_nat_nat,B: nat] :
      ( pp(aa_nat_bool(sums_nat(F),A))
     => ( pp(aa_nat_bool(sums_nat(G),B))
       => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_if(F),G)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% sums_add
tff(fact_7138_sums__add,axiom,
    ! [F: fun_nat_int,A: int,G: fun_nat_int,B: int] :
      ( pp(aa_int_bool(sums_int(F),A))
     => ( pp(aa_int_bool(sums_int(G),B))
       => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_pv(F),G)),aa_int_int(plus_plus_int(A),B))) ) ) ).

% sums_add
tff(fact_7139_sums__mult,axiom,
    ! [F: fun_nat_real,A: real,C: real] :
      ( pp(aa_real_bool(sums_real(F),A))
     => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_pw(F),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A))) ) ).

% sums_mult
tff(fact_7140_sums__mult2,axiom,
    ! [F: fun_nat_real,A: real,C: real] :
      ( pp(aa_real_bool(sums_real(F),A))
     => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_hy(F),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ).

% sums_mult2
tff(fact_7141_sums__diff,axiom,
    ! [F: fun_nat_real,A: real,G: fun_nat_real,B: real] :
      ( pp(aa_real_bool(sums_real(F),A))
     => ( pp(aa_real_bool(sums_real(G),B))
       => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ij(F),G)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ) ) ).

% sums_diff
tff(fact_7142_sums__divide,axiom,
    ! [F: fun_nat_complex,A: complex,C: complex] :
      ( pp(aa_complex_bool(sums_complex(F),A))
     => pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_pm(F),C)),aa_complex_complex(divide1717551699836669952omplex(A),C))) ) ).

% sums_divide
tff(fact_7143_sums__divide,axiom,
    ! [F: fun_nat_real,A: real,C: real] :
      ( pp(aa_real_bool(sums_real(F),A))
     => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_il(F),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C))) ) ).

% sums_divide
tff(fact_7144_summable__mult,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_pw(F),C)) ) ).

% summable_mult
tff(fact_7145_summable__mult2,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_hy(F),C)) ) ).

% summable_mult2
tff(fact_7146_summable__diff,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ij(F),G)) ) ) ).

% summable_diff
tff(fact_7147_summable__divide,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_pm(F),C)) ) ).

% summable_divide
tff(fact_7148_summable__divide,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_il(F),C)) ) ).

% summable_divide
tff(fact_7149_sums__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,S: real,T: real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G,N2)))
     => ( pp(aa_real_bool(sums_real(F),S))
       => ( pp(aa_real_bool(sums_real(G),T))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,S),T)) ) ) ) ).

% sums_le
tff(fact_7150_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_7151_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_7152_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_7153_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_7154_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_7155_summable__finite,axiom,
    ! [N3: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N3))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N3))
           => ( aa_nat_complex(F,N2) = zero_zero_complex ) )
       => summable_complex(F) ) ) ).

% summable_finite
tff(fact_7156_summable__finite,axiom,
    ! [N3: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N3))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N3))
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => summable_real(F) ) ) ).

% summable_finite
tff(fact_7157_summable__finite,axiom,
    ! [N3: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N3))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N3))
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => summable_nat(F) ) ) ).

% summable_finite
tff(fact_7158_summable__finite,axiom,
    ! [N3: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N3))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N3))
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => summable_int(F) ) ) ).

% summable_finite
tff(fact_7159_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_pl(C),F)),aa_complex_complex(times_times_complex(C),D)))
      <=> pp(aa_complex_bool(sums_complex(F),D)) ) ) ).

% sums_mult_iff
tff(fact_7160_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_ic(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_7161_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_px(C),F)),aa_complex_complex(times_times_complex(D),C)))
      <=> pp(aa_complex_bool(sums_complex(F),D)) ) ) ).

% sums_mult2_iff
tff(fact_7162_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_py(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_7163_summable__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_pl(C),F))
     => ( ( C != zero_zero_complex )
       => summable_complex(F) ) ) ).

% summable_mult_D
tff(fact_7164_summable__mult__D,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ic(C),F))
     => ( ( C != zero_zero_real )
       => summable_real(F) ) ) ).

% summable_mult_D
tff(fact_7165_summable__zero__power,axiom,
    summable_real(aa_real_fun_nat_real(power_power_real,zero_zero_real)) ).

% summable_zero_power
tff(fact_7166_summable__zero__power,axiom,
    summable_complex(power_power_complex(zero_zero_complex)) ).

% summable_zero_power
tff(fact_7167_summable__zero__power,axiom,
    summable_int(power_power_int(zero_zero_int)) ).

% summable_zero_power
tff(fact_7168_suminf__mult2,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(F)),C) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_hy(F),C)) ) ) ).

% suminf_mult2
tff(fact_7169_suminf__mult,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_pw(F),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),suminf_real(F)) ) ) ).

% suminf_mult
tff(fact_7170_suminf__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_ig(F),G)) ) ) ) ).

% suminf_add
tff(fact_7171_suminf__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,suminf_nat(F)),suminf_nat(G)) = suminf_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_if(F),G)) ) ) ) ).

% suminf_add
tff(fact_7172_suminf__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => ( aa_int_int(plus_plus_int(suminf_int(F)),suminf_int(G)) = suminf_int(aa_fun5565043122120104496at_int(aTP_Lamm_pv(F),G)) ) ) ) ).

% suminf_add
tff(fact_7173_suminf__diff,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_ij(F),G)) ) ) ) ).

% suminf_diff
tff(fact_7174_suminf__divide,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_pm(F),C)) = aa_complex_complex(divide1717551699836669952omplex(suminf_complex(F)),C) ) ) ).

% suminf_divide
tff(fact_7175_suminf__divide,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_il(F),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,suminf_real(F)),C) ) ) ).

% suminf_divide
tff(fact_7176_powser__insidea,axiom,
    ! [F: fun_nat_real,X: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_pz(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_qa(F),Z)) ) ) ).

% powser_insidea
tff(fact_7177_powser__insidea,axiom,
    ! [F: fun_nat_complex,X: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_qb(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_qc(F),Z)) ) ) ).

% powser_insidea
tff(fact_7178_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 )
        <=> ! [N5: nat] : aa_nat_real(F,N5) = zero_zero_real ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7179_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 )
        <=> ! [N5: nat] : aa_nat_nat(F,N5) = zero_zero_nat ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7180_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 )
        <=> ! [N5: nat] : aa_nat_int(F,N5) = zero_zero_int ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7181_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_7182_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_7183_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_7184_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_7185_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_7186_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_7187_sums__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex,A: complex] :
      ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_pl(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_7188_sums__mult__D,axiom,
    ! [C: real,F: fun_nat_real,A: real] :
      ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ic(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_7189_sums__Suc__iff,axiom,
    ! [F: fun_nat_real,S: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_mk(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_7190_sums__Suc,axiom,
    ! [F: fun_nat_real,L: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_mk(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_7191_sums__Suc,axiom,
    ! [F: fun_nat_nat,L: nat] :
      ( pp(aa_nat_bool(sums_nat(aTP_Lamm_mj(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_7192_sums__Suc,axiom,
    ! [F: fun_nat_int,L: int] :
      ( pp(aa_int_bool(sums_int(aTP_Lamm_mi(F)),L))
     => pp(aa_int_bool(sums_int(F),aa_int_int(plus_plus_int(L),aa_nat_int(F,zero_zero_nat)))) ) ).

% sums_Suc
tff(fact_7193_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_lo(F)) ).

% summable_zero_power'
tff(fact_7194_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_lq(F)) ).

% summable_zero_power'
tff(fact_7195_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_qd(F)) ).

% summable_zero_power'
tff(fact_7196_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_lo(F)) ).

% summable_0_powser
tff(fact_7197_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_lq(F)) ).

% summable_0_powser
tff(fact_7198_sums__zero__iff__shift,axiom,
    ! [N: nat,F: fun_nat_complex,S: complex] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => ( aa_nat_complex(F,I3) = zero_zero_complex ) )
     => ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qe(N),F)),S))
      <=> pp(aa_complex_bool(sums_complex(F),S)) ) ) ).

% sums_zero_iff_shift
tff(fact_7199_sums__zero__iff__shift,axiom,
    ! [N: nat,F: fun_nat_real,S: real] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => ( aa_nat_real(F,I3) = zero_zero_real ) )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_qf(N),F)),S))
      <=> pp(aa_real_bool(sums_real(F),S)) ) ) ).

% sums_zero_iff_shift
tff(fact_7200_summable__powser__split__head,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_qg(F),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_qb(F),Z)) ) ).

% summable_powser_split_head
tff(fact_7201_summable__powser__split__head,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_qh(F),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_pz(F),Z)) ) ).

% summable_powser_split_head
tff(fact_7202_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_qb(F),Z))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_qg(F),Z)) ) ).

% powser_split_head(3)
tff(fact_7203_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_pz(F),Z))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_qh(F),Z)) ) ).

% powser_split_head(3)
tff(fact_7204_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,M: nat,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_qi(F),M),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_qb(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7205_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,M: nat,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_qj(F),M),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_pz(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7206_pi__gt__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),pi)) ).

% pi_gt_zero
tff(fact_7207_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_7208_pi__ge__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),pi)) ).

% pi_ge_zero
tff(fact_7209_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_pe(F)) ) ) ).

% summable_norm_comparison_test
tff(fact_7210_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_pr(A2),F)),groups2073611262835488442omplex(F,A2))) ) ).

% sums_If_finite_set
tff(fact_7211_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_pu(A2),F)),groups3539618377306564664at_int(F,A2))) ) ).

% sums_If_finite_set
tff(fact_7212_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_pt(A2),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2))) ) ).

% sums_If_finite_set
tff(fact_7213_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_ps(A2),F)),groups6591440286371151544t_real(F,A2))) ) ).

% sums_If_finite_set
tff(fact_7214_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_pn(P),F)),groups2073611262835488442omplex(F,collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_7215_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_pq(P),F)),groups3539618377306564664at_int(F,collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_7216_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_pp(P),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_7217_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_po(P),F)),groups6591440286371151544t_real(F,collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_7218_sums__finite,axiom,
    ! [N3: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N3))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N3))
           => ( aa_nat_complex(F,N2) = zero_zero_complex ) )
       => pp(aa_complex_bool(sums_complex(F),groups2073611262835488442omplex(F,N3))) ) ) ).

% sums_finite
tff(fact_7219_sums__finite,axiom,
    ! [N3: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N3))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N3))
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => pp(aa_int_bool(sums_int(F),groups3539618377306564664at_int(F,N3))) ) ) ).

% sums_finite
tff(fact_7220_sums__finite,axiom,
    ! [N3: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N3))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N3))
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => pp(aa_nat_bool(sums_nat(F),aa_set_nat_nat(groups3542108847815614940at_nat(F),N3))) ) ) ).

% sums_finite
tff(fact_7221_sums__finite,axiom,
    ! [N3: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N3))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N3))
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => pp(aa_real_bool(sums_real(F),groups6591440286371151544t_real(F,N3))) ) ) ).

% sums_finite
tff(fact_7222_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_hj(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_7223_summable__rabs,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_hj(F))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,suminf_real(F))),suminf_real(aTP_Lamm_hj(F)))) ) ).

% summable_rabs
tff(fact_7224_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_7225_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_7226_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_7227_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_7228_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_7229_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_7230_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,aa_nat_set_nat(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_7231_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),aa_nat_set_nat(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_7232_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,aa_nat_set_nat(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_7233_powser__sums__if,axiom,
    ! [M: nat,Z: complex] : pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_qk(M),Z)),aa_nat_complex(power_power_complex(Z),M))) ).

% powser_sums_if
tff(fact_7234_powser__sums__if,axiom,
    ! [M: nat,Z: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_ql(M),Z)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),M))) ).

% powser_sums_if
tff(fact_7235_powser__sums__if,axiom,
    ! [M: nat,Z: int] : pp(aa_int_bool(sums_int(aa_int_fun_nat_int(aTP_Lamm_qm(M),Z)),aa_nat_int(power_power_int(Z),M))) ).

% powser_sums_if
tff(fact_7236_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,aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
       => summable_int(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_7237_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),aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
       => summable_nat(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_7238_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,aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
       => summable_real(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_7239_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_7240_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_7241_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_7242_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_7243_sums__iff__shift,axiom,
    ! [F: fun_nat_real,N: nat,S: real] :
      ( pp(aa_real_bool(sums_real(aa_nat_fun_nat_real(aTP_Lamm_ma(F),N)),S))
    <=> pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N))))) ) ).

% sums_iff_shift
tff(fact_7244_sums__split__initial__segment,axiom,
    ! [F: fun_nat_real,S: real,N: nat] :
      ( pp(aa_real_bool(sums_real(F),S))
     => pp(aa_real_bool(sums_real(aa_nat_fun_nat_real(aTP_Lamm_ma(F),N)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N))))) ) ).

% sums_split_initial_segment
tff(fact_7245_sums__iff__shift_H,axiom,
    ! [F: fun_nat_real,N: nat,S: real] :
      ( pp(aa_real_bool(sums_real(aa_nat_fun_nat_real(aTP_Lamm_ma(F),N)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N)))))
    <=> pp(aa_real_bool(sums_real(F),S)) ) ).

% sums_iff_shift'
tff(fact_7246_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_mk(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_7247_sums__If__finite__set_H,axiom,
    ! [G: fun_nat_real,S3: real,A2: set_nat,S4: real,F: fun_nat_real] :
      ( pp(aa_real_bool(sums_real(G),S3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( ( S4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S3),groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_qn(G),F),A2)) )
         => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_qo(G),A2),F)),S4)) ) ) ) ).

% sums_If_finite_set'
tff(fact_7248_summable__norm,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_pg(F))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(F))),suminf_real(aTP_Lamm_pg(F)))) ) ).

% summable_norm
tff(fact_7249_summable__norm,axiom,
    ! [F: fun_nat_complex] :
      ( summable_real(aTP_Lamm_pe(F))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(F))),suminf_real(aTP_Lamm_pe(F)))) ) ).

% summable_norm
tff(fact_7250_sum__le__suminf,axiom,
    ! [F: fun_nat_int,I5: set_nat] :
      ( summable_int(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I5))
       => ( ! [N2: nat] :
              ( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I5)))
             => 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,I5)),suminf_int(F))) ) ) ) ).

% sum_le_suminf
tff(fact_7251_sum__le__suminf,axiom,
    ! [F: fun_nat_nat,I5: set_nat] :
      ( summable_nat(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I5))
       => ( ! [N2: nat] :
              ( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I5)))
             => 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),I5)),suminf_nat(F))) ) ) ) ).

% sum_le_suminf
tff(fact_7252_sum__le__suminf,axiom,
    ! [F: fun_nat_real,I5: set_nat] :
      ( summable_real(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I5))
       => ( ! [N2: nat] :
              ( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I5)))
             => 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,I5)),suminf_real(F))) ) ) ) ).

% sum_le_suminf
tff(fact_7253_suminf__split__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(F) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_ma(F),K))),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_split_initial_segment
tff(fact_7254_suminf__minus__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(aa_nat_fun_nat_real(aTP_Lamm_ma(F),K)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_minus_initial_segment
tff(fact_7255_powser__sums__zero,axiom,
    ! [A: fun_nat_complex] : pp(aa_complex_bool(sums_complex(aTP_Lamm_lo(A)),aa_nat_complex(A,zero_zero_nat))) ).

% powser_sums_zero
tff(fact_7256_powser__sums__zero,axiom,
    ! [A: fun_nat_real] : pp(aa_real_bool(sums_real(aTP_Lamm_lq(A)),aa_nat_real(A,zero_zero_nat))) ).

% powser_sums_zero
tff(fact_7257_powser__inside,axiom,
    ! [F: fun_nat_real,X: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_pz(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_pz(F),Z)) ) ) ).

% powser_inside
tff(fact_7258_powser__inside,axiom,
    ! [F: fun_nat_complex,X: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_qb(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_qb(F),Z)) ) ) ).

% powser_inside
tff(fact_7259_sum__less__suminf,axiom,
    ! [F: fun_nat_int,N: nat] :
      ( summable_int(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,M3))) )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_int(F))) ) ) ).

% sum_less_suminf
tff(fact_7260_sum__less__suminf,axiom,
    ! [F: fun_nat_nat,N: nat] :
      ( summable_nat(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,M3))) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_nat(F))) ) ) ).

% sum_less_suminf
tff(fact_7261_sum__less__suminf,axiom,
    ! [F: fun_nat_real,N: nat] :
      ( summable_real(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,M3))) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_real(F))) ) ) ).

% sum_less_suminf
tff(fact_7262_pi__less__4,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) ).

% pi_less_4
tff(fact_7263_pi__ge__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)) ).

% pi_ge_two
tff(fact_7264_pi__half__neq__two,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) != numeral_numeral_real(aa_num_num(bit0,one)) ).

% pi_half_neq_two
tff(fact_7265_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_qb(F),Z))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_qb(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_qg(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_7266_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_pz(F),Z))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_pz(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_qh(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_7267_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_qb(F),Z))
     => ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_qg(F),Z))),Z) = aa_complex_complex(minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_qb(F),Z))),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_7268_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_pz(F),Z))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_qh(F),Z))),Z) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_pz(F),Z))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_7269_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_complex,E: real] :
      ( summable_complex(F)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
       => ~ ! [N8: nat] :
              ~ ! [M2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),M2))
                 => ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N6)))),E)) ) ) ) ).

% summable_partial_sum_bound
tff(fact_7270_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_real,E: real] :
      ( summable_real(F)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
       => ~ ! [N8: nat] :
              ~ ! [M2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),M2))
                 => ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N6)))),E)) ) ) ) ).

% summable_partial_sum_bound
tff(fact_7271_suminf__exist__split,axiom,
    ! [R: real,F: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R))
     => ( summable_real(F)
       => ? [N8: nat] :
          ! [N6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N6))
           => 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_ma(F),N6)))),R)) ) ) ) ).

% suminf_exist_split
tff(fact_7272_suminf__exist__split,axiom,
    ! [R: real,F: fun_nat_complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R))
     => ( summable_complex(F)
       => ? [N8: nat] :
          ! [N6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_qp(F),N6)))),R)) ) ) ) ).

% suminf_exist_split
tff(fact_7273_summable__power__series,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,I3)),one_one_real))
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I3)))
       => ( 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_pz(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_7274_Abel__lemma,axiom,
    ! [R: real,R0: real,A: fun_nat_complex,M4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),R))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R),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_qq(R),A)) ) ) ) ).

% Abel_lemma
tff(fact_7275_summable__ratio__test,axiom,
    ! [C: real,N3: 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,N3),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_7276_summable__ratio__test,axiom,
    ! [C: real,N3: 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,N3),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_7277_sum__less__suminf2,axiom,
    ! [F: fun_nat_int,N: nat,I: nat] :
      ( summable_int(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,M3))) )
       => ( 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,aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_int(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_7278_sum__less__suminf2,axiom,
    ! [F: fun_nat_nat,N: nat,I: nat] :
      ( summable_nat(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,M3))) )
       => ( 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),aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_nat(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_7279_sum__less__suminf2,axiom,
    ! [F: fun_nat_real,N: nat,I: nat] :
      ( summable_real(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,M3))) )
       => ( 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,aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_real(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_7280_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_7281_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_7282_power__half__series,axiom,
    pp(aa_real_bool(sums_real(aTP_Lamm_qr),one_one_real)) ).

% power_half_series
tff(fact_7283_pi__half__neq__zero,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_7284_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),numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% pi_half_less_two
tff(fact_7285_pi__half__le__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% pi_half_le_two
tff(fact_7286_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),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% pi_half_gt_zero
tff(fact_7287_pi__half__ge__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% pi_half_ge_zero
tff(fact_7288_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,numeral_numeral_real(aa_num_num(bit0,one))),pi))),pi)) ).

% m2pi_less_pi
tff(fact_7289_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),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% arctan_ubound
tff(fact_7290_arctan__one,axiom,
    aa_real_real(arctan,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% arctan_one
tff(fact_7291_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),numeral_numeral_real(aa_num_num(bit0,one))))),zero_zero_real)) ).

% minus_pi_half_less_zero
tff(fact_7292_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),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arctan,Y))) ).

% arctan_lbound
tff(fact_7293_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),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),numeral_numeral_real(aa_num_num(bit0,one))))) ) ).

% arctan_bounded
tff(fact_7294_sums__if_H,axiom,
    ! [G: fun_nat_real,X: real] :
      ( pp(aa_real_bool(sums_real(G),X))
     => pp(aa_real_bool(sums_real(aTP_Lamm_qs(G)),X)) ) ).

% sums_if'
tff(fact_7295_sums__if,axiom,
    ! [G: fun_nat_real,X: real,F: fun_nat_real,Y: real] :
      ( pp(aa_real_bool(sums_real(G),X))
     => ( pp(aa_real_bool(sums_real(F),Y))
       => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_qt(G),F)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))) ) ) ).

% sums_if
tff(fact_7296_machin__Euler,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% machin_Euler
tff(fact_7297_machin,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))))))) ).

% machin
tff(fact_7298_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D2: 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))),D2)))),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))),D2)),one_one_nat))))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(F))) ) ) ).

% sum_pos_lt_pair
tff(fact_7299_sin__cos__npi,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),pi)),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% sin_cos_npi
tff(fact_7300_diffs__equiv,axiom,
    ! [C: fun_nat_real,X: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_qu(C),X))
     => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_qv(C),X)),suminf_real(aa_real_fun_nat_real(aTP_Lamm_qu(C),X)))) ) ).

% diffs_equiv
tff(fact_7301_diffs__equiv,axiom,
    ! [C: fun_nat_complex,X: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_qw(C),X))
     => pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_qx(C),X)),suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_qw(C),X)))) ) ).

% diffs_equiv
tff(fact_7302_cos__pi__eq__zero,axiom,
    ! [M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))))),numeral_numeral_real(aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_7303_monoseq__def,axiom,
    ! [X5: fun_nat_real] :
      ( topolo6980174941875973593q_real(X5)
    <=> ( ! [M7: nat,N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,M7)),aa_nat_real(X5,N5))) )
        | ! [M7: nat,N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,N5)),aa_nat_real(X5,M7))) ) ) ) ).

% monoseq_def
tff(fact_7304_monoseq__def,axiom,
    ! [X5: fun_nat_set_nat] :
      ( topolo7278393974255667507et_nat(X5)
    <=> ( ! [M7: nat,N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
           => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,M7)),aa_nat_set_nat(X5,N5))) )
        | ! [M7: nat,N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
           => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,N5)),aa_nat_set_nat(X5,M7))) ) ) ) ).

% monoseq_def
tff(fact_7305_monoseq__def,axiom,
    ! [X5: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X5)
    <=> ( ! [M7: nat,N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,M7)),aa_nat_rat(X5,N5))) )
        | ! [M7: nat,N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,N5)),aa_nat_rat(X5,M7))) ) ) ) ).

% monoseq_def
tff(fact_7306_monoseq__def,axiom,
    ! [X5: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X5)
    <=> ( ! [M7: nat,N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
           => pp(aa_num_bool(ord_less_eq_num(aa_nat_num(X5,M7)),aa_nat_num(X5,N5))) )
        | ! [M7: nat,N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
           => pp(aa_num_bool(ord_less_eq_num(aa_nat_num(X5,N5)),aa_nat_num(X5,M7))) ) ) ) ).

% monoseq_def
tff(fact_7307_monoseq__def,axiom,
    ! [X5: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X5)
    <=> ( ! [M7: nat,N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,M7)),aa_nat_nat(X5,N5))) )
        | ! [M7: nat,N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,N5)),aa_nat_nat(X5,M7))) ) ) ) ).

% monoseq_def
tff(fact_7308_monoseq__def,axiom,
    ! [X5: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X5)
    <=> ( ! [M7: nat,N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,M7)),aa_nat_int(X5,N5))) )
        | ! [M7: nat,N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,N5)),aa_nat_int(X5,M7))) ) ) ) ).

% monoseq_def
tff(fact_7309_monoI2,axiom,
    ! [X5: fun_nat_real] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,N2)),aa_nat_real(X5,M3))) )
     => topolo6980174941875973593q_real(X5) ) ).

% monoI2
tff(fact_7310_monoI2,axiom,
    ! [X5: fun_nat_set_nat] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,N2)),aa_nat_set_nat(X5,M3))) )
     => topolo7278393974255667507et_nat(X5) ) ).

% monoI2
tff(fact_7311_monoI2,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,N2)),aa_nat_rat(X5,M3))) )
     => topolo4267028734544971653eq_rat(X5) ) ).

% monoI2
tff(fact_7312_monoI2,axiom,
    ! [X5: fun_nat_num] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_num_bool(ord_less_eq_num(aa_nat_num(X5,N2)),aa_nat_num(X5,M3))) )
     => topolo1459490580787246023eq_num(X5) ) ).

% monoI2
tff(fact_7313_monoI2,axiom,
    ! [X5: fun_nat_nat] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,N2)),aa_nat_nat(X5,M3))) )
     => topolo4902158794631467389eq_nat(X5) ) ).

% monoI2
tff(fact_7314_monoI2,axiom,
    ! [X5: fun_nat_int] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,N2)),aa_nat_int(X5,M3))) )
     => topolo4899668324122417113eq_int(X5) ) ).

% monoI2
tff(fact_7315_monoI1,axiom,
    ! [X5: fun_nat_real] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,M3)),aa_nat_real(X5,N2))) )
     => topolo6980174941875973593q_real(X5) ) ).

% monoI1
tff(fact_7316_monoI1,axiom,
    ! [X5: fun_nat_set_nat] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,M3)),aa_nat_set_nat(X5,N2))) )
     => topolo7278393974255667507et_nat(X5) ) ).

% monoI1
tff(fact_7317_monoI1,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,M3)),aa_nat_rat(X5,N2))) )
     => topolo4267028734544971653eq_rat(X5) ) ).

% monoI1
tff(fact_7318_monoI1,axiom,
    ! [X5: fun_nat_num] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_num_bool(ord_less_eq_num(aa_nat_num(X5,M3)),aa_nat_num(X5,N2))) )
     => topolo1459490580787246023eq_num(X5) ) ).

% monoI1
tff(fact_7319_monoI1,axiom,
    ! [X5: fun_nat_nat] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,M3)),aa_nat_nat(X5,N2))) )
     => topolo4902158794631467389eq_nat(X5) ) ).

% monoI1
tff(fact_7320_monoI1,axiom,
    ! [X5: fun_nat_int] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,M3)),aa_nat_int(X5,N2))) )
     => topolo4899668324122417113eq_int(X5) ) ).

% monoI1
tff(fact_7321_sin__pi__minus,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),X)) = aa_real_real(sin_real,X) ).

% sin_pi_minus
tff(fact_7322_cos__periodic__pi,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X)) ).

% cos_periodic_pi
tff(fact_7323_cos__periodic__pi2,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),X)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X)) ).

% cos_periodic_pi2
tff(fact_7324_sin__periodic__pi,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_periodic_pi
tff(fact_7325_sin__periodic__pi2,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),X)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_periodic_pi2
tff(fact_7326_cos__minus__pi,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X)) ).

% cos_minus_pi
tff(fact_7327_cos__pi__minus,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),X)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X)) ).

% cos_pi_minus
tff(fact_7328_sin__minus__pi,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_minus_pi
tff(fact_7329_sin__cos__squared__add3,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,X))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(sin_complex,X))) = one_one_complex ).

% sin_cos_squared_add3
tff(fact_7330_sin__cos__squared__add3,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,X))) = one_one_real ).

% sin_cos_squared_add3
tff(fact_7331_sin__npi2,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))) = zero_zero_real ).

% sin_npi2
tff(fact_7332_sin__npi,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).

% sin_npi
tff(fact_7333_sin__npi__int,axiom,
    ! [N: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,N))) = zero_zero_real ).

% sin_npi_int
tff(fact_7334_cos__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_half
tff(fact_7335_sin__two__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)) = zero_zero_real ).

% sin_two_pi
tff(fact_7336_sin__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) = one_one_real ).

% sin_pi_half
tff(fact_7337_cos__two__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)) = one_one_real ).

% cos_two_pi
tff(fact_7338_cos__periodic,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(cos_real,X) ).

% cos_periodic
tff(fact_7339_sin__periodic,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(sin_real,X) ).

% sin_periodic
tff(fact_7340_cos__2pi__minus,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),X)) = aa_real_real(cos_real,X) ).

% cos_2pi_minus
tff(fact_7341_cos__npi,axiom,
    ! [N: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% cos_npi
tff(fact_7342_cos__npi2,axiom,
    ! [N: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% cos_npi2
tff(fact_7343_sin__cos__squared__add2,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_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(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add2
tff(fact_7344_sin__cos__squared__add2,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add2
tff(fact_7345_sin__cos__squared__add,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_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(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add
tff(fact_7346_sin__cos__squared__add,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add
tff(fact_7347_sin__2npi,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_7348_cos__2npi,axiom,
    ! [N: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)) = one_one_real ).

% cos_2npi
tff(fact_7349_sin__2pi__minus,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),X)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_2pi_minus
tff(fact_7350_sin__int__2pin,axiom,
    ! [N: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,N))) = zero_zero_real ).

% sin_int_2pin
tff(fact_7351_cos__int__2pin,axiom,
    ! [N: int] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,N))) = one_one_real ).

% cos_int_2pin
tff(fact_7352_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_7353_sin__3over2__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sin_3over2_pi
tff(fact_7354_cos__npi__int,axiom,
    ! [N: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,N))) = one_one_real ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,N))) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% cos_npi_int
tff(fact_7355_sin__diff,axiom,
    ! [X: real,Y: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(sin_real,Y))) ).

% sin_diff
tff(fact_7356_polar__Ex,axiom,
    ! [X: real,Y: real] :
    ? [R3: real,A4: real] :
      ( ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),aa_real_real(cos_real,A4)) )
      & ( Y = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),aa_real_real(sin_real,A4)) ) ) ).

% polar_Ex
tff(fact_7357_sin__add,axiom,
    ! [X: real,Y: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(sin_real,Y))) ).

% sin_add
tff(fact_7358_cos__add,axiom,
    ! [X: real,Y: real] : aa_real_real(cos_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(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))) ).

% cos_add
tff(fact_7359_cos__diff,axiom,
    ! [X: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))) ).

% cos_diff
tff(fact_7360_sin__double,axiom,
    ! [X: complex] : aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,X))),aa_complex_complex(cos_complex,X)) ).

% sin_double
tff(fact_7361_sin__double,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(sin_real,X))),aa_real_real(cos_real,X)) ).

% sin_double
tff(fact_7362_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_7363_sin__x__le__x,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_eq_real,aa_real_real(sin_real,X)),X)) ) ).

% sin_x_le_x
tff(fact_7364_sin__le__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X)),one_one_real)) ).

% sin_le_one
tff(fact_7365_cos__le__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X)),one_one_real)) ).

% cos_le_one
tff(fact_7366_abs__sin__x__le__abs__x,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,X))),aa_real_real(abs_abs_real,X))) ).

% abs_sin_x_le_abs_x
tff(fact_7367_sin__cos__le1,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))))),one_one_real)) ).

% sin_cos_le1
tff(fact_7368_sin__squared__eq,axiom,
    ! [X: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_7369_sin__squared__eq,axiom,
    ! [X: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_7370_cos__squared__eq,axiom,
    ! [X: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_7371_cos__squared__eq,axiom,
    ! [X: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = 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(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_7372_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_7373_sin__x__ge__neg__x,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_eq_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(sin_real,X))) ) ).

% sin_x_ge_neg_x
tff(fact_7374_sin__ge__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_eq_real,X),pi))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).

% sin_ge_zero
tff(fact_7375_sin__ge__minus__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(sin_real,X))) ).

% sin_ge_minus_one
tff(fact_7376_cos__inj__pi,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))
           => ( ( aa_real_real(cos_real,X) = aa_real_real(cos_real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_7377_cos__mono__le__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_eq_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_7378_cos__monotone__0__pi__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,Y),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,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_7379_cos__ge__minus__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(cos_real,X))) ).

% cos_ge_minus_one
tff(fact_7380_abs__sin__le__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,X))),one_one_real)) ).

% abs_sin_le_one
tff(fact_7381_abs__cos__le__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(cos_real,X))),one_one_real)) ).

% abs_cos_le_one
tff(fact_7382_sin__times__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W),Z))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_7383_sin__times__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_7384_sin__times__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W),Z))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_7385_sin__times__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_7386_cos__times__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W),Z))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_7387_cos__times__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_7388_sin__plus__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_7389_sin__plus__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_7390_sin__diff__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_7391_sin__diff__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_7392_cos__diff__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Z),W)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_7393_cos__diff__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one)))))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),W)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_7394_cos__double,axiom,
    ! [X: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_7395_cos__double,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_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(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_7396_cos__double__sin,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_7397_cos__double__sin,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),W)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_7398_cos__two__neq__zero,axiom,
    aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one))) != zero_zero_real ).

% cos_two_neq_zero
tff(fact_7399_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_7400_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_7401_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_7402_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_7403_cos__monotone__minus__pi__0_H,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_eq_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_eq_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,X))) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_7404_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ? [I4: int] : X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I4)),pi) ) ).

% sin_zero_iff_int2
tff(fact_7405_diffs__def,axiom,
    ! [C: fun_nat_rat,X4: nat] : aa_nat_rat(diffs_rat(C),X4) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,X4))),aa_nat_rat(C,aa_nat_nat(suc,X4))) ).

% diffs_def
tff(fact_7406_diffs__def,axiom,
    ! [C: fun_nat_int,X4: nat] : aa_nat_int(diffs_int(C),X4) = aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,X4))),aa_nat_int(C,aa_nat_nat(suc,X4))) ).

% diffs_def
tff(fact_7407_diffs__def,axiom,
    ! [C: fun_nat_real,X4: nat] : aa_nat_real(diffs_real(C),X4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,X4))),aa_nat_real(C,aa_nat_nat(suc,X4))) ).

% diffs_def
tff(fact_7408_diffs__def,axiom,
    ! [C: fun_nat_complex,X4: nat] : aa_nat_complex(diffs_complex(C),X4) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,X4))),aa_nat_complex(C,aa_nat_nat(suc,X4))) ).

% diffs_def
tff(fact_7409_sincos__total__pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),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)))) = one_one_real )
       => ? [T4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),pi))
            & ( X = aa_real_real(cos_real,T4) )
            & ( Y = aa_real_real(sin_real,T4) ) ) ) ) ).

% sincos_total_pi
tff(fact_7410_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),numeral_numeral_real(aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),pi)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_7411_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),numeral_numeral_real(aa_num_num(bit0,one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),pi)),numeral_numeral_real(aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_7412_termdiff__converges__all,axiom,
    ! [C: fun_nat_complex,X: complex] :
      ( ! [X2: complex] : summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_qb(C),X2))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_qw(C),X)) ) ).

% termdiff_converges_all
tff(fact_7413_termdiff__converges__all,axiom,
    ! [C: fun_nat_real,X: real] :
      ( ! [X2: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_pz(C),X2))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_qu(C),X)) ) ).

% termdiff_converges_all
tff(fact_7414_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),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_7415_cos__two__less__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one)))),zero_zero_real)) ).

% cos_two_less_zero
tff(fact_7416_cos__is__zero,axiom,
    ? [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),numeral_numeral_real(aa_num_num(bit0,one))))
      & ( aa_real_real(cos_real,X2) = zero_zero_real )
      & ! [Y3: real] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),numeral_numeral_real(aa_num_num(bit0,one))))
            & ( aa_real_real(cos_real,Y3) = zero_zero_real ) )
         => ( Y3 = X2 ) ) ) ).

% cos_is_zero
tff(fact_7417_cos__two__le__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one)))),zero_zero_real)) ).

% cos_two_le_zero
tff(fact_7418_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_7419_cos__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ? [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),pi))
            & ( aa_real_real(cos_real,X2) = Y )
            & ! [Y3: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),pi))
                  & ( aa_real_real(cos_real,Y3) = Y ) )
               => ( Y3 = X2 ) ) ) ) ) ).

% cos_total
tff(fact_7420_sincos__total__pi__half,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,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 )
         => ? [T4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T4))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
              & ( X = aa_real_real(cos_real,T4) )
              & ( Y = aa_real_real(sin_real,T4) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_7421_sincos__total__2pi__le,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 )
     => ? [T4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T4))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)))
          & ( X = aa_real_real(cos_real,T4) )
          & ( Y = aa_real_real(sin_real,T4) ) ) ) ).

% sincos_total_2pi_le
tff(fact_7422_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 )
     => ~ ! [T4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)))
             => ( ( X = aa_real_real(cos_real,T4) )
               => ( Y != aa_real_real(sin_real,T4) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_7423_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_7424_cos__times__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W),Z))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_7425_cos__times__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_7426_cos__plus__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_7427_cos__plus__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_7428_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),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_7429_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,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_7430_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),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,numeral_numeral_real(aa_num_num(bit0,one))),X))),one_one_real)) ) ) ).

% cos_double_less_one
tff(fact_7431_sin__30,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_30
tff(fact_7432_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),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_7433_sin__monotone__2pi__le,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),numeral_numeral_real(aa_num_num(bit0,one))))),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_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),numeral_numeral_real(aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X))) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_7434_sin__mono__le__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),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),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),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),numeral_numeral_real(aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_7435_sin__inj__pi,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),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),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),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),numeral_numeral_real(aa_num_num(bit0,one)))))
           => ( ( aa_real_real(sin_real,X) = aa_real_real(sin_real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_7436_cos__60,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_60
tff(fact_7437_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
    <=> ? [X3: int] : X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,X3)),numeral_numeral_real(aa_num_num(bit0,one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_7438_cos__double__cos,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_complex) ).

% cos_double_cos
tff(fact_7439_cos__double__cos,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),W)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_real) ).

% cos_double_cos
tff(fact_7440_cos__treble__cos,axiom,
    ! [X: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),X)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),aa_complex_complex(cos_complex,X))) ).

% cos_treble_cos
tff(fact_7441_cos__treble__cos,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,one))),X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,one))),aa_real_real(cos_real,X))) ).

% cos_treble_cos
tff(fact_7442_termdiff__converges,axiom,
    ! [X: real,K5: real,C: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),K5))
     => ( ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X2)),K5))
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_pz(C),X2)) )
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_qy(X),C)) ) ) ).

% termdiff_converges
tff(fact_7443_termdiff__converges,axiom,
    ! [X: complex,K5: real,C: fun_nat_complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),K5))
     => ( ! [X2: complex] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X2)),K5))
           => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_qb(C),X2)) )
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qz(X),C)) ) ) ).

% termdiff_converges
tff(fact_7444_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,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_7445_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)),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_7446_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),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),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_7447_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),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),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),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),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_7448_sin__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ? [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
            & ( aa_real_real(sin_real,X2) = Y )
            & ! [Y3: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
                  & ( aa_real_real(sin_real,Y3) = Y ) )
               => ( Y3 = X2 ) ) ) ) ) ).

% sin_total
tff(fact_7449_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),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),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_7450_cos__ge__zero,axiom,
    ! [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),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),numeral_numeral_real(aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,X))) ) ) ).

% cos_ge_zero
tff(fact_7451_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
    <=> ( ? [X3: nat] : X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X3)),numeral_numeral_real(aa_num_num(bit0,one)))),pi)
        | ? [X3: nat] : X = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X3)),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_7452_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_7453_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ? [I4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I4))
          & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_7454_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = zero_zero_real )
    <=> ? [I4: int] :
          ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I4))
          & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_7455_sin__zero__lemma,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( aa_real_real(sin_real,X) = zero_zero_real )
       => ? [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_7456_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ( ? [N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5))
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N5)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) )
        | ? [N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5))
            & ( X = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N5)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_7457_cos__zero__lemma,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( aa_real_real(cos_real,X) = zero_zero_real )
       => ? [N2: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_7458_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = zero_zero_real )
    <=> ( ? [N5: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5))
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N5)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) )
        | ? [N5: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5))
            & ( X = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N5)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_7459_mono__SucI1,axiom,
    ! [X5: fun_nat_real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,N2)),aa_nat_real(X5,aa_nat_nat(suc,N2))))
     => topolo6980174941875973593q_real(X5) ) ).

% mono_SucI1
tff(fact_7460_mono__SucI1,axiom,
    ! [X5: fun_nat_set_nat] :
      ( ! [N2: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,N2)),aa_nat_set_nat(X5,aa_nat_nat(suc,N2))))
     => topolo7278393974255667507et_nat(X5) ) ).

% mono_SucI1
tff(fact_7461_mono__SucI1,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,N2)),aa_nat_rat(X5,aa_nat_nat(suc,N2))))
     => topolo4267028734544971653eq_rat(X5) ) ).

% mono_SucI1
tff(fact_7462_mono__SucI1,axiom,
    ! [X5: fun_nat_num] :
      ( ! [N2: nat] : pp(aa_num_bool(ord_less_eq_num(aa_nat_num(X5,N2)),aa_nat_num(X5,aa_nat_nat(suc,N2))))
     => topolo1459490580787246023eq_num(X5) ) ).

% mono_SucI1
tff(fact_7463_mono__SucI1,axiom,
    ! [X5: fun_nat_nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,N2)),aa_nat_nat(X5,aa_nat_nat(suc,N2))))
     => topolo4902158794631467389eq_nat(X5) ) ).

% mono_SucI1
tff(fact_7464_mono__SucI1,axiom,
    ! [X5: fun_nat_int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,N2)),aa_nat_int(X5,aa_nat_nat(suc,N2))))
     => topolo4899668324122417113eq_int(X5) ) ).

% mono_SucI1
tff(fact_7465_mono__SucI2,axiom,
    ! [X5: fun_nat_real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,aa_nat_nat(suc,N2))),aa_nat_real(X5,N2)))
     => topolo6980174941875973593q_real(X5) ) ).

% mono_SucI2
tff(fact_7466_mono__SucI2,axiom,
    ! [X5: fun_nat_set_nat] :
      ( ! [N2: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,aa_nat_nat(suc,N2))),aa_nat_set_nat(X5,N2)))
     => topolo7278393974255667507et_nat(X5) ) ).

% mono_SucI2
tff(fact_7467_mono__SucI2,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,aa_nat_nat(suc,N2))),aa_nat_rat(X5,N2)))
     => topolo4267028734544971653eq_rat(X5) ) ).

% mono_SucI2
tff(fact_7468_mono__SucI2,axiom,
    ! [X5: fun_nat_num] :
      ( ! [N2: nat] : pp(aa_num_bool(ord_less_eq_num(aa_nat_num(X5,aa_nat_nat(suc,N2))),aa_nat_num(X5,N2)))
     => topolo1459490580787246023eq_num(X5) ) ).

% mono_SucI2
tff(fact_7469_mono__SucI2,axiom,
    ! [X5: fun_nat_nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,aa_nat_nat(suc,N2))),aa_nat_nat(X5,N2)))
     => topolo4902158794631467389eq_nat(X5) ) ).

% mono_SucI2
tff(fact_7470_mono__SucI2,axiom,
    ! [X5: fun_nat_int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,aa_nat_nat(suc,N2))),aa_nat_int(X5,N2)))
     => topolo4899668324122417113eq_int(X5) ) ).

% mono_SucI2
tff(fact_7471_monoseq__Suc,axiom,
    ! [X5: fun_nat_real] :
      ( topolo6980174941875973593q_real(X5)
    <=> ( ! [N5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,N5)),aa_nat_real(X5,aa_nat_nat(suc,N5))))
        | ! [N5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,aa_nat_nat(suc,N5))),aa_nat_real(X5,N5))) ) ) ).

% monoseq_Suc
tff(fact_7472_monoseq__Suc,axiom,
    ! [X5: fun_nat_set_nat] :
      ( topolo7278393974255667507et_nat(X5)
    <=> ( ! [N5: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,N5)),aa_nat_set_nat(X5,aa_nat_nat(suc,N5))))
        | ! [N5: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,aa_nat_nat(suc,N5))),aa_nat_set_nat(X5,N5))) ) ) ).

% monoseq_Suc
tff(fact_7473_monoseq__Suc,axiom,
    ! [X5: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X5)
    <=> ( ! [N5: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,N5)),aa_nat_rat(X5,aa_nat_nat(suc,N5))))
        | ! [N5: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,aa_nat_nat(suc,N5))),aa_nat_rat(X5,N5))) ) ) ).

% monoseq_Suc
tff(fact_7474_monoseq__Suc,axiom,
    ! [X5: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X5)
    <=> ( ! [N5: nat] : pp(aa_num_bool(ord_less_eq_num(aa_nat_num(X5,N5)),aa_nat_num(X5,aa_nat_nat(suc,N5))))
        | ! [N5: nat] : pp(aa_num_bool(ord_less_eq_num(aa_nat_num(X5,aa_nat_nat(suc,N5))),aa_nat_num(X5,N5))) ) ) ).

% monoseq_Suc
tff(fact_7475_monoseq__Suc,axiom,
    ! [X5: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X5)
    <=> ( ! [N5: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,N5)),aa_nat_nat(X5,aa_nat_nat(suc,N5))))
        | ! [N5: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,aa_nat_nat(suc,N5))),aa_nat_nat(X5,N5))) ) ) ).

% monoseq_Suc
tff(fact_7476_monoseq__Suc,axiom,
    ! [X5: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X5)
    <=> ( ! [N5: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,N5)),aa_nat_int(X5,aa_nat_nat(suc,N5))))
        | ! [N5: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,aa_nat_nat(suc,N5))),aa_nat_int(X5,N5))) ) ) ).

% monoseq_Suc
tff(fact_7477_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))
       => ? [T4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),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_ra,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),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),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_7478_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))
       => ? [T4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),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_ra,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),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),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_7479_Maclaurin__cos__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,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_ra,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),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),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_expansion
tff(fact_7480_sin__paired,axiom,
    ! [X: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_rb,X)),aa_real_real(sin_real,X))) ).

% sin_paired
tff(fact_7481_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_7482_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,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,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,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_7483_tan__periodic__pi,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),pi)) = aa_real_real(tan_real,X) ).

% tan_periodic_pi
tff(fact_7484_fact__Suc,axiom,
    ! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N))),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).

% fact_Suc
tff(fact_7485_fact__Suc,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,N)) = aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))),aa_nat_int(semiri1406184849735516958ct_int,N)) ).

% fact_Suc
tff(fact_7486_fact__Suc,axiom,
    ! [N: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,N))),aa_nat_complex(semiri5044797733671781792omplex,N)) ).

% fact_Suc
tff(fact_7487_fact__Suc,axiom,
    ! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))),aa_nat_real(semiri2265585572941072030t_real,N)) ).

% fact_Suc
tff(fact_7488_fact__Suc,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N))),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ).

% fact_Suc
tff(fact_7489_tan__npi,axiom,
    ! [N: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).

% tan_npi
tff(fact_7490_tan__periodic__n,axiom,
    ! [X: real,N: num] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(N)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_n
tff(fact_7491_tan__periodic__nat,axiom,
    ! [X: real,N: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_nat
tff(fact_7492_tan__periodic__int,axiom,
    ! [X: real,I: int] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_int
tff(fact_7493_fact__2,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% fact_2
tff(fact_7494_fact__2,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% fact_2
tff(fact_7495_fact__2,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_7496_fact__2,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% fact_2
tff(fact_7497_fact__2,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_7498_tan__periodic,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic
tff(fact_7499_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_7500_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_7501_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_7502_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_7503_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_7504_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_7505_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_7506_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_7507_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_7508_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_7509_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_7510_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_7511_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_7512_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_7513_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_7514_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_7515_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_7516_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_7517_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_7518_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_7519_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_7520_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_7521_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_7522_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_7523_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_7524_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_7525_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_7526_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_7527_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_7528_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_7529_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_7530_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_7531_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_7532_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_7533_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_7534_tan__def,axiom,
    ! [X4: complex] : aa_complex_complex(tan_complex,X4) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,X4)),aa_complex_complex(cos_complex,X4)) ).

% tan_def
tff(fact_7535_tan__def,axiom,
    ! [X4: real] : aa_real_real(tan_real,X4) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,X4)),aa_real_real(cos_real,X4)) ).

% tan_def
tff(fact_7536_fact__numeral,axiom,
    ! [K: num] : aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,K)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(K)),aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_7537_fact__numeral,axiom,
    ! [K: num] : aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,K)) = aa_rat_rat(times_times_rat(numeral_numeral_rat(K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_7538_fact__numeral,axiom,
    ! [K: num] : aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(numeral_numeral_nat,K)) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_7539_fact__numeral,axiom,
    ! [K: num] : aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(K)),aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_7540_fact__numeral,axiom,
    ! [K: num] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_7541_square__fact__le__2__fact,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% square_fact_le_2_fact
tff(fact_7542_tan__45,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) = one_one_real ).

% tan_45
tff(fact_7543_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(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_7544_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(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_7545_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_7546_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_7547_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_7548_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(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_7549_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(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_7550_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_7551_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_7552_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_7553_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),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_7554_lemma__tan__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ? [X2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(tan_real,X2))) ) ) ).

% lemma_tan_total
tff(fact_7555_tan__total,axiom,
    ! [Y: real] :
    ? [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X2))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,X2) = 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),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),numeral_numeral_real(aa_num_num(bit0,one)))))
            & ( aa_real_real(tan_real,Y3) = Y ) )
         => ( Y3 = X2 ) ) ) ).

% tan_total
tff(fact_7556_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),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),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_7557_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),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),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),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),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_7558_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),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),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),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),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_7559_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X2))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,X2) = Y ) ) ).

% lemma_tan_total1
tff(fact_7560_tan__minus__45,axiom,
    aa_real_real(tan_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% tan_minus_45
tff(fact_7561_tan__inverse,axiom,
    ! [Y: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(tan_real,Y)) = aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),Y)) ).

% tan_inverse
tff(fact_7562_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_7563_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_7564_tan__total__pos,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ? [X2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
          & ( aa_real_real(tan_real,X2) = Y ) ) ) ).

% tan_total_pos
tff(fact_7565_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),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_7566_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)),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_7567_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),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),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_7568_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),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),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),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),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_7569_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),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_7570_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),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),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_7571_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),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),numeral_numeral_real(aa_num_num(bit0,one)))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,X)) = X ) ) ) ).

% arctan_tan
tff(fact_7572_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),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),numeral_numeral_real(aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y)) = Y ) ) ).

% arctan
tff(fact_7573_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_rc(X),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_real(aa_nat480766339200631653x_real(Diff,zero_zero_nat),zero_zero_complex) ) ) ) ).

% Maclaurin_zero
tff(fact_7574_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_rd(X),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(aa_nat_fun_real_real(Diff,zero_zero_nat),zero_zero_real) ) ) ) ).

% Maclaurin_zero
tff(fact_7575_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_re(X),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_rat_real(aa_nat_fun_rat_real(Diff,zero_zero_nat),zero_zero_rat) ) ) ) ).

% Maclaurin_zero
tff(fact_7576_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_rf(X),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_real(aa_nat_fun_nat_real(Diff,zero_zero_nat),zero_zero_nat) ) ) ) ).

% Maclaurin_zero
tff(fact_7577_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_rg(X),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_int_real(aa_nat_fun_int_real(Diff,zero_zero_nat),zero_zero_int) ) ) ) ).

% Maclaurin_zero
tff(fact_7578_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ? [B8: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_rh(H),J),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B8),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N)),aa_nat_real(semiri2265585572941072030t_real,N)))) ) ).

% Maclaurin_lemma
tff(fact_7579_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_7580_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_7581_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_7582_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_7583_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_7584_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_7585_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))
     => ? [Z4: 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),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))))),Z4))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))))
          & ( aa_real_real(tan_real,Z4) = X ) ) ) ).

% tan_total_pi4
tff(fact_7586_tan__half,axiom,
    ! [X: complex] : aa_complex_complex(tan_complex,X) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X))),aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X))),one_one_complex)) ).

% tan_half
tff(fact_7587_tan__half,axiom,
    ! [X: real] : aa_real_real(tan_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X))),one_one_real)) ).

% tan_half
tff(fact_7588_cos__coeff__def,axiom,
    ! [X4: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X4))
       => ( aa_nat_real(cos_coeff,X4) = 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,X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X4)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X4))
       => ( aa_nat_real(cos_coeff,X4) = zero_zero_real ) ) ) ).

% cos_coeff_def
tff(fact_7589_cos__paired,axiom,
    ! [X: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_ri,X)),aa_real_real(cos_real,X))) ).

% cos_paired
tff(fact_7590_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))
       => ? [T4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),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_rj,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),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),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_7591_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [T4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T4))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),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_rj,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),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),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_7592_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N: nat] :
    ? [T4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,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_rj,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),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),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_expansion2
tff(fact_7593_Maclaurin__sin__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T4: real] : aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rj,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),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),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_expansion
tff(fact_7594_sin__coeff__def,axiom,
    ! [X4: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X4))
       => ( aa_nat_real(sin_coeff,X4) = 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))),X4))
       => ( aa_nat_real(sin_coeff,X4) = 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,X4),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X4)) ) ) ) ).

% sin_coeff_def
tff(fact_7595_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_7596_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_7597_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_7598_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_7599_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_7600_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_7601_fact__div__fact__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R),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),R)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R))) ) ).

% fact_div_fact_le_pow
tff(fact_7602_sin__coeff__Suc,axiom,
    ! [N: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(cos_coeff,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))) ).

% sin_coeff_Suc
tff(fact_7603_cos__coeff__Suc,axiom,
    ! [N: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,N))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))) ).

% cos_coeff_Suc
tff(fact_7604_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_7605_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(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_7606_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(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_7607_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_7608_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_7609_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : 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,N))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).

% fact_fact_dvd_fact
tff(fact_7610_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,N))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).

% fact_fact_dvd_fact
tff(fact_7611_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,N))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).

% fact_fact_dvd_fact
tff(fact_7612_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : 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,N))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).

% fact_fact_dvd_fact
tff(fact_7613_fact__fact__dvd__fact,axiom,
    ! [K: 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,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,N))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).

% fact_fact_dvd_fact
tff(fact_7614_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),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_7615_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),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_7616_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)))
             => ( Z != aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(cos_real,T4)),aa_real_real(sin_real,T4)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_7617_real__sqrt__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(sqrt,X) = aa_real_real(sqrt,Y) )
    <=> ( X = Y ) ) ).

% real_sqrt_eq_iff
tff(fact_7618_real__sqrt__zero,axiom,
    aa_real_real(sqrt,zero_zero_real) = zero_zero_real ).

% real_sqrt_zero
tff(fact_7619_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sqrt,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_7620_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_7621_real__sqrt__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% real_sqrt_le_iff
tff(fact_7622_real__sqrt__one,axiom,
    aa_real_real(sqrt,one_one_real) = one_one_real ).

% real_sqrt_one
tff(fact_7623_real__sqrt__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sqrt,X) = one_one_real )
    <=> ( X = one_one_real ) ) ).

% real_sqrt_eq_1_iff
tff(fact_7624_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_7625_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_7626_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% real_sqrt_le_0_iff
tff(fact_7627_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y)) ) ).

% real_sqrt_ge_0_iff
tff(fact_7628_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_7629_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_7630_real__sqrt__le__1__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ).

% real_sqrt_le_1_iff
tff(fact_7631_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y)) ) ).

% real_sqrt_ge_1_iff
tff(fact_7632_real__sqrt__abs2,axiom,
    ! [X: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)) = aa_real_real(abs_abs_real,X) ).

% real_sqrt_abs2
tff(fact_7633_real__sqrt__mult__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,A)),aa_real_real(sqrt,A)) = aa_real_real(abs_abs_real,A) ).

% real_sqrt_mult_self
tff(fact_7634_real__sqrt__four,axiom,
    aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% real_sqrt_four
tff(fact_7635_real__sqrt__abs,axiom,
    ! [X: real] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,X) ).

% real_sqrt_abs
tff(fact_7636_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( 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(sqrt,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X ) ) ).

% real_sqrt_pow2
tff(fact_7637_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% real_sqrt_pow2_iff
tff(fact_7638_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,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))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,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))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_7639_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_7640_real__sqrt__le__mono,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_eq_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y))) ) ).

% real_sqrt_le_mono
tff(fact_7641_real__sqrt__minus,axiom,
    ! [X: real] : aa_real_real(sqrt,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(sqrt,X)) ).

% real_sqrt_minus
tff(fact_7642_real__sqrt__power,axiom,
    ! [X: real,K: nat] : aa_real_real(sqrt,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(sqrt,X)),K) ).

% real_sqrt_power
tff(fact_7643_real__sqrt__mult,axiom,
    ! [X: real,Y: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)) ).

% real_sqrt_mult
tff(fact_7644_real__sqrt__divide,axiom,
    ! [X: real,Y: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)) ).

% real_sqrt_divide
tff(fact_7645_complex__diff,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_complex_complex(minus_minus_complex(aa_real_complex(aa_rea2649881322057434177omplex(complex2,A),B)),aa_real_complex(aa_rea2649881322057434177omplex(complex2,C),D)) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ).

% complex_diff
tff(fact_7646_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_7647_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( aa_real_real(sqrt,X) = zero_zero_real )
       => ( X = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_7648_real__sqrt__ge__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_eq_real,zero_zero_real),aa_real_real(sqrt,X))) ) ).

% real_sqrt_ge_zero
tff(fact_7649_real__sqrt__ge__one,axiom,
    ! [X: 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_eq_real,one_one_real),aa_real_real(sqrt,X))) ) ).

% real_sqrt_ge_one
tff(fact_7650_Complex__eq__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( aa_real_complex(aa_rea2649881322057434177omplex(complex2,A),B) = numera6690914467698888265omplex(W) )
    <=> ( ( A = numeral_numeral_real(W) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
tff(fact_7651_complex__add,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_complex_complex(plus_plus_complex(aa_real_complex(aa_rea2649881322057434177omplex(complex2,A),B)),aa_real_complex(aa_rea2649881322057434177omplex(complex2,C),D)) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ).

% complex_add
tff(fact_7652_complex__norm,axiom,
    ! [X: real,Y: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(aa_rea2649881322057434177omplex(complex2,X),Y)) = 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))))) ).

% complex_norm
tff(fact_7653_real__div__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(sqrt,X)) = aa_real_real(sqrt,X) ) ) ).

% real_div_sqrt
tff(fact_7654_sqrt__add__le__add__sqrt,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,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_7655_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(sqrt,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))))) ).

% le_real_sqrt_sumsq
tff(fact_7656_Complex__eq__neg__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( aa_real_complex(aa_rea2649881322057434177omplex(complex2,A),B) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_7657_complex__mult,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_complex_complex(times_times_complex(aa_real_complex(aa_rea2649881322057434177omplex(complex2,A),B)),aa_real_complex(aa_rea2649881322057434177omplex(complex2,C),D)) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ).

% complex_mult
tff(fact_7658_sqrt2__less__2,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% sqrt2_less_2
tff(fact_7659_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_7660_sqrt__le__D,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_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))))) ) ).

% sqrt_le_D
tff(fact_7661_real__le__rsqrt,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)))),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(sqrt,Y))) ) ).

% real_le_rsqrt
tff(fact_7662_real__le__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_eq_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_eq_real,aa_real_real(sqrt,X)),Y)) ) ) ) ).

% real_le_lsqrt
tff(fact_7663_real__sqrt__unique,axiom,
    ! [Y: real,X: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( aa_real_real(sqrt,X) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_7664_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,numeral_numeral_real(aa_num_num(bit0,one))))),U)) ) ).

% lemma_real_divide_sqrt_less
tff(fact_7665_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X: real,Y: 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))))) = X )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_7666_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X: real,Y: 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))))) = Y )
     => ( X = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_7667_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_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)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_7668_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),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))))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_7669_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_7670_sqrt__ge__absD,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(sqrt,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)))),Y)) ) ).

% sqrt_ge_absD
tff(fact_7671_cos__45,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_45
tff(fact_7672_sin__45,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_45
tff(fact_7673_tan__60,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one))) ).

% tan_60
tff(fact_7674_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_7675_sqrt__sum__squares__le__sum,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,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)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_7676_sqrt__even__pow2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% sqrt_even_pow2
tff(fact_7677_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,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)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,Y)))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_7678_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),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))))))) ).

% real_sqrt_ge_abs2
tff(fact_7679_real__sqrt__ge__abs1,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(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))))))) ).

% real_sqrt_ge_abs1
tff(fact_7680_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)),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_7681_cos__30,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_30
tff(fact_7682_sin__60,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_60
tff(fact_7683_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_7684_real__sqrt__power__even,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))
     => ( 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(sqrt,X)),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_7685_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,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))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_7686_arith__geo__mean__sqrt,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,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_7687_tan__30,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))) ).

% tan_30
tff(fact_7688_cos__x__y__le__one,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,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)))),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)) ).

% cos_x_y_le_one
tff(fact_7689_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,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,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_7690_cos__arctan,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(arctan,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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% cos_arctan
tff(fact_7691_sin__arctan,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(arctan,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sin_arctan
tff(fact_7692_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),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),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_7693_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,X)))
     => ( aa_real_real(sin_real,X) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_7694_arctan__half,axiom,
    ! [X: real] : aa_real_real(arctan,X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% arctan_half
tff(fact_7695_arcosh__real__def,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => ( aa_real_real(arcosh_real,X) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),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)))) ) ) ).

% arcosh_real_def
tff(fact_7696_arsinh__real__def,axiom,
    ! [X: real] : aa_real_real(arsinh_real,X) = aa_real_real(ln_ln_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_def
tff(fact_7697_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))
       => ? [T4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T4)),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_rk,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T4)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_7698_cos__arcsin,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))
       => ( aa_real_real(cos_real,aa_real_real(arcsin,X)) = 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))))) ) ) ) ).

% cos_arcsin
tff(fact_7699_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
     => ( aa_real_real(sin_real,aa_real_real(arccos,Y)) = 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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_arccos_abs
tff(fact_7700_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_7701_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_7702_exp__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% exp_le_cancel_iff
tff(fact_7703_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_7704_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_7705_one__le__exp__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(exp_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% one_le_exp_iff
tff(fact_7706_exp__le__one__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% exp_le_one_iff
tff(fact_7707_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_7708_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_7709_cos__arccos,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_7710_sin__arcsin,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_7711_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) ).

% arccos_0
tff(fact_7712_arcsin__1,axiom,
    aa_real_real(arcsin,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) ).

% arcsin_1
tff(fact_7713_arcsin__minus__1,axiom,
    aa_real_real(arcsin,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% arcsin_minus_1
tff(fact_7714_norm__exp,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,X))),aa_real_real(exp_real,aa_real_real(real_V7735802525324610683m_real,X)))) ).

% norm_exp
tff(fact_7715_norm__exp,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,X))),aa_real_real(exp_real,aa_complex_real(real_V1022390504157884413omplex,X)))) ).

% norm_exp
tff(fact_7716_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_7717_exp__times__arg__commute,axiom,
    ! [A2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,A2)),A2) = aa_complex_complex(times_times_complex(A2),aa_complex_complex(exp_complex,A2)) ).

% exp_times_arg_commute
tff(fact_7718_exp__times__arg__commute,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,A2)),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(exp_real,A2)) ).

% exp_times_arg_commute
tff(fact_7719_exp__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ? [X2: real] : aa_real_real(exp_real,X2) = Y ) ).

% exp_total
tff(fact_7720_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_7721_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_7722_not__exp__le__zero,axiom,
    ! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),zero_zero_real)) ).

% not_exp_le_zero
tff(fact_7723_exp__ge__zero,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,X))) ).

% exp_ge_zero
tff(fact_7724_mult__exp__exp,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,Y)) = aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(X),Y)) ).

% mult_exp_exp
tff(fact_7725_mult__exp__exp,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)) = aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) ).

% mult_exp_exp
tff(fact_7726_exp__add__commuting,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(times_times_complex(X),Y) = aa_complex_complex(times_times_complex(Y),X) )
     => ( aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,Y)) ) ) ).

% exp_add_commuting
tff(fact_7727_exp__add__commuting,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X) )
     => ( aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)) ) ) ).

% exp_add_commuting
tff(fact_7728_exp__diff,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(exp_complex,aa_complex_complex(minus_minus_complex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,Y)) ).

% exp_diff
tff(fact_7729_exp__diff,axiom,
    ! [X: real,Y: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)) ).

% exp_diff
tff(fact_7730_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_7731_exp__ge__add__one__self,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(exp_real,X))) ).

% exp_ge_add_one_self
tff(fact_7732_exp__minus__inverse,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X))) = one_one_real ).

% exp_minus_inverse
tff(fact_7733_exp__minus__inverse,axiom,
    ! [X: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X))) = one_one_complex ).

% exp_minus_inverse
tff(fact_7734_exp__of__nat2__mult,axiom,
    ! [X: real,N: nat] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,X)),N) ).

% exp_of_nat2_mult
tff(fact_7735_exp__of__nat2__mult,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(X),aa_nat_complex(semiri8010041392384452111omplex,N))) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,X)),N) ).

% exp_of_nat2_mult
tff(fact_7736_exp__of__nat__mult,axiom,
    ! [N: nat,X: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,X)),N) ).

% exp_of_nat_mult
tff(fact_7737_exp__of__nat__mult,axiom,
    ! [N: nat,X: complex] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),X)) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,X)),N) ).

% exp_of_nat_mult
tff(fact_7738_arccos__le__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_eq_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_eq_real,aa_real_real(arccos,Y)),aa_real_real(arccos,X))) ) ) ) ).

% arccos_le_arccos
tff(fact_7739_arccos__le__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_eq_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ) ).

% arccos_le_mono
tff(fact_7740_arccos__eq__iff,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)) )
     => ( ( aa_real_real(arccos,X) = aa_real_real(arccos,Y) )
      <=> ( X = Y ) ) ) ).

% arccos_eq_iff
tff(fact_7741_arcsin__le__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_eq_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_eq_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y))) ) ) ) ).

% arcsin_le_arcsin
tff(fact_7742_arcsin__minus,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))
       => ( aa_real_real(arcsin,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(arcsin,X)) ) ) ) ).

% arcsin_minus
tff(fact_7743_arcsin__le__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_eq_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ).

% arcsin_le_mono
tff(fact_7744_arcsin__eq__iff,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))
       => ( ( aa_real_real(arcsin,X) = aa_real_real(arcsin,Y) )
        <=> ( X = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_7745_exp__ge__add__one__self__aux,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_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(exp_real,X))) ) ).

% exp_ge_add_one_self_aux
tff(fact_7746_lemma__exp__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y))
     => ? [X2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),one_one_real)))
          & ( aa_real_real(exp_real,X2) = Y ) ) ) ).

% lemma_exp_total
tff(fact_7747_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_7748_ln__x__over__x__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,one_one_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(divide_divide_real,aa_real_real(ln_ln_real,Y)),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,X)),X))) ) ) ).

% ln_x_over_x_mono
tff(fact_7749_arccos__lbound,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_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,zero_zero_real),aa_real_real(arccos,Y))) ) ) ).

% arccos_lbound
tff(fact_7750_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_7751_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_7752_arccos__ubound,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_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(arccos,Y)),pi)) ) ) ).

% arccos_ubound
tff(fact_7753_exp__le,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),numeral_numeral_real(aa_num_num(bit1,one)))) ).

% exp_le
tff(fact_7754_arccos__cos,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_eq_real,X),pi))
       => ( aa_real_real(arccos,aa_real_real(cos_real,X)) = X ) ) ) ).

% arccos_cos
tff(fact_7755_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_7756_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_7757_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
     => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ).

% cos_arccos_abs
tff(fact_7758_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Theta)),pi))
     => ( aa_real_real(arccos,aa_real_real(cos_real,Theta)) = aa_real_real(abs_abs_real,Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_7759_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_7760_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_7761_tanh__altdef,axiom,
    ! [X: real] : aa_real_real(tanh_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))) ).

% tanh_altdef
tff(fact_7762_tanh__altdef,axiom,
    ! [X: complex] : aa_complex_complex(tanh_complex,X) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X)))),aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X)))) ).

% tanh_altdef
tff(fact_7763_exp__half__le2,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% exp_half_le2
tff(fact_7764_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_7765_arccos__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_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,zero_zero_real),aa_real_real(arccos,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),pi)) ) ) ) ).

% arccos_bounded
tff(fact_7766_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_7767_exp__double,axiom,
    ! [Z: complex] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z)) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% exp_double
tff(fact_7768_exp__double,axiom,
    ! [Z: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Z)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% exp_double
tff(fact_7769_arccos__cos2,axiom,
    ! [X: 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,aa_real_real(uminus_uminus_real,pi)),X))
       => ( aa_real_real(arccos,aa_real_real(cos_real,X)) = aa_real_real(uminus_uminus_real,X) ) ) ) ).

% arccos_cos2
tff(fact_7770_arccos__minus,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))
       => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,X)) ) ) ) ).

% arccos_minus
tff(fact_7771_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_7772_arccos,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_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,zero_zero_real),aa_real_real(arccos,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),pi))
          & ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_7773_exp__bound__half,axiom,
    ! [Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Z))),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).

% exp_bound_half
tff(fact_7774_exp__bound__half,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z))),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).

% exp_bound_half
tff(fact_7775_arccos__minus__abs,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,X)) ) ) ).

% arccos_minus_abs
tff(fact_7776_exp__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_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_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)))))) ) ) ).

% exp_bound
tff(fact_7777_real__exp__bound__lemma,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_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X)))) ) ) ).

% real_exp_bound_lemma
tff(fact_7778_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_7779_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_7780_arccos__le__pi2,axiom,
    ! [Y: 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,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ).

% arccos_le_pi2
tff(fact_7781_exp__bound__lemma,axiom,
    ! [Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(real_V7735802525324610683m_real,Z))))) ) ).

% exp_bound_lemma
tff(fact_7782_exp__bound__lemma,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_complex_real(real_V1022390504157884413omplex,Z))))) ) ).

% exp_bound_lemma
tff(fact_7783_Maclaurin__exp__le,axiom,
    ! [X: real,N: nat] :
    ? [T4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),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_rk,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T4)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ).

% Maclaurin_exp_le
tff(fact_7784_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),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),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_7785_arcsin__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_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(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ).

% arcsin_bounded
tff(fact_7786_arcsin__ubound,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_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(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ).

% arcsin_ubound
tff(fact_7787_arcsin__lbound,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_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(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))) ) ) ).

% arcsin_lbound
tff(fact_7788_exp__lower__Taylor__quadratic,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_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(exp_real,X))) ) ).

% exp_lower_Taylor_quadratic
tff(fact_7789_arcsin__sin,axiom,
    ! [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),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),numeral_numeral_real(aa_num_num(bit0,one)))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,X)) = X ) ) ) ).

% arcsin_sin
tff(fact_7790_tanh__real__altdef,axiom,
    ! [X: real] : aa_real_real(tanh_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one)))),X)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one)))),X)))) ).

% tanh_real_altdef
tff(fact_7791_le__arcsin__iff,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_eq_real,X),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),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),numeral_numeral_real(aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(arcsin,X)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,Y)),X)) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_7792_arcsin__le__iff,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_eq_real,X),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),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),numeral_numeral_real(aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,X)),Y))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(sin_real,Y))) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_7793_arcsin__pi,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_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(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y)),pi))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_7794_arcsin,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_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(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_7795_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K3: int] : aa_real_real(arccos,aa_real_real(cos_real,Theta)) != aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Theta),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,K3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_7796_sin__arccos,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))
       => ( aa_real_real(sin_real,aa_real_real(arccos,X)) = 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))))) ) ) ) ).

% sin_arccos
tff(fact_7797_pochhammer__double,axiom,
    ! [Z: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N))),aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(Z,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),N)) ).

% pochhammer_double
tff(fact_7798_pochhammer__double,axiom,
    ! [Z: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),N))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),N)) ).

% pochhammer_double
tff(fact_7799_pochhammer__double,axiom,
    ! [Z: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_nat_complex(comm_s2602460028002588243omplex(Z),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))))),N)) ).

% pochhammer_double
tff(fact_7800_of__nat__code,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,N) = semiri7787848453975740701ux_rat(aTP_Lamm_rl,N,zero_zero_rat) ).

% of_nat_code
tff(fact_7801_of__nat__code,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_int_int(aa_nat_fun_int_int(semiri8420488043553186161ux_int(aTP_Lamm_rm),N),zero_zero_int) ).

% of_nat_code
tff(fact_7802_of__nat__code,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(aa_nat_fun_real_real(semiri7260567687927622513x_real(aTP_Lamm_rn),N),zero_zero_real) ).

% of_nat_code
tff(fact_7803_of__nat__code,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_ro),N),zero_zero_nat) ).

% of_nat_code
tff(fact_7804_of__nat__code,axiom,
    ! [N: nat] : aa_nat_complex(semiri8010041392384452111omplex,N) = aa_complex_complex(aa_nat3829984526574172007omplex(semiri2816024913162550771omplex(aTP_Lamm_rp),N),zero_zero_complex) ).

% of_nat_code
tff(fact_7805_gchoose__row__sum__weighted,axiom,
    ! [R: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_rq(R),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),M)) = aa_rat_rat(times_times_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)),numeral_numeral_rat(aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(R),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_7806_gchoose__row__sum__weighted,axiom,
    ! [R: complex,M: nat] : groups2073611262835488442omplex(aTP_Lamm_rr(R),aa_nat_set_nat(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(R),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_7807_gchoose__row__sum__weighted,axiom,
    ! [R: real,M: nat] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rs,R),aa_nat_set_nat(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))),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,R),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_7808_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,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_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_7809_Maclaurin__sin__bound,axiom,
    ! [X: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,X)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rj,X),aa_nat_set_nat(set_ord_lessThan_nat,N))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,X)),N)))) ).

% Maclaurin_sin_bound
tff(fact_7810_inverse__eq__iff__eq,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B) )
    <=> ( A = B ) ) ).

% inverse_eq_iff_eq
tff(fact_7811_inverse__eq__iff__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(invers8013647133539491842omplex,B) )
    <=> ( A = B ) ) ).

% inverse_eq_iff_eq
tff(fact_7812_inverse__eq__iff__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( inverse_inverse_rat(A) = inverse_inverse_rat(B) )
    <=> ( A = B ) ) ).

% inverse_eq_iff_eq
tff(fact_7813_inverse__inverse__eq,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,aa_real_real(inverse_inverse_real,A)) = A ).

% inverse_inverse_eq
tff(fact_7814_inverse__inverse__eq,axiom,
    ! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = A ).

% inverse_inverse_eq
tff(fact_7815_inverse__inverse__eq,axiom,
    ! [A: rat] : inverse_inverse_rat(inverse_inverse_rat(A)) = A ).

% inverse_inverse_eq
tff(fact_7816_inverse__zero,axiom,
    aa_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

% inverse_zero
tff(fact_7817_inverse__zero,axiom,
    aa_complex_complex(invers8013647133539491842omplex,zero_zero_complex) = zero_zero_complex ).

% inverse_zero
tff(fact_7818_inverse__zero,axiom,
    inverse_inverse_rat(zero_zero_rat) = zero_zero_rat ).

% inverse_zero
tff(fact_7819_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_7820_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_7821_inverse__nonzero__iff__nonzero,axiom,
    ! [A: rat] :
      ( ( inverse_inverse_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_7822_inverse__mult__distrib,axiom,
    ! [A: real,B: 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,A)),aa_real_real(inverse_inverse_real,B)) ).

% inverse_mult_distrib
tff(fact_7823_inverse__mult__distrib,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) ).

% inverse_mult_distrib
tff(fact_7824_inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] : inverse_inverse_rat(aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(inverse_inverse_rat(A)),inverse_inverse_rat(B)) ).

% inverse_mult_distrib
tff(fact_7825_inverse__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(inverse_inverse_real,X) = one_one_real )
    <=> ( X = one_one_real ) ) ).

% inverse_eq_1_iff
tff(fact_7826_inverse__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,X) = one_one_complex )
    <=> ( X = one_one_complex ) ) ).

% inverse_eq_1_iff
tff(fact_7827_inverse__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( inverse_inverse_rat(X) = one_one_rat )
    <=> ( X = one_one_rat ) ) ).

% inverse_eq_1_iff
tff(fact_7828_inverse__1,axiom,
    aa_real_real(inverse_inverse_real,one_one_real) = one_one_real ).

% inverse_1
tff(fact_7829_inverse__1,axiom,
    aa_complex_complex(invers8013647133539491842omplex,one_one_complex) = one_one_complex ).

% inverse_1
tff(fact_7830_inverse__1,axiom,
    inverse_inverse_rat(one_one_rat) = one_one_rat ).

% inverse_1
tff(fact_7831_inverse__divide,axiom,
    ! [A: real,B: real] : aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) ).

% inverse_divide
tff(fact_7832_inverse__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(B),A) ).

% inverse_divide
tff(fact_7833_inverse__divide,axiom,
    ! [A: rat,B: rat] : inverse_inverse_rat(divide_divide_rat(A,B)) = divide_divide_rat(B,A) ).

% inverse_divide
tff(fact_7834_inverse__minus__eq,axiom,
    ! [A: 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)) ).

% inverse_minus_eq
tff(fact_7835_inverse__minus__eq,axiom,
    ! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(invers8013647133539491842omplex,A)) ).

% inverse_minus_eq
tff(fact_7836_inverse__minus__eq,axiom,
    ! [A: rat] : inverse_inverse_rat(uminus_uminus_rat(A)) = uminus_uminus_rat(inverse_inverse_rat(A)) ).

% inverse_minus_eq
tff(fact_7837_abs__inverse,axiom,
    ! [A: 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)) ).

% abs_inverse
tff(fact_7838_abs__inverse,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(invers8013647133539491842omplex,A)) = aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(abs_abs_complex,A)) ).

% abs_inverse
tff(fact_7839_abs__inverse,axiom,
    ! [A: rat] : abs_abs_rat(inverse_inverse_rat(A)) = inverse_inverse_rat(abs_abs_rat(A)) ).

% abs_inverse
tff(fact_7840_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_7841_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_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_7842_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_7843_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_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_7844_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_7845_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,inverse_inverse_rat(A)),inverse_inverse_rat(B)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% inverse_less_iff_less
tff(fact_7846_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_7847_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,inverse_inverse_rat(A)),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_7848_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_7849_inverse__negative__iff__negative,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_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_7850_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_7851_inverse__positive__iff__positive,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_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_7852_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_7853_binomial__Suc__Suc,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K))) ).

% binomial_Suc_Suc
tff(fact_7854_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_7855_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,inverse_inverse_rat(A)),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_7856_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_7857_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,inverse_inverse_rat(A)),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_7858_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_7859_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_7860_right__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(times_times_rat(A),inverse_inverse_rat(A)) = one_one_rat ) ) ).

% right_inverse
tff(fact_7861_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_7862_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_7863_left__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(times_times_rat(inverse_inverse_rat(A)),A) = one_one_rat ) ) ).

% left_inverse
tff(fact_7864_inverse__eq__divide__numeral,axiom,
    ! [W: num] : aa_real_real(inverse_inverse_real,numeral_numeral_real(W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(W)) ).

% inverse_eq_divide_numeral
tff(fact_7865_inverse__eq__divide__numeral,axiom,
    ! [W: num] : aa_complex_complex(invers8013647133539491842omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(W)) ).

% inverse_eq_divide_numeral
tff(fact_7866_inverse__eq__divide__numeral,axiom,
    ! [W: num] : inverse_inverse_rat(numeral_numeral_rat(W)) = divide_divide_rat(one_one_rat,numeral_numeral_rat(W)) ).

% inverse_eq_divide_numeral
tff(fact_7867_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_7868_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_7869_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_7870_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : inverse_inverse_rat(uminus_uminus_rat(numeral_numeral_rat(W))) = divide_divide_rat(one_one_rat,uminus_uminus_rat(numeral_numeral_rat(W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_7871_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: real,X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,Y)),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(inverse_inverse_real,Y)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_7872_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: complex,X: complex] :
      ( ( aa_complex_complex(times_times_complex(Y),X) = aa_complex_complex(times_times_complex(X),Y) )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,Y)),X) = aa_complex_complex(times_times_complex(X),aa_complex_complex(invers8013647133539491842omplex,Y)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_7873_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: rat,X: rat] :
      ( ( aa_rat_rat(times_times_rat(Y),X) = aa_rat_rat(times_times_rat(X),Y) )
     => ( aa_rat_rat(times_times_rat(inverse_inverse_rat(Y)),X) = aa_rat_rat(times_times_rat(X),inverse_inverse_rat(Y)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_7874_power__inverse,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,A)),N) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).

% power_inverse
tff(fact_7875_power__inverse,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,A)),N) = aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(power_power_complex(A),N)) ).

% power_inverse
tff(fact_7876_power__inverse,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(inverse_inverse_rat(A)),N) = inverse_inverse_rat(aa_nat_rat(power_power_rat(A),N)) ).

% power_inverse
tff(fact_7877_inverse__eq__imp__eq,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B) )
     => ( A = B ) ) ).

% inverse_eq_imp_eq
tff(fact_7878_inverse__eq__imp__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(invers8013647133539491842omplex,B) )
     => ( A = B ) ) ).

% inverse_eq_imp_eq
tff(fact_7879_inverse__eq__imp__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( inverse_inverse_rat(A) = inverse_inverse_rat(B) )
     => ( A = B ) ) ).

% inverse_eq_imp_eq
tff(fact_7880_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_7881_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_7882_nonzero__imp__inverse__nonzero,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( inverse_inverse_rat(A) != zero_zero_rat ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_7883_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_7884_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_7885_nonzero__inverse__inverse__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( inverse_inverse_rat(inverse_inverse_rat(A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_7886_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_7887_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_7888_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( inverse_inverse_rat(A) = inverse_inverse_rat(B) )
     => ( ( A != zero_zero_rat )
       => ( ( B != zero_zero_rat )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_7889_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_7890_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_7891_inverse__zero__imp__zero,axiom,
    ! [A: rat] :
      ( ( inverse_inverse_rat(A) = zero_zero_rat )
     => ( A = zero_zero_rat ) ) ).

% inverse_zero_imp_zero
tff(fact_7892_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_7893_field__class_Ofield__inverse__zero,axiom,
    aa_complex_complex(invers8013647133539491842omplex,zero_zero_complex) = zero_zero_complex ).

% field_class.field_inverse_zero
tff(fact_7894_field__class_Ofield__inverse__zero,axiom,
    inverse_inverse_rat(zero_zero_rat) = zero_zero_rat ).

% field_class.field_inverse_zero
tff(fact_7895_real__sqrt__inverse,axiom,
    ! [X: real] : aa_real_real(sqrt,aa_real_real(inverse_inverse_real,X)) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)) ).

% real_sqrt_inverse
tff(fact_7896_norm__inverse__le__norm,axiom,
    ! [R: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R),aa_real_real(real_V7735802525324610683m_real,X)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R))
       => 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,R))) ) ) ).

% norm_inverse_le_norm
tff(fact_7897_norm__inverse__le__norm,axiom,
    ! [R: real,X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R),aa_complex_real(real_V1022390504157884413omplex,X)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R))
       => 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,R))) ) ) ).

% norm_inverse_le_norm
tff(fact_7898_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_7899_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_7900_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),inverse_inverse_rat(A))) ) ).

% positive_imp_inverse_positive
tff(fact_7901_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_7902_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,inverse_inverse_rat(A)),zero_zero_rat)) ) ).

% negative_imp_inverse_negative
tff(fact_7903_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_7904_inverse__positive__imp__positive,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_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_7905_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_7906_inverse__negative__imp__negative,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_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_7907_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_7908_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,inverse_inverse_rat(B)),inverse_inverse_rat(A))) ) ) ).

% less_imp_inverse_less_neg
tff(fact_7909_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_7910_inverse__less__imp__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,inverse_inverse_rat(A)),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_7911_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_7912_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,inverse_inverse_rat(B)),inverse_inverse_rat(A))) ) ) ).

% less_imp_inverse_less
tff(fact_7913_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_7914_inverse__less__imp__less,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,inverse_inverse_rat(A)),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_7915_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_7916_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_7917_nonzero__inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( inverse_inverse_rat(aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(inverse_inverse_rat(B)),inverse_inverse_rat(A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_7918_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_7919_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_7920_nonzero__inverse__minus__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( inverse_inverse_rat(uminus_uminus_rat(A)) = uminus_uminus_rat(inverse_inverse_rat(A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_7921_Suc__times__binomial__eq,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).

% Suc_times_binomial_eq
tff(fact_7922_Suc__times__binomial,axiom,
    ! [K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) ).

% Suc_times_binomial
tff(fact_7923_inverse__numeral__1,axiom,
    aa_real_real(inverse_inverse_real,numeral_numeral_real(one)) = numeral_numeral_real(one) ).

% inverse_numeral_1
tff(fact_7924_inverse__numeral__1,axiom,
    aa_complex_complex(invers8013647133539491842omplex,numera6690914467698888265omplex(one)) = numera6690914467698888265omplex(one) ).

% inverse_numeral_1
tff(fact_7925_inverse__numeral__1,axiom,
    inverse_inverse_rat(numeral_numeral_rat(one)) = numeral_numeral_rat(one) ).

% inverse_numeral_1
tff(fact_7926_inverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = one_one_real )
     => ( aa_real_real(inverse_inverse_real,A) = B ) ) ).

% inverse_unique
tff(fact_7927_inverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = one_one_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) = B ) ) ).

% inverse_unique
tff(fact_7928_inverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = one_one_rat )
     => ( inverse_inverse_rat(A) = B ) ) ).

% inverse_unique
tff(fact_7929_divide__inverse__commute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,B)),A) ).

% divide_inverse_commute
tff(fact_7930_divide__inverse__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,B)),A) ).

% divide_inverse_commute
tff(fact_7931_divide__inverse__commute,axiom,
    ! [A: rat,B: rat] : divide_divide_rat(A,B) = aa_rat_rat(times_times_rat(inverse_inverse_rat(B)),A) ).

% divide_inverse_commute
tff(fact_7932_divide__inverse,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(inverse_inverse_real,B)) ).

% divide_inverse
tff(fact_7933_divide__inverse,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,B)) ).

% divide_inverse
tff(fact_7934_divide__inverse,axiom,
    ! [A: rat,B: rat] : divide_divide_rat(A,B) = aa_rat_rat(times_times_rat(A),inverse_inverse_rat(B)) ).

% divide_inverse
tff(fact_7935_field__class_Ofield__divide__inverse,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(inverse_inverse_real,B)) ).

% field_class.field_divide_inverse
tff(fact_7936_field__class_Ofield__divide__inverse,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,B)) ).

% field_class.field_divide_inverse
tff(fact_7937_field__class_Ofield__divide__inverse,axiom,
    ! [A: rat,B: rat] : divide_divide_rat(A,B) = aa_rat_rat(times_times_rat(A),inverse_inverse_rat(B)) ).

% field_class.field_divide_inverse
tff(fact_7938_inverse__eq__divide,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ).

% inverse_eq_divide
tff(fact_7939_inverse__eq__divide,axiom,
    ! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ).

% inverse_eq_divide
tff(fact_7940_inverse__eq__divide,axiom,
    ! [A: rat] : inverse_inverse_rat(A) = divide_divide_rat(one_one_rat,A) ).

% inverse_eq_divide
tff(fact_7941_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_7942_power__mult__power__inverse__commute,axiom,
    ! [X: real,M: nat,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_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,aa_real_real(inverse_inverse_real,X)),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,X)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M)) ).

% power_mult_power_inverse_commute
tff(fact_7943_power__mult__power__inverse__commute,axiom,
    ! [X: complex,M: nat,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,X)),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,X)),N)),aa_nat_complex(power_power_complex(X),M)) ).

% power_mult_power_inverse_commute
tff(fact_7944_power__mult__power__inverse__commute,axiom,
    ! [X: rat,M: nat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M)),aa_nat_rat(power_power_rat(inverse_inverse_rat(X)),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(inverse_inverse_rat(X)),N)),aa_nat_rat(power_power_rat(X),M)) ).

% power_mult_power_inverse_commute
tff(fact_7945_power__mult__inverse__distrib,axiom,
    ! [X: real,M: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M)),aa_real_real(inverse_inverse_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M)) ).

% power_mult_inverse_distrib
tff(fact_7946_power__mult__inverse__distrib,axiom,
    ! [X: complex,M: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),aa_complex_complex(invers8013647133539491842omplex,X)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,X)),aa_nat_complex(power_power_complex(X),M)) ).

% power_mult_inverse_distrib
tff(fact_7947_power__mult__inverse__distrib,axiom,
    ! [X: rat,M: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M)),inverse_inverse_rat(X)) = aa_rat_rat(times_times_rat(inverse_inverse_rat(X)),aa_nat_rat(power_power_rat(X),M)) ).

% power_mult_inverse_distrib
tff(fact_7948_choose__mult__lemma,axiom,
    ! [M: nat,R: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R)),M)) ).

% choose_mult_lemma
tff(fact_7949_mult__inverse__of__nat__commute,axiom,
    ! [Xa2: nat,X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,Xa2))),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,Xa2))) ).

% mult_inverse_of_nat_commute
tff(fact_7950_mult__inverse__of__nat__commute,axiom,
    ! [Xa2: nat,X: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri8010041392384452111omplex,Xa2))),X) = aa_complex_complex(times_times_complex(X),aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri8010041392384452111omplex,Xa2))) ).

% mult_inverse_of_nat_commute
tff(fact_7951_mult__inverse__of__nat__commute,axiom,
    ! [Xa2: nat,X: rat] : aa_rat_rat(times_times_rat(inverse_inverse_rat(aa_nat_rat(semiri681578069525770553at_rat,Xa2))),X) = aa_rat_rat(times_times_rat(X),inverse_inverse_rat(aa_nat_rat(semiri681578069525770553at_rat,Xa2))) ).

% mult_inverse_of_nat_commute
tff(fact_7952_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_7953_nonzero__abs__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( abs_abs_rat(inverse_inverse_rat(A)) = inverse_inverse_rat(abs_abs_rat(A)) ) ) ).

% nonzero_abs_inverse
tff(fact_7954_binomial__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),R)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R))) ) ).

% binomial_le_pow
tff(fact_7955_mult__inverse__of__int__commute,axiom,
    ! [Xa2: int,X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_int_real(ring_1_of_int_real,Xa2))),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(inverse_inverse_real,aa_int_real(ring_1_of_int_real,Xa2))) ).

% mult_inverse_of_int_commute
tff(fact_7956_mult__inverse__of__int__commute,axiom,
    ! [Xa2: int,X: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(ring_17405671764205052669omplex,Xa2))),X) = aa_complex_complex(times_times_complex(X),aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(ring_17405671764205052669omplex,Xa2))) ).

% mult_inverse_of_int_commute
tff(fact_7957_mult__inverse__of__int__commute,axiom,
    ! [Xa2: int,X: rat] : aa_rat_rat(times_times_rat(inverse_inverse_rat(aa_int_rat(ring_1_of_int_rat,Xa2))),X) = aa_rat_rat(times_times_rat(X),inverse_inverse_rat(aa_int_rat(ring_1_of_int_rat,Xa2))) ).

% mult_inverse_of_int_commute
tff(fact_7958_divide__real__def,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(inverse_inverse_real,Y)) ).

% divide_real_def
tff(fact_7959_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_7960_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_7961_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_7962_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_7963_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_7964_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_7965_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_7966_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_7967_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_7968_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_7969_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_7970_inverse__le__imp__le,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,inverse_inverse_rat(A)),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_7971_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_7972_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,inverse_inverse_rat(B)),inverse_inverse_rat(A))) ) ) ).

% le_imp_inverse_le
tff(fact_7973_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_7974_inverse__le__imp__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,inverse_inverse_rat(A)),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_7975_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_7976_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,inverse_inverse_rat(B)),inverse_inverse_rat(A))) ) ) ).

% le_imp_inverse_le_neg
tff(fact_7977_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_7978_inverse__le__1__iff,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_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_7979_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_7980_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_7981_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),inverse_inverse_rat(A))) ) ) ).

% one_less_inverse
tff(fact_7982_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_7983_one__less__inverse__iff,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_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_7984_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_7985_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_7986_field__class_Ofield__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(times_times_rat(inverse_inverse_rat(A)),A) = one_one_rat ) ) ).

% field_class.field_inverse
tff(fact_7987_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_7988_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_7989_division__ring__inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( plus_plus_rat(inverse_inverse_rat(A),inverse_inverse_rat(B)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(inverse_inverse_rat(A)),plus_plus_rat(A,B))),inverse_inverse_rat(B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_7990_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_7991_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_7992_inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( plus_plus_rat(inverse_inverse_rat(A),inverse_inverse_rat(B)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),inverse_inverse_rat(A))),inverse_inverse_rat(B)) ) ) ) ).

% inverse_add
tff(fact_7993_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_7994_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_7995_division__ring__inverse__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( minus_minus_rat(inverse_inverse_rat(A),inverse_inverse_rat(B)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(inverse_inverse_rat(A)),minus_minus_rat(B,A))),inverse_inverse_rat(B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_7996_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_7997_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_7998_nonzero__inverse__eq__divide,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( inverse_inverse_rat(A) = divide_divide_rat(one_one_rat,A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_7999_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,A)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(suc,A))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,B)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),A)) ).

% Suc_times_binomial_add
tff(fact_8000_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))),aa_nat_nat(suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_8001_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_8002_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ).

% binomial_absorb_comp
tff(fact_8003_gbinomial__pochhammer,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),K))),aa_nat_complex(semiri5044797733671781792omplex,K)) ).

% gbinomial_pochhammer
tff(fact_8004_gbinomial__pochhammer,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(A)),K)),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).

% gbinomial_pochhammer
tff(fact_8005_gbinomial__pochhammer,axiom,
    ! [A: real,K: 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_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)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,A)),K))),aa_nat_real(semiri2265585572941072030t_real,K)) ).

% gbinomial_pochhammer
tff(fact_8006_gbinomial__pochhammer_H,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = divide_divide_rat(aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(minus_minus_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K)),one_one_rat)),K),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).

% gbinomial_pochhammer'
tff(fact_8007_gbinomial__pochhammer_H,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)),aa_nat_complex(semiri5044797733671781792omplex,K)) ).

% gbinomial_pochhammer'
tff(fact_8008_gbinomial__pochhammer_H,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)),aa_nat_real(semiri2265585572941072030t_real,K)) ).

% gbinomial_pochhammer'
tff(fact_8009_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_8010_gbinomial__Suc__Suc,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_8011_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(plus_plus_rat(A,one_one_rat)),aa_nat_nat(suc,K)) = plus_plus_rat(aa_nat_rat(gbinomial_rat(A),K),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_8012_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_8013_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_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),K) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_8014_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_8015_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_8016_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_8017_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_8018_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_8019_inverse__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,inverse_inverse_rat(A)),inverse_inverse_rat(B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_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(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_8020_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_8021_inverse__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,inverse_inverse_rat(A)),inverse_inverse_rat(B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_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(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_8022_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_8023_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),inverse_inverse_rat(A))) ) ) ).

% one_le_inverse
tff(fact_8024_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_8025_inverse__less__1__iff,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_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_8026_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_8027_one__le__inverse__iff,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_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_8028_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_8029_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_8030_inverse__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( minus_minus_rat(inverse_inverse_rat(A),inverse_inverse_rat(B)) = uminus_uminus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(inverse_inverse_rat(A)),minus_minus_rat(A,B))),inverse_inverse_rat(B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_8031_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_8032_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,inverse_inverse_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)))),X)) ) ).

% reals_Archimedean
tff(fact_8033_binomial__absorption,axiom,
    ! [K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ).

% binomial_absorption
tff(fact_8034_gbinomial__addition__formula,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,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(suc,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).

% gbinomial_addition_formula
tff(fact_8035_gbinomial__addition__formula,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).

% gbinomial_addition_formula
tff(fact_8036_gbinomial__addition__formula,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = plus_plus_rat(aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),aa_nat_nat(suc,K)),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K)) ).

% gbinomial_addition_formula
tff(fact_8037_gbinomial__absorb__comp,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_rat(minus_minus_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K)) ).

% gbinomial_absorb_comp
tff(fact_8038_gbinomial__absorb__comp,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),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_absorb_comp
tff(fact_8039_gbinomial__absorb__comp,axiom,
    ! [A: complex,K: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(gbinomial_complex(A),K)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).

% gbinomial_absorb_comp
tff(fact_8040_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_8041_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(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_8042_gbinomial__mult__1_H,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),A) = plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_8043_gbinomial__mult__1_H,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_8044_gbinomial__mult__1_H,axiom,
    ! [A: complex,K: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),A) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,K)),aa_nat_complex(gbinomial_complex(A),K))),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_8045_gbinomial__mult__1,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(A),K)) = plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_8046_gbinomial__mult__1,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_8047_gbinomial__mult__1,axiom,
    ! [A: complex,K: nat] : aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(A),K)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,K)),aa_nat_complex(gbinomial_complex(A),K))),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_8048_forall__pos__mono__1,axiom,
    ! [P: fun_real_bool,E: real] :
      ( ! [D2: real,E2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D2),E2))
         => ( pp(aa_real_bool(P,D2))
           => pp(aa_real_bool(P,E2)) ) )
     => ( ! [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),E))
         => pp(aa_real_bool(P,E)) ) ) ) ).

% forall_pos_mono_1
tff(fact_8049_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_8050_forall__pos__mono,axiom,
    ! [P: fun_real_bool,E: real] :
      ( ! [D2: real,E2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D2),E2))
         => ( pp(aa_real_bool(P,D2))
           => pp(aa_real_bool(P,E2)) ) )
     => ( ! [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),E))
         => pp(aa_real_bool(P,E)) ) ) ) ).

% forall_pos_mono
tff(fact_8051_real__arch__inverse,axiom,
    ! [E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
    <=> ? [N5: nat] :
          ( ( N5 != 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,N5))))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N5))),E)) ) ) ).

% real_arch_inverse
tff(fact_8052_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,X)),X) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)) ) ) ).

% sqrt_divide_self_eq
tff(fact_8053_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_8054_pochhammer__rec,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),N)) ).

% pochhammer_rec
tff(fact_8055_pochhammer__rec,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),N)) ).

% pochhammer_rec
tff(fact_8056_pochhammer__rec,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(A,one_one_rat)),N)) ).

% pochhammer_rec
tff(fact_8057_pochhammer__rec,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),N)) ).

% pochhammer_rec
tff(fact_8058_pochhammer__rec,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_int_int(times_times_int(A),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(plus_plus_int(A),one_one_int)),N)) ).

% pochhammer_rec
tff(fact_8059_pochhammer__rec_H,axiom,
    ! [Z: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(plus_plus_rat(Z,aa_nat_rat(semiri681578069525770553at_rat,N))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)) ).

% pochhammer_rec'
tff(fact_8060_pochhammer__rec_H,axiom,
    ! [Z: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(suc,N)) = aa_int_int(times_times_int(aa_int_int(plus_plus_int(Z),aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(comm_s4660882817536571857er_int(Z),N)) ).

% pochhammer_rec'
tff(fact_8061_pochhammer__rec_H,axiom,
    ! [Z: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,N))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),N)) ).

% pochhammer_rec'
tff(fact_8062_pochhammer__rec_H,axiom,
    ! [Z: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)) ).

% pochhammer_rec'
tff(fact_8063_pochhammer__rec_H,axiom,
    ! [Z: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,N))),aa_nat_complex(comm_s2602460028002588243omplex(Z),N)) ).

% pochhammer_rec'
tff(fact_8064_pochhammer__Suc,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(A),N)),plus_plus_rat(A,aa_nat_rat(semiri681578069525770553at_rat,N))) ).

% pochhammer_Suc
tff(fact_8065_pochhammer__Suc,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_int_int(times_times_int(aa_nat_int(comm_s4660882817536571857er_int(A),N)),aa_int_int(plus_plus_int(A),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% pochhammer_Suc
tff(fact_8066_pochhammer__Suc,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,N))) ).

% pochhammer_Suc
tff(fact_8067_pochhammer__Suc,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).

% pochhammer_Suc
tff(fact_8068_pochhammer__Suc,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(A),N)),aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,N))) ).

% pochhammer_Suc
tff(fact_8069_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = zero_zero_rat )
    <=> ? [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),N))
          & ( A = uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,K2)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_8070_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 )
    <=> ? [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),N))
          & ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K2)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_8071_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = zero_zero_complex )
    <=> ? [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),N))
          & ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K2)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_8072_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_8073_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_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_8074_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_8075_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_8076_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_8077_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_8078_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(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,N))),K) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8079_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_8080_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_8081_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_8082_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_8083_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(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,N))),K) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8084_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_8085_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_8086_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_8087_pochhammer__product_H,axiom,
    ! [Z: rat,N: nat,M: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)),aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(Z,aa_nat_rat(semiri681578069525770553at_rat,N))),M)) ).

% pochhammer_product'
tff(fact_8088_pochhammer__product_H,axiom,
    ! [Z: int,N: nat,M: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_int_int(times_times_int(aa_nat_int(comm_s4660882817536571857er_int(Z),N)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(plus_plus_int(Z),aa_nat_int(semiri1314217659103216013at_int,N))),M)) ).

% pochhammer_product'
tff(fact_8089_pochhammer__product_H,axiom,
    ! [Z: real,N: nat,M: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),N)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,N))),M)) ).

% pochhammer_product'
tff(fact_8090_pochhammer__product_H,axiom,
    ! [Z: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,N))),M)) ).

% pochhammer_product'
tff(fact_8091_pochhammer__product_H,axiom,
    ! [Z: complex,N: nat,M: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),N)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,N))),M)) ).

% pochhammer_product'
tff(fact_8092_summable__exp,axiom,
    ! [X: complex] : summable_complex(aTP_Lamm_rt(X)) ).

% summable_exp
tff(fact_8093_summable__exp,axiom,
    ! [X: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_ru,X)) ).

% summable_exp
tff(fact_8094_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_8095_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(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_8096_binomial__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K6))
     => ( 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))),K6)),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),K6))) ) ) ).

% binomial_mono
tff(fact_8097_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_8098_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_8099_binomial__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K6))
     => ( 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,K6),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ) ) ).

% binomial_antimono
tff(fact_8100_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_8101_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_8102_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_8103_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,inverse_inverse_rat(aa_nat_rat(semiri681578069525770553at_rat,N2))),X)) ) ) ).

% ex_inverse_of_nat_less
tff(fact_8104_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_8105_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_8106_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_8107_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(times_times_rat(aa_nat_rat(power_power_rat(X),N)),aa_nat_rat(power_power_rat(inverse_inverse_rat(X)),M)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_8108_Suc__times__gbinomial,axiom,
    ! [K: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(plus_plus_rat(A,one_one_rat)),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(plus_plus_rat(A,one_one_rat)),aa_nat_rat(gbinomial_rat(A),K)) ).

% Suc_times_gbinomial
tff(fact_8109_Suc__times__gbinomial,axiom,
    ! [K: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) ).

% Suc_times_gbinomial
tff(fact_8110_Suc__times__gbinomial,axiom,
    ! [K: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(gbinomial_complex(A),K)) ).

% Suc_times_gbinomial
tff(fact_8111_gbinomial__absorption,axiom,
    ! [K: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K)) ).

% gbinomial_absorption
tff(fact_8112_gbinomial__absorption,axiom,
    ! [K: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),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_absorption
tff(fact_8113_gbinomial__absorption,axiom,
    ! [K: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).

% gbinomial_absorption
tff(fact_8114_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_8115_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(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(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_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_8116_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_8117_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_8118_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),N) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),M)),aa_nat_rat(comm_s4028243227959126397er_rat(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_8119_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(times_times_int(aa_nat_int(comm_s4660882817536571857er_int(Z),M)),aa_nat_int(comm_s4660882817536571857er_int(aa_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_8120_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_8121_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_8122_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_8123_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_8124_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K6))
     => ( 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))),K6)),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),K6))) ) ) ).

% binomial_strict_mono
tff(fact_8125_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K6))
     => ( 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,K6),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ) ) ).

% binomial_strict_antimono
tff(fact_8126_central__binomial__odd,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_8127_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_8128_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(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))) = 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_8129_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_8130_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_8131_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)) = divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,N),aa_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_8132_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_8133_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_8134_gbinomial__factors,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(plus_plus_rat(A,one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(divide_divide_rat(plus_plus_rat(A,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))),aa_nat_rat(gbinomial_rat(A),K)) ).

% gbinomial_factors
tff(fact_8135_gbinomial__factors,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) ).

% gbinomial_factors
tff(fact_8136_gbinomial__factors,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))),aa_nat_complex(gbinomial_complex(A),K)) ).

% gbinomial_factors
tff(fact_8137_gbinomial__rec,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(plus_plus_rat(A,one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),divide_divide_rat(plus_plus_rat(A,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8138_gbinomial__rec,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8139_gbinomial__rec,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8140_gbinomial__negated__upper,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K)),aa_nat_rat(gbinomial_rat(minus_minus_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,K),A),one_one_rat)),K)) ).

% gbinomial_negated_upper
tff(fact_8141_gbinomial__negated__upper,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = 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)),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,K)),A)),one_one_real)),K)) ).

% gbinomial_negated_upper
tff(fact_8142_gbinomial__negated__upper,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,K)),A)),one_one_complex)),K)) ).

% gbinomial_negated_upper
tff(fact_8143_gbinomial__index__swap,axiom,
    ! [K: nat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K)),aa_nat_rat(gbinomial_rat(minus_minus_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)),K)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)),aa_nat_rat(gbinomial_rat(minus_minus_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),one_one_rat)),N)) ).

% gbinomial_index_swap
tff(fact_8144_gbinomial__index__swap,axiom,
    ! [K: nat,N: nat] : 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)),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),one_one_real)),K)) = 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)),N)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),N)) ).

% gbinomial_index_swap
tff(fact_8145_gbinomial__index__swap,axiom,
    ! [K: nat,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),one_one_complex)),K)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),N)) ).

% gbinomial_index_swap
tff(fact_8146_exp__plus__inverse__exp,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X))))) ).

% exp_plus_inverse_exp
tff(fact_8147_pochhammer__absorb__comp,axiom,
    ! [R: code_integer,K: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(R,aa_nat_Code_integer(semiri4939895301339042750nteger,K))),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R)),K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R),one_one_Code_integer)),K)) ).

% pochhammer_absorb_comp
tff(fact_8148_pochhammer__absorb__comp,axiom,
    ! [R: rat,K: nat] : aa_rat_rat(times_times_rat(minus_minus_rat(R,aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(R)),K)) = aa_rat_rat(times_times_rat(R),aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(uminus_uminus_rat(R),one_one_rat)),K)) ).

% pochhammer_absorb_comp
tff(fact_8149_pochhammer__absorb__comp,axiom,
    ! [R: int,K: nat] : aa_int_int(times_times_int(aa_int_int(minus_minus_int(R),aa_nat_int(semiri1314217659103216013at_int,K))),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,R)),K)) = aa_int_int(times_times_int(R),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,R)),one_one_int)),K)) ).

% pochhammer_absorb_comp
tff(fact_8150_pochhammer__absorb__comp,axiom,
    ! [R: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,R)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,R)),one_one_real)),K)) ).

% pochhammer_absorb_comp
tff(fact_8151_pochhammer__absorb__comp,axiom,
    ! [R: complex,K: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(R),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,R)),K)) = aa_complex_complex(times_times_complex(R),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,R)),one_one_complex)),K)) ).

% pochhammer_absorb_comp
tff(fact_8152_pochhammer__same,axiom,
    ! [N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(semiri3624122377584611663nteger,N)) ).

% pochhammer_same
tff(fact_8153_pochhammer__same,axiom,
    ! [N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,N))),N) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).

% pochhammer_same
tff(fact_8154_pochhammer__same,axiom,
    ! [N: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),N) = aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(semiri1406184849735516958ct_int,N)) ).

% pochhammer_same
tff(fact_8155_pochhammer__same,axiom,
    ! [N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),N) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(semiri5044797733671781792omplex,N)) ).

% pochhammer_same
tff(fact_8156_pochhammer__same,axiom,
    ! [N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),N) = 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)),N)),aa_nat_real(semiri2265585572941072030t_real,N)) ).

% pochhammer_same
tff(fact_8157_choose__two,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% choose_two
tff(fact_8158_gbinomial__minus,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(uminus_uminus_rat(A)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K)),aa_nat_rat(gbinomial_rat(minus_minus_rat(plus_plus_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K)),one_one_rat)),K)) ).

% gbinomial_minus
tff(fact_8159_gbinomial__minus,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(uminus_uminus_real,A)),K) = 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)),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% gbinomial_minus
tff(fact_8160_gbinomial__minus,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% gbinomial_minus
tff(fact_8161_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,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_8162_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_8163_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_8164_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_8165_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) = plus_plus_rat(aa_nat_rat(gbinomial_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(minus_minus_rat(A,one_one_rat)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8166_tan__cot,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),X)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,X)) ).

% tan_cot
tff(fact_8167_pochhammer__minus_H,axiom,
    ! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K)),one_one_Code_integer)),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K)) ).

% pochhammer_minus'
tff(fact_8168_pochhammer__minus_H,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(minus_minus_rat(B,aa_nat_rat(semiri681578069525770553at_rat,K)),one_one_rat)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(B)),K)) ).

% pochhammer_minus'
tff(fact_8169_pochhammer__minus_H,axiom,
    ! [B: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K) = aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K)) ).

% pochhammer_minus'
tff(fact_8170_pochhammer__minus_H,axiom,
    ! [B: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K) = 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)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K)) ).

% pochhammer_minus'
tff(fact_8171_pochhammer__minus_H,axiom,
    ! [B: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K)) ).

% pochhammer_minus'
tff(fact_8172_pochhammer__minus,axiom,
    ! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K)),one_one_Code_integer)),K)) ).

% pochhammer_minus
tff(fact_8173_pochhammer__minus,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(B)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(minus_minus_rat(B,aa_nat_rat(semiri681578069525770553at_rat,K)),one_one_rat)),K)) ).

% pochhammer_minus
tff(fact_8174_pochhammer__minus,axiom,
    ! [B: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K) = aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K)) ).

% pochhammer_minus
tff(fact_8175_pochhammer__minus,axiom,
    ! [B: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K) = 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)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% pochhammer_minus
tff(fact_8176_pochhammer__minus,axiom,
    ! [B: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% pochhammer_minus
tff(fact_8177_real__le__x__sinh,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_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X)))),numeral_numeral_real(aa_num_num(bit0,one))))) ) ).

% real_le_x_sinh
tff(fact_8178_real__le__abs__sinh,axiom,
    ! [X: 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,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X)))),numeral_numeral_real(aa_num_num(bit0,one)))))) ).

% real_le_abs_sinh
tff(fact_8179_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : groups2906978787729119204at_rat(aTP_Lamm_rv(K),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_rat(gbinomial_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_8180_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : groups2073611262835488442omplex(aTP_Lamm_rw(K),aa_nat_set_nat(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_8181_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_rx,K),aa_nat_set_nat(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_8182_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_8183_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_8184_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(times_times_rat(divide_divide_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_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_8185_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_8186_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_8187_fact__double,axiom,
    ! [N: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one)))),N))),aa_nat_complex(semiri5044797733671781792omplex,N)) ).

% fact_double
tff(fact_8188_fact__double,axiom,
    ! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_nat_rat(comm_s4028243227959126397er_rat(divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))),N))),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).

% fact_double
tff(fact_8189_fact__double,axiom,
    ! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = 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,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),N))),aa_nat_real(semiri2265585572941072030t_real,N)) ).

% fact_double
tff(fact_8190_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_8191_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) = divide_divide_rat(set_fo1949268297981939178at_rat(aTP_Lamm_ry(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_8192_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_rz(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_8193_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_sa(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_8194_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N))),aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(Z,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = groups73079841787564623at_rat(aTP_Lamm_sb(Z),aa_nat_set_nat(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_8195_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),numeral_numeral_real(aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_sc,Z),aa_nat_set_nat(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_8196_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_sd(Z),aa_nat_set_nat(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_8197_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) = set_fo1949268297981939178at_rat(aTP_Lamm_se(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_8198_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(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_sf(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_8199_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_sg(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_8200_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_sh(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_8201_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_si(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_8202_gbinomial__partial__row__sum,axiom,
    ! [A: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_rq(A),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_rat_rat(times_times_rat(divide_divide_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),one_one_rat),numeral_numeral_rat(aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8203_gbinomial__partial__row__sum,axiom,
    ! [A: complex,M: nat] : groups2073611262835488442omplex(aTP_Lamm_rr(A),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),one_one_complex)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8204_gbinomial__partial__row__sum,axiom,
    ! [A: real,M: nat] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rs,A),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real)),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8205_atMost__iff,axiom,
    ! [I: real,K: real] :
      ( 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_8206_atMost__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( member_set_nat(I,set_or4236626031148496127et_nat(K))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(I),K)) ) ).

% atMost_iff
tff(fact_8207_atMost__iff,axiom,
    ! [I: rat,K: rat] :
      ( 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_8208_atMost__iff,axiom,
    ! [I: num,K: num] :
      ( member_num(I,set_ord_atMost_num(K))
    <=> pp(aa_num_bool(ord_less_eq_num(I),K)) ) ).

% atMost_iff
tff(fact_8209_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_8210_atMost__iff,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_ord_atMost_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K)) ) ).

% atMost_iff
tff(fact_8211_finite__atMost,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_ord_atMost_nat,K))) ).

% finite_atMost
tff(fact_8212_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_8213_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_8214_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_8215_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_8216_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_8217_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_8218_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_8219_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_8220_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_8221_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_8222_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_8223_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_8224_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_8225_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_8226_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_8227_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_8228_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_8229_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_8230_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_8231_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_8232_atMost__subset__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4236626031148496127et_nat(X),set_or4236626031148496127et_nat(Y))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y)) ) ).

% atMost_subset_iff
tff(fact_8233_atMost__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( 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_8234_atMost__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ord_less_eq_set_num(set_ord_atMost_num(X),set_ord_atMost_num(Y))
    <=> pp(aa_num_bool(ord_less_eq_num(X),Y)) ) ).

% atMost_subset_iff
tff(fact_8235_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_8236_atMost__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_atMost_nat,X)),aa_nat_set_nat(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_8237_dvd__prod__eqI,axiom,
    ! [A2: set_real,A: real,B: nat,F: fun_real_nat] :
      ( finite_finite_real(A2)
     => ( 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_8238_dvd__prod__eqI,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B: nat,F: fun_VEBT_VEBT_nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
       => ( ( B = aa_VEBT_VEBT_nat(F,A) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),groups6361806394783013919BT_nat(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_8239_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_8240_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_8241_dvd__prod__eqI,axiom,
    ! [A2: set_real,A: real,B: int,F: fun_real_int] :
      ( finite_finite_real(A2)
     => ( 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_8242_dvd__prod__eqI,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B: int,F: fun_VEBT_VEBT_int] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
       => ( ( B = aa_VEBT_VEBT_int(F,A) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),groups6359315924273963643BT_int(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_8243_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_8244_dvd__prod__eqI,axiom,
    ! [A2: set_real,A: real,B: code_integer,F: fun_re6978948838062957830nteger] :
      ( finite_finite_real(A2)
     => ( 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_8245_dvd__prod__eqI,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B: code_integer,F: fun_VE9219535225822561060nteger] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
       => ( ( B = aa_VEB5935388676523804373nteger(F,A) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),groups3770682396051356844nteger(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_8246_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_8247_dvd__prodI,axiom,
    ! [A2: set_real,A: real,F: fun_real_nat] :
      ( finite_finite_real(A2)
     => ( 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_8248_dvd__prodI,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_VEBT_VEBT_nat(F,A)),groups6361806394783013919BT_nat(F,A2))) ) ) ).

% dvd_prodI
tff(fact_8249_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_8250_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_8251_dvd__prodI,axiom,
    ! [A2: set_real,A: real,F: fun_real_int] :
      ( finite_finite_real(A2)
     => ( 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_8252_dvd__prodI,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,F: fun_VEBT_VEBT_int] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_VEBT_VEBT_int(F,A)),groups6359315924273963643BT_int(F,A2))) ) ) ).

% dvd_prodI
tff(fact_8253_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_8254_dvd__prodI,axiom,
    ! [A2: set_real,A: real,F: fun_re6978948838062957830nteger] :
      ( finite_finite_real(A2)
     => ( 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_8255_dvd__prodI,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,F: fun_VE9219535225822561060nteger] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_VEB5935388676523804373nteger(F,A)),groups3770682396051356844nteger(F,A2))) ) ) ).

% dvd_prodI
tff(fact_8256_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_8257_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_complex] :
      ( finite_finite_real(S3)
     => ( ( member_real(A,S3)
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_sj(A),B),S3) = aa_real_complex(B,A) ) )
        & ( ~ member_real(A,S3)
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_sj(A),B),S3) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_8258_prod_Odelta_H,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VE5569070535936029429omplex] :
      ( finite5795047828879050333T_VEBT(S3)
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S3))
         => ( groups127312072573709053omplex(aa_fun8982142231764154876omplex(aTP_Lamm_sk(A),B),S3) = aa_VEBT_VEBT_complex(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S3))
         => ( groups127312072573709053omplex(aa_fun8982142231764154876omplex(aTP_Lamm_sk(A),B),S3) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_8259_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_sl(A),B),S3) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_sl(A),B),S3) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_8260_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_sm(A),B),S3) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_sm(A),B),S3) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_8261_prod_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_sn(A),B),S3) = aa_complex_complex(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_sn(A),B),S3) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_8262_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( finite_finite_real(S3)
     => ( ( member_real(A,S3)
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_so(A),B),S3) = aa_real_real(B,A) ) )
        & ( ~ member_real(A,S3)
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_so(A),B),S3) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_8263_prod_Odelta_H,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(S3)
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S3))
         => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aTP_Lamm_sp(A),B),S3) = aa_VEBT_VEBT_real(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S3))
         => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aTP_Lamm_sp(A),B),S3) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_8264_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_sq(A),B),S3) = aa_nat_real(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_sq(A),B),S3) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_8265_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_sr(A),B),S3) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_sr(A),B),S3) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_8266_prod_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_ss(A),B),S3) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_ss(A),B),S3) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_8267_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_complex] :
      ( finite_finite_real(S3)
     => ( ( member_real(A,S3)
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_st(A),B),S3) = aa_real_complex(B,A) ) )
        & ( ~ member_real(A,S3)
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_st(A),B),S3) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_8268_prod_Odelta,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VE5569070535936029429omplex] :
      ( finite5795047828879050333T_VEBT(S3)
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S3))
         => ( groups127312072573709053omplex(aa_fun8982142231764154876omplex(aTP_Lamm_su(A),B),S3) = aa_VEBT_VEBT_complex(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S3))
         => ( groups127312072573709053omplex(aa_fun8982142231764154876omplex(aTP_Lamm_su(A),B),S3) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_8269_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_sv(A),B),S3) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_sv(A),B),S3) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_8270_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_sw(A),B),S3) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_sw(A),B),S3) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_8271_prod_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_sx(A),B),S3) = aa_complex_complex(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_sx(A),B),S3) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_8272_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( finite_finite_real(S3)
     => ( ( member_real(A,S3)
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_sy(A),B),S3) = aa_real_real(B,A) ) )
        & ( ~ member_real(A,S3)
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_sy(A),B),S3) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_8273_prod_Odelta,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(S3)
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S3))
         => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aTP_Lamm_sz(A),B),S3) = aa_VEBT_VEBT_real(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S3))
         => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aTP_Lamm_sz(A),B),S3) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_8274_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ta(A),B),S3) = aa_nat_real(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ta(A),B),S3) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_8275_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_tb(A),B),S3) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_tb(A),B),S3) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_8276_prod_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_tc(A),B),S3) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_tc(A),B),S3) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_8277_Icc__subset__Iic__iff,axiom,
    ! [L: set_nat,H: set_nat,H2: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(L,H),set_or4236626031148496127et_nat(H2))
    <=> ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(L),H))
        | pp(aa_set_nat_bool(ord_less_eq_set_nat(H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8278_Icc__subset__Iic__iff,axiom,
    ! [L: rat,H: rat,H2: rat] :
      ( 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_8279_Icc__subset__Iic__iff,axiom,
    ! [L: num,H: num,H2: num] :
      ( ord_less_eq_set_num(set_or7049704709247886629st_num(L,H),set_ord_atMost_num(H2))
    <=> ( ~ pp(aa_num_bool(ord_less_eq_num(L),H))
        | pp(aa_num_bool(ord_less_eq_num(H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8280_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H: nat,H2: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(L),H)),aa_nat_set_nat(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_8281_Icc__subset__Iic__iff,axiom,
    ! [L: int,H: int,H2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_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_8282_Icc__subset__Iic__iff,axiom,
    ! [L: real,H: real,H2: real] :
      ( 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_8283_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,N)),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_8284_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_8285_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_8286_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_8287_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_real(G,N)) ).

% prod.lessThan_Suc
tff(fact_8288_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_rat(G,N)) ).

% prod.lessThan_Suc
tff(fact_8289_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_nat(G,N)) ).

% prod.lessThan_Suc
tff(fact_8290_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(times_times_int(groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_int(G,N)) ).

% prod.lessThan_Suc
tff(fact_8291_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_8292_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_8293_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_8294_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(times_times_int(groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_8295_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,aa_nat_set_nat(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,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_complex(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_8296_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,aa_nat_set_nat(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,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_8297_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,aa_nat_set_nat(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,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_8298_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_one_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_8299_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,aa_nat_set_nat(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,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_int_int(times_times_int(groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_8300_divide__complex__def,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(divide1717551699836669952omplex(X),Y) = aa_complex_complex(times_times_complex(X),aa_complex_complex(invers8013647133539491842omplex,Y)) ).

% divide_complex_def
tff(fact_8301_infinite__Iic,axiom,
    ! [A: int] : ~ pp(aa_set_int_bool(finite_finite_int,set_ord_atMost_int(A))) ).

% infinite_Iic
tff(fact_8302_prod_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A2: set_nat] : groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_td(G),H),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,A2)),groups708209901874060359at_nat(H,A2)) ).

% prod.distrib
tff(fact_8303_prod_Odistrib,axiom,
    ! [G: fun_nat_int,H: fun_nat_int,A2: set_nat] : groups705719431365010083at_int(aa_fun5565043122120104496at_int(aTP_Lamm_te(G),H),A2) = aa_int_int(times_times_int(groups705719431365010083at_int(G,A2)),groups705719431365010083at_int(H,A2)) ).

% prod.distrib
tff(fact_8304_prod_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A2: set_int] : groups1705073143266064639nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_tf(G),H),A2) = aa_int_int(times_times_int(groups1705073143266064639nt_int(G,A2)),groups1705073143266064639nt_int(H,A2)) ).

% prod.distrib
tff(fact_8305_prod__power__distrib,axiom,
    ! [F: fun_nat_nat,A2: set_nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,groups708209901874060359at_nat(F,A2)),N) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tg(F),N),A2) ).

% prod_power_distrib
tff(fact_8306_prod__power__distrib,axiom,
    ! [F: fun_nat_int,A2: set_nat,N: nat] : aa_nat_int(power_power_int(groups705719431365010083at_int(F,A2)),N) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_th(F),N),A2) ).

% prod_power_distrib
tff(fact_8307_prod__power__distrib,axiom,
    ! [F: fun_int_int,A2: set_int,N: nat] : aa_nat_int(power_power_int(groups1705073143266064639nt_int(F,A2)),N) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_ti(F),N),A2) ).

% prod_power_distrib
tff(fact_8308_prod_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_nat,G: fun_VE597728521808591204at_nat,R2: fun_VE3763956218935105199t_bool] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups6361806394783013919BT_nat(aa_fun1096289392377091562BT_nat(aa_fun7476968357888788569BT_nat(aTP_Lamm_tj(B2),G),R2),A2) = groups708209901874060359at_nat(aa_fun5562694175296226570at_nat(aa_fun8318684123515291377at_nat(aTP_Lamm_tk(A2),G),R2),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_8309_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_nat,G: fun_real_fun_nat_nat,R2: fun_re9121486324101951821t_bool] :
      ( finite_finite_real(A2)
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups4696554848551431203al_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_tl(B2),G),R2),A2) = groups708209901874060359at_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_tm(A2),G),R2),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_8310_prod_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_nat,G: fun_int_fun_nat_nat,R2: 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_to(B2),G),R2),A2) = groups708209901874060359at_nat(aa_fun4133023758092209198at_nat(aa_fun7384697781603784697at_nat(aTP_Lamm_tq(A2),G),R2),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_8311_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B2: set_nat,G: fun_co636942507509095880at_nat,R2: 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_ts(B2),G),R2),A2) = groups708209901874060359at_nat(aa_fun996456248398975532at_nat(aa_fun8754082201773474805at_nat(aTP_Lamm_tv(A2),G),R2),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_8312_prod_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_nat,G: fun_VE7331017413731640256at_int,R2: fun_VE3763956218935105199t_bool] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups6359315924273963643BT_int(aa_fun6141810409722670662BT_int(aa_fun6474440919679335769BT_int(aTP_Lamm_tw(B2),G),R2),A2) = groups705719431365010083at_int(aa_fun1384843155787029862at_int(aa_fun6066863168309489065at_int(aTP_Lamm_ty(A2),G),R2),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_8313_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_nat,G: fun_real_fun_nat_int,R2: fun_re9121486324101951821t_bool] :
      ( finite_finite_real(A2)
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups4694064378042380927al_int(aa_fun1626300625876492518al_int(aa_fun1560246819506852821al_int(aTP_Lamm_tz(B2),G),R2),A2) = groups705719431365010083at_int(aa_fun1494899438858969354at_int(aa_fun6942549246588167161at_int(aTP_Lamm_ub(A2),G),R2),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_8314_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B2: set_nat,G: fun_co7370231399432144932at_int,R2: fun_co2841405903858895691t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups858564598930262913ex_int(aa_fun4872694516007147366ex_int(aa_fun2110580046701903059ex_int(aTP_Lamm_uc(B2),G),R2),A2) = groups705719431365010083at_int(aa_fun6041977265744554632at_int(aa_fun3475035881565545205at_int(aTP_Lamm_ue(A2),G),R2),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_8315_prod_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_int,G: fun_VE294566951569886236nt_int,R2: fun_VE3303673267288458579t_bool] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( groups6359315924273963643BT_int(aa_fun4615758601980977314BT_int(aa_fun627869447348354137BT_int(aTP_Lamm_uf(B2),G),R2),A2) = groups1705073143266064639nt_int(aa_fun5422038952152218214nt_int(aa_fun3638225402958239493nt_int(aTP_Lamm_ug(A2),G),R2),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_8316_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_int,G: fun_real_fun_int_int,R2: fun_re8661203372455305201t_bool] :
      ( finite_finite_real(A2)
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( groups4694064378042380927al_int(aa_fun529217645232493962al_int(aa_fun5832317237790237469al_int(aTP_Lamm_uh(B2),G),R2),A2) = groups1705073143266064639nt_int(aa_fun5532095235224157706nt_int(aa_fun4786959800906006429nt_int(aTP_Lamm_ui(A2),G),R2),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_8317_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B2: set_int,G: fun_co333780937270390912nt_int,R2: fun_co2381122952212249071t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( groups858564598930262913ex_int(aa_fun3700423334941611530ex_int(aa_fun3520494530470439451ex_int(aTP_Lamm_uj(B2),G),R2),A2) = groups1705073143266064639nt_int(aa_fun855801025254967176nt_int(aa_fun315996177946036377nt_int(aTP_Lamm_uk(A2),G),R2),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_8318_mod__prod__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kd(F),A),A2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,groups708209901874060359at_nat(F,A2)),A) ).

% mod_prod_eq
tff(fact_8319_mod__prod__eq,axiom,
    ! [F: fun_nat_int,A: int,A2: set_nat] : aa_int_int(modulo_modulo_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_ul(F),A),A2)),A) = aa_int_int(modulo_modulo_int(groups705719431365010083at_int(F,A2)),A) ).

% mod_prod_eq
tff(fact_8320_mod__prod__eq,axiom,
    ! [F: fun_int_int,A: int,A2: set_int] : aa_int_int(modulo_modulo_int(groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_kc(F),A),A2)),A) = aa_int_int(modulo_modulo_int(groups1705073143266064639nt_int(F,A2)),A) ).

% mod_prod_eq
tff(fact_8321_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),groups129246275422532515t_real(aTP_Lamm_mk(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_8322_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_mh(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_8323_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),groups708209901874060359at_nat(aTP_Lamm_mj(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_8324_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(times_times_int(aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_mi(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_8325_atMost__def,axiom,
    ! [U: real] : set_ord_atMost_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_um,U)) ).

% atMost_def
tff(fact_8326_atMost__def,axiom,
    ! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aTP_Lamm_bk(U)) ).

% atMost_def
tff(fact_8327_atMost__def,axiom,
    ! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_un,U)) ).

% atMost_def
tff(fact_8328_atMost__def,axiom,
    ! [U: num] : set_ord_atMost_num(U) = collect_num(aTP_Lamm_uo(U)) ).

% atMost_def
tff(fact_8329_atMost__def,axiom,
    ! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_up,U)) ).

% atMost_def
tff(fact_8330_atMost__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bg,U)) ).

% atMost_def
tff(fact_8331_prod__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups708209901874060359at_nat(F,A2))) ) ).

% prod_nonneg
tff(fact_8332_prod__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X2))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups705719431365010083at_int(F,A2))) ) ).

% prod_nonneg
tff(fact_8333_prod__nonneg,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(F,X2))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F,A2))) ) ).

% prod_nonneg
tff(fact_8334_prod__mono,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,I3)),aa_nat_real(G,I3))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups129246275422532515t_real(F,A2)),groups129246275422532515t_real(G,A2))) ) ).

% prod_mono
tff(fact_8335_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( ! [I3: real] :
          ( member_real(I3,A2)
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I3)),aa_real_real(G,I3))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F,A2)),groups1681761925125756287l_real(G,A2))) ) ).

% prod_mono
tff(fact_8336_prod__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ! [I3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,I3)),aa_VEBT_VEBT_real(G,I3))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2703838992350267259T_real(F,A2)),groups2703838992350267259T_real(G,A2))) ) ).

% prod_mono
tff(fact_8337_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I3)),aa_int_real(G,I3))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F,A2)),groups2316167850115554303t_real(G,A2))) ) ).

% prod_mono
tff(fact_8338_prod__mono,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I3)),aa_complex_real(G,I3))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups766887009212190081x_real(F,A2)),groups766887009212190081x_real(G,A2))) ) ).

% prod_mono
tff(fact_8339_prod__mono,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3))) ) )
     => 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_8340_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I3: real] :
          ( member_real(I3,A2)
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3))) ) )
     => 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_8341_prod__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( ! [I3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I3)),aa_VEBT_VEBT_rat(G,I3))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5726676334696518183BT_rat(F,A2)),groups5726676334696518183BT_rat(G,A2))) ) ).

% prod_mono
tff(fact_8342_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3))) ) )
     => 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_8343_prod__mono,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I3)),aa_complex_rat(G,I3))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups225925009352817453ex_rat(F,A2)),groups225925009352817453ex_rat(G,A2))) ) ).

% prod_mono
tff(fact_8344_prod__pos,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups708209901874060359at_nat(F,A2))) ) ).

% prod_pos
tff(fact_8345_prod__pos,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,X2))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),groups705719431365010083at_int(F,A2))) ) ).

% prod_pos
tff(fact_8346_prod__pos,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(F,X2))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(F,A2))) ) ).

% prod_pos
tff(fact_8347_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(F,X2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups129246275422532515t_real(F,A2))) ) ).

% prod_ge_1
tff(fact_8348_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,X2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups1681761925125756287l_real(F,A2))) ) ).

% prod_ge_1
tff(fact_8349_prod__ge__1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_VEBT_VEBT_real(F,X2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups2703838992350267259T_real(F,A2))) ) ).

% prod_ge_1
tff(fact_8350_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(F,X2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups2316167850115554303t_real(F,A2))) ) ).

% prod_ge_1
tff(fact_8351_prod__ge__1,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X2: complex] :
          ( pp(aa_set_complex_bool(member_complex(X2),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_complex_real(F,X2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups766887009212190081x_real(F,A2))) ) ).

% prod_ge_1
tff(fact_8352_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,X2))) )
     => 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_8353_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_real_rat(F,X2))) )
     => 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_8354_prod__ge__1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_VEBT_VEBT_rat(F,X2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups5726676334696518183BT_rat(F,A2))) ) ).

% prod_ge_1
tff(fact_8355_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(F,X2))) )
     => 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_8356_prod__ge__1,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X2: complex] :
          ( pp(aa_set_complex_bool(member_complex(X2),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,X2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups225925009352817453ex_rat(F,A2))) ) ).

% prod_ge_1
tff(fact_8357_prod__zero,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
            & ( aa_nat_complex(F,X4) = zero_zero_complex ) )
       => ( groups6464643781859351333omplex(F,A2) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_8358_prod__zero,axiom,
    ! [A2: set_int,F: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
            & ( aa_int_complex(F,X4) = zero_zero_complex ) )
       => ( groups7440179247065528705omplex(F,A2) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_8359_prod__zero,axiom,
    ! [A2: set_complex,F: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ? [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
            & ( aa_complex_complex(F,X4) = zero_zero_complex ) )
       => ( groups3708469109370488835omplex(F,A2) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_8360_prod__zero,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
            & ( aa_nat_real(F,X4) = zero_zero_real ) )
       => ( groups129246275422532515t_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_8361_prod__zero,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))
            & ( aa_int_real(F,X4) = zero_zero_real ) )
       => ( groups2316167850115554303t_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_8362_prod__zero,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))
            & ( aa_complex_real(F,X4) = zero_zero_real ) )
       => ( groups766887009212190081x_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_8363_prod__zero,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))
            & ( aa_nat_rat(F,X4) = zero_zero_rat ) )
       => ( groups73079841787564623at_rat(F,A2) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_8364_prod__zero,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))
            & ( aa_int_rat(F,X4) = zero_zero_rat ) )
       => ( groups1072433553688619179nt_rat(F,A2) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_8365_prod__zero,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))
            & ( aa_complex_rat(F,X4) = zero_zero_rat ) )
       => ( groups225925009352817453ex_rat(F,A2) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_8366_prod__zero,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))
            & ( aa_int_nat(F,X4) = zero_zero_nat ) )
       => ( groups1707563613775114915nt_nat(F,A2) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_8367_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,A: nat,B: nat] : groups6464643781859351333omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_uq(F),A),B),one_one_complex) ).

% prod_atLeastAtMost_code
tff(fact_8368_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B: nat] : groups129246275422532515t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_ur(F),A),B),one_one_real) ).

% prod_atLeastAtMost_code
tff(fact_8369_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups73079841787564623at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = set_fo1949268297981939178at_rat(aTP_Lamm_us(F),A,B,one_one_rat) ).

% prod_atLeastAtMost_code
tff(fact_8370_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : groups708209901874060359at_nat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_ut(F),A),B),one_one_nat) ).

% prod_atLeastAtMost_code
tff(fact_8371_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : groups705719431365010083at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_uu(F),A),B),one_one_int) ).

% prod_atLeastAtMost_code
tff(fact_8372_prod_OatMost__shift,axiom,
    ! [G: fun_nat_real,N: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),groups129246275422532515t_real(aTP_Lamm_mk(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_8373_prod_OatMost__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_mh(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_8374_prod_OatMost__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),groups708209901874060359at_nat(aTP_Lamm_mj(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_8375_prod_OatMost__shift,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_int_int(times_times_int(aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_mi(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_8376_prod_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,P: fun_VEBT_VEBT_bool] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( groups127312072573709053omplex(G,collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_ke(A2),P))) = groups127312072573709053omplex(aa_fun6792494735270171281omplex(aTP_Lamm_uv(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8377_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_complex,P: fun_real_bool] :
      ( finite_finite_real(A2)
     => ( groups713298508707869441omplex(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_kg(A2),P))) = groups713298508707869441omplex(aa_fun6813637976755515871omplex(aTP_Lamm_uw(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8378_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_ki(A2),P))) = groups6464643781859351333omplex(aa_fun8588439219086927967omplex(aTP_Lamm_ux(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8379_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_kk(A2),P))) = groups7440179247065528705omplex(aa_fun3606824691168040799omplex(aTP_Lamm_uy(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8380_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_uz(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8381_prod_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,P: fun_VEBT_VEBT_bool] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( groups2703838992350267259T_real(G,collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_ke(A2),P))) = groups2703838992350267259T_real(aa_fun2072160908118572431T_real(aTP_Lamm_va(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8382_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_real,P: fun_real_bool] :
      ( finite_finite_real(A2)
     => ( groups1681761925125756287l_real(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_kg(A2),P))) = groups1681761925125756287l_real(aa_fun6348708079608492509l_real(aTP_Lamm_vb(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8383_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_ki(A2),P))) = groups129246275422532515t_real(aa_fun1750223370215434333t_real(aTP_Lamm_vc(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8384_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_kk(A2),P))) = groups2316167850115554303t_real(aa_fun6974987029372589917t_real(aTP_Lamm_vd(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8385_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_ve(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8386_power__sum,axiom,
    ! [C: real,F: fun_nat_nat,A2: set_nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups129246275422532515t_real(aa_fun1538629612289296340t_real(aTP_Lamm_vf(C),F),A2) ).

% power_sum
tff(fact_8387_power__sum,axiom,
    ! [C: complex,F: fun_nat_nat,A2: set_nat] : aa_nat_complex(power_power_complex(C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups6464643781859351333omplex(aa_fun1958354139826840918omplex(aTP_Lamm_vg(C),F),A2) ).

% power_sum
tff(fact_8388_power__sum,axiom,
    ! [C: nat,F: fun_nat_nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_vh(C),F),A2) ).

% power_sum
tff(fact_8389_power__sum,axiom,
    ! [C: int,F: fun_nat_nat,A2: set_nat] : aa_nat_int(power_power_int(C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups705719431365010083at_int(aa_fun1613598493704395220at_int(aTP_Lamm_vi(C),F),A2) ).

% power_sum
tff(fact_8390_power__sum,axiom,
    ! [C: int,F: fun_int_nat,A2: set_int] : aa_nat_int(power_power_int(C),groups4541462559716669496nt_nat(F,A2)) = groups1705073143266064639nt_int(aa_fun2328901131235339916nt_int(aTP_Lamm_vj(C),F),A2) ).

% power_sum
tff(fact_8391_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lz(G),K),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_8392_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_int,M: nat,K: nat,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_vk(G),K),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_8393_finite__nat__iff__bounded__le,axiom,
    ! [S3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
    <=> ? [K2: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(S3),aa_nat_set_nat(set_ord_atMost_nat,K2))) ) ).

% finite_nat_iff_bounded_le
tff(fact_8394_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,X2)),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_8395_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X2)),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_8396_prod__le__1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,X2)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2703838992350267259T_real(F,A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_8397_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X2)),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_8398_prod__le__1,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X2: complex] :
          ( pp(aa_set_complex_bool(member_complex(X2),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X2)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups766887009212190081x_real(F,A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_8399_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X2)),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_8400_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,X2)),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_8401_prod__le__1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,X2)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5726676334696518183BT_rat(F,A2)),one_one_rat)) ) ).

% prod_le_1
tff(fact_8402_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X2)),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_8403_prod__le__1,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X2: complex] :
          ( pp(aa_set_complex_bool(member_complex(X2),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X2)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups225925009352817453ex_rat(F,A2)),one_one_rat)) ) ).

% prod_le_1
tff(fact_8404_prod_Orelated,axiom,
    ! [R2: fun_co1378794544173401837x_bool,S3: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,one_one_complex),one_one_complex))
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,X1),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,Y1),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_complex_complex(times_times_complex(X1),Y1)),aa_complex_complex(times_times_complex(X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S3))
         => ( ! [X2: nat] :
                ( pp(aa_set_nat_bool(member_nat(X2),S3))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_nat_complex(H,X2)),aa_nat_complex(G,X2))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,groups6464643781859351333omplex(H,S3)),groups6464643781859351333omplex(G,S3))) ) ) ) ) ).

% prod.related
tff(fact_8405_prod_Orelated,axiom,
    ! [R2: fun_co1378794544173401837x_bool,S3: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,one_one_complex),one_one_complex))
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,X1),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,Y1),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_complex_complex(times_times_complex(X1),Y1)),aa_complex_complex(times_times_complex(X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S3))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_int_complex(H,X2)),aa_int_complex(G,X2))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,groups7440179247065528705omplex(H,S3)),groups7440179247065528705omplex(G,S3))) ) ) ) ) ).

% prod.related
tff(fact_8406_prod_Orelated,axiom,
    ! [R2: fun_co1378794544173401837x_bool,S3: set_complex,H: fun_complex_complex,G: fun_complex_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,one_one_complex),one_one_complex))
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,X1),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,Y1),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_complex_complex(times_times_complex(X1),Y1)),aa_complex_complex(times_times_complex(X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
         => ( ! [X2: complex] :
                ( pp(aa_set_complex_bool(member_complex(X2),S3))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_complex_complex(H,X2)),aa_complex_complex(G,X2))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,groups3708469109370488835omplex(H,S3)),groups3708469109370488835omplex(G,S3))) ) ) ) ) ).

% prod.related
tff(fact_8407_prod_Orelated,axiom,
    ! [R2: fun_re1255401033344146673l_bool,S3: set_nat,H: fun_nat_real,G: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,one_one_real),one_one_real))
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S3))
         => ( ! [X2: nat] :
                ( pp(aa_set_nat_bool(member_nat(X2),S3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_nat_real(H,X2)),aa_nat_real(G,X2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,groups129246275422532515t_real(H,S3)),groups129246275422532515t_real(G,S3))) ) ) ) ) ).

% prod.related
tff(fact_8408_prod_Orelated,axiom,
    ! [R2: fun_re1255401033344146673l_bool,S3: set_int,H: fun_int_real,G: fun_int_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,one_one_real),one_one_real))
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_int_real(H,X2)),aa_int_real(G,X2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,groups2316167850115554303t_real(H,S3)),groups2316167850115554303t_real(G,S3))) ) ) ) ) ).

% prod.related
tff(fact_8409_prod_Orelated,axiom,
    ! [R2: fun_re1255401033344146673l_bool,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,one_one_real),one_one_real))
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
         => ( ! [X2: complex] :
                ( pp(aa_set_complex_bool(member_complex(X2),S3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_complex_real(H,X2)),aa_complex_real(G,X2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,groups766887009212190081x_real(H,S3)),groups766887009212190081x_real(G,S3))) ) ) ) ) ).

% prod.related
tff(fact_8410_prod_Orelated,axiom,
    ! [R2: fun_rat_fun_rat_bool,S3: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,one_one_rat),one_one_rat))
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,X1),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,aa_rat_rat(times_times_rat(X1),Y1)),aa_rat_rat(times_times_rat(X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S3))
         => ( ! [X2: nat] :
                ( pp(aa_set_nat_bool(member_nat(X2),S3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,aa_nat_rat(H,X2)),aa_nat_rat(G,X2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,groups73079841787564623at_rat(H,S3)),groups73079841787564623at_rat(G,S3))) ) ) ) ) ).

% prod.related
tff(fact_8411_prod_Orelated,axiom,
    ! [R2: fun_rat_fun_rat_bool,S3: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,one_one_rat),one_one_rat))
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,X1),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,aa_rat_rat(times_times_rat(X1),Y1)),aa_rat_rat(times_times_rat(X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,aa_int_rat(H,X2)),aa_int_rat(G,X2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,groups1072433553688619179nt_rat(H,S3)),groups1072433553688619179nt_rat(G,S3))) ) ) ) ) ).

% prod.related
tff(fact_8412_prod_Orelated,axiom,
    ! [R2: fun_rat_fun_rat_bool,S3: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,one_one_rat),one_one_rat))
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,X1),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,aa_rat_rat(times_times_rat(X1),Y1)),aa_rat_rat(times_times_rat(X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
         => ( ! [X2: complex] :
                ( pp(aa_set_complex_bool(member_complex(X2),S3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,aa_complex_rat(H,X2)),aa_complex_rat(G,X2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,groups225925009352817453ex_rat(H,S3)),groups225925009352817453ex_rat(G,S3))) ) ) ) ) ).

% prod.related
tff(fact_8413_prod_Orelated,axiom,
    ! [R2: fun_nat_fun_nat_bool,S3: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,one_one_nat),one_one_nat))
     => ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X1),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,Y1),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,aa_int_nat(H,X2)),aa_int_nat(G,X2))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,groups1707563613775114915nt_nat(H,S3)),groups1707563613775114915nt_nat(G,S3))) ) ) ) ) ).

% prod.related
tff(fact_8414_prod__dvd__prod__subset2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( finite_finite_real(B2)
     => ( ord_less_eq_set_real(A2,B2)
       => ( ! [A4: real] :
              ( member_real(A4,A2)
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_real_nat(F,A4)),aa_real_nat(G,A4))) )
         => 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_8415_prod__dvd__prod__subset2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( finite5795047828879050333T_VEBT(B2)
     => ( ord_le4337996190870823476T_VEBT(A2,B2)
       => ( ! [A4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_VEBT_VEBT_nat(F,A4)),aa_VEBT_VEBT_nat(G,A4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups6361806394783013919BT_nat(F,A2)),groups6361806394783013919BT_nat(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8416_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))
       => ( ! [A4: int] :
              ( pp(aa_set_int_bool(member_int(A4),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(F,A4)),aa_int_nat(G,A4))) )
         => 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_8417_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))
       => ( ! [A4: complex] :
              ( pp(aa_set_complex_bool(member_complex(A4),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_complex_nat(F,A4)),aa_complex_nat(G,A4))) )
         => 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_8418_prod__dvd__prod__subset2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_int,G: fun_real_int] :
      ( finite_finite_real(B2)
     => ( ord_less_eq_set_real(A2,B2)
       => ( ! [A4: real] :
              ( member_real(A4,A2)
             => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_real_int(F,A4)),aa_real_int(G,A4))) )
         => 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_8419_prod__dvd__prod__subset2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_int,G: fun_VEBT_VEBT_int] :
      ( finite5795047828879050333T_VEBT(B2)
     => ( ord_le4337996190870823476T_VEBT(A2,B2)
       => ( ! [A4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),A2))
             => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_VEBT_VEBT_int(F,A4)),aa_VEBT_VEBT_int(G,A4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups6359315924273963643BT_int(F,A2)),groups6359315924273963643BT_int(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8420_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))
       => ( ! [A4: complex] :
              ( pp(aa_set_complex_bool(member_complex(A4),A2))
             => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_complex_int(F,A4)),aa_complex_int(G,A4))) )
         => 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_8421_prod__dvd__prod__subset2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_re6978948838062957830nteger,G: fun_re6978948838062957830nteger] :
      ( finite_finite_real(B2)
     => ( ord_less_eq_set_real(A2,B2)
       => ( ! [A4: real] :
              ( member_real(A4,A2)
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_real_Code_integer(F,A4)),aa_real_Code_integer(G,A4))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,groups6225526099057966256nteger(F,A2)),groups6225526099057966256nteger(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8422_prod__dvd__prod__subset2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VE9219535225822561060nteger,G: fun_VE9219535225822561060nteger] :
      ( finite5795047828879050333T_VEBT(B2)
     => ( ord_le4337996190870823476T_VEBT(A2,B2)
       => ( ! [A4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),A2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_VEB5935388676523804373nteger(F,A4)),aa_VEB5935388676523804373nteger(G,A4))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,groups3770682396051356844nteger(F,A2)),groups3770682396051356844nteger(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8423_prod__dvd__prod__subset2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_Code_integer,G: 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))
       => ( ! [A4: int] :
              ( pp(aa_set_int_bool(member_int(A4),A2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_int_Code_integer(F,A4)),aa_int_Code_integer(G,A4))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,groups3827104343326376752nteger(F,A2)),groups3827104343326376752nteger(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8424_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_8425_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_8426_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_8427_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_8428_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_8429_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(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_8430_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(ord_less_eq_set_nat(A2),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups708209901874060359at_nat(F,A2)),groups708209901874060359at_nat(F,B2))) ) ) ).

% prod_dvd_prod_subset
tff(fact_8431_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(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_8432_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_8433_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_real,S3: set_real,I: fun_real_real,J: fun_real_real,T2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(S4)
     => ( finite_finite_real(T3)
       => ( ! [A4: real] :
              ( member_real(A4,minus_minus_set_real(S3,S4))
             => ( aa_real_real(I,aa_real_real(J,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( member_real(A4,minus_minus_set_real(S3,S4))
               => member_real(aa_real_real(J,A4),minus_minus_set_real(T2,T3)) )
           => ( ! [B4: real] :
                  ( member_real(B4,minus_minus_set_real(T2,T3))
                 => ( aa_real_real(J,aa_real_real(I,B4)) = B4 ) )
             => ( ! [B4: real] :
                    ( member_real(B4,minus_minus_set_real(T2,T3))
                   => member_real(aa_real_real(I,B4),minus_minus_set_real(S3,S4)) )
               => ( ! [A4: real] :
                      ( member_real(A4,S4)
                     => ( aa_real_complex(G,A4) = one_one_complex ) )
                 => ( ! [B4: real] :
                        ( member_real(B4,T3)
                       => ( aa_real_complex(H,B4) = one_one_complex ) )
                   => ( ! [A4: real] :
                          ( member_real(A4,S3)
                         => ( aa_real_complex(H,aa_real_real(J,A4)) = aa_real_complex(G,A4) ) )
                     => ( groups713298508707869441omplex(G,S3) = groups713298508707869441omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8434_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_VEBT_VEBT,S3: set_real,I: fun_VEBT_VEBT_real,J: fun_real_VEBT_VEBT,T2: set_VEBT_VEBT,G: fun_real_complex,H: fun_VE5569070535936029429omplex] :
      ( finite_finite_real(S4)
     => ( finite5795047828879050333T_VEBT(T3)
       => ( ! [A4: real] :
              ( member_real(A4,minus_minus_set_real(S3,S4))
             => ( aa_VEBT_VEBT_real(I,aa_real_VEBT_VEBT(J,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( member_real(A4,minus_minus_set_real(S3,S4))
               => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_real_VEBT_VEBT(J,A4)),minus_5127226145743854075T_VEBT(T2,T3))) )
           => ( ! [B4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(T2,T3)))
                 => ( aa_real_VEBT_VEBT(J,aa_VEBT_VEBT_real(I,B4)) = B4 ) )
             => ( ! [B4: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(T2,T3)))
                   => member_real(aa_VEBT_VEBT_real(I,B4),minus_minus_set_real(S3,S4)) )
               => ( ! [A4: real] :
                      ( member_real(A4,S4)
                     => ( aa_real_complex(G,A4) = one_one_complex ) )
                 => ( ! [B4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),T3))
                       => ( aa_VEBT_VEBT_complex(H,B4) = one_one_complex ) )
                   => ( ! [A4: real] :
                          ( member_real(A4,S3)
                         => ( aa_VEBT_VEBT_complex(H,aa_real_VEBT_VEBT(J,A4)) = aa_real_complex(G,A4) ) )
                     => ( groups713298508707869441omplex(G,S3) = groups127312072573709053omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8435_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T3: set_real,S3: set_VEBT_VEBT,I: fun_real_VEBT_VEBT,J: fun_VEBT_VEBT_real,T2: set_real,G: fun_VE5569070535936029429omplex,H: fun_real_complex] :
      ( finite5795047828879050333T_VEBT(S4)
     => ( finite_finite_real(T3)
       => ( ! [A4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(S3,S4)))
             => ( aa_real_VEBT_VEBT(I,aa_VEBT_VEBT_real(J,A4)) = A4 ) )
         => ( ! [A4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(S3,S4)))
               => member_real(aa_VEBT_VEBT_real(J,A4),minus_minus_set_real(T2,T3)) )
           => ( ! [B4: real] :
                  ( member_real(B4,minus_minus_set_real(T2,T3))
                 => ( aa_VEBT_VEBT_real(J,aa_real_VEBT_VEBT(I,B4)) = B4 ) )
             => ( ! [B4: real] :
                    ( member_real(B4,minus_minus_set_real(T2,T3))
                   => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_real_VEBT_VEBT(I,B4)),minus_5127226145743854075T_VEBT(S3,S4))) )
               => ( ! [A4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),S4))
                     => ( aa_VEBT_VEBT_complex(G,A4) = one_one_complex ) )
                 => ( ! [B4: real] :
                        ( member_real(B4,T3)
                       => ( aa_real_complex(H,B4) = one_one_complex ) )
                   => ( ! [A4: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),S3))
                         => ( aa_real_complex(H,aa_VEBT_VEBT_real(J,A4)) = aa_VEBT_VEBT_complex(G,A4) ) )
                     => ( groups127312072573709053omplex(G,S3) = groups713298508707869441omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8436_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T3: set_VEBT_VEBT,S3: set_VEBT_VEBT,I: fun_VE3165209603783583043T_VEBT,J: fun_VE3165209603783583043T_VEBT,T2: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( finite5795047828879050333T_VEBT(S4)
     => ( finite5795047828879050333T_VEBT(T3)
       => ( ! [A4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(S3,S4)))
             => ( aa_VEB5832953974749077948T_VEBT(I,aa_VEB5832953974749077948T_VEBT(J,A4)) = A4 ) )
         => ( ! [A4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(S3,S4)))
               => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_VEB5832953974749077948T_VEBT(J,A4)),minus_5127226145743854075T_VEBT(T2,T3))) )
           => ( ! [B4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(T2,T3)))
                 => ( aa_VEB5832953974749077948T_VEBT(J,aa_VEB5832953974749077948T_VEBT(I,B4)) = B4 ) )
             => ( ! [B4: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(T2,T3)))
                   => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_VEB5832953974749077948T_VEBT(I,B4)),minus_5127226145743854075T_VEBT(S3,S4))) )
               => ( ! [A4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),S4))
                     => ( aa_VEBT_VEBT_complex(G,A4) = one_one_complex ) )
                 => ( ! [B4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),T3))
                       => ( aa_VEBT_VEBT_complex(H,B4) = one_one_complex ) )
                   => ( ! [A4: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),S3))
                         => ( aa_VEBT_VEBT_complex(H,aa_VEB5832953974749077948T_VEBT(J,A4)) = aa_VEBT_VEBT_complex(G,A4) ) )
                     => ( groups127312072573709053omplex(G,S3) = groups127312072573709053omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8437_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_int,S3: set_real,I: fun_int_real,J: fun_real_int,T2: set_int,G: fun_real_complex,H: fun_int_complex] :
      ( finite_finite_real(S4)
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A4: real] :
              ( member_real(A4,minus_minus_set_real(S3,S4))
             => ( aa_int_real(I,aa_real_int(J,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( member_real(A4,minus_minus_set_real(S3,S4))
               => pp(aa_set_int_bool(member_int(aa_real_int(J,A4)),minus_minus_set_int(T2,T3))) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T2,T3)))
                 => ( aa_real_int(J,aa_int_real(I,B4)) = B4 ) )
             => ( ! [B4: int] :
                    ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T2,T3)))
                   => member_real(aa_int_real(I,B4),minus_minus_set_real(S3,S4)) )
               => ( ! [A4: real] :
                      ( member_real(A4,S4)
                     => ( aa_real_complex(G,A4) = one_one_complex ) )
                 => ( ! [B4: int] :
                        ( pp(aa_set_int_bool(member_int(B4),T3))
                       => ( aa_int_complex(H,B4) = one_one_complex ) )
                   => ( ! [A4: real] :
                          ( member_real(A4,S3)
                         => ( aa_int_complex(H,aa_real_int(J,A4)) = aa_real_complex(G,A4) ) )
                     => ( groups713298508707869441omplex(G,S3) = groups7440179247065528705omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8438_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T3: set_int,S3: set_VEBT_VEBT,I: fun_int_VEBT_VEBT,J: fun_VEBT_VEBT_int,T2: set_int,G: fun_VE5569070535936029429omplex,H: fun_int_complex] :
      ( finite5795047828879050333T_VEBT(S4)
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(S3,S4)))
             => ( aa_int_VEBT_VEBT(I,aa_VEBT_VEBT_int(J,A4)) = A4 ) )
         => ( ! [A4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(S3,S4)))
               => pp(aa_set_int_bool(member_int(aa_VEBT_VEBT_int(J,A4)),minus_minus_set_int(T2,T3))) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T2,T3)))
                 => ( aa_VEBT_VEBT_int(J,aa_int_VEBT_VEBT(I,B4)) = B4 ) )
             => ( ! [B4: int] :
                    ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T2,T3)))
                   => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_int_VEBT_VEBT(I,B4)),minus_5127226145743854075T_VEBT(S3,S4))) )
               => ( ! [A4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),S4))
                     => ( aa_VEBT_VEBT_complex(G,A4) = one_one_complex ) )
                 => ( ! [B4: int] :
                        ( pp(aa_set_int_bool(member_int(B4),T3))
                       => ( aa_int_complex(H,B4) = one_one_complex ) )
                   => ( ! [A4: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),S3))
                         => ( aa_int_complex(H,aa_VEBT_VEBT_int(J,A4)) = aa_VEBT_VEBT_complex(G,A4) ) )
                     => ( groups127312072573709053omplex(G,S3) = groups7440179247065528705omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8439_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_complex,S3: set_real,I: fun_complex_real,J: fun_real_complex,T2: set_complex,G: fun_real_complex,H: fun_complex_complex] :
      ( finite_finite_real(S4)
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
       => ( ! [A4: real] :
              ( member_real(A4,minus_minus_set_real(S3,S4))
             => ( aa_complex_real(I,aa_real_complex(J,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( member_real(A4,minus_minus_set_real(S3,S4))
               => pp(aa_set_complex_bool(member_complex(aa_real_complex(J,A4)),minus_811609699411566653omplex(T2,T3))) )
           => ( ! [B4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(T2,T3)))
                 => ( aa_real_complex(J,aa_complex_real(I,B4)) = B4 ) )
             => ( ! [B4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(T2,T3)))
                   => member_real(aa_complex_real(I,B4),minus_minus_set_real(S3,S4)) )
               => ( ! [A4: real] :
                      ( member_real(A4,S4)
                     => ( aa_real_complex(G,A4) = one_one_complex ) )
                 => ( ! [B4: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B4),T3))
                       => ( aa_complex_complex(H,B4) = one_one_complex ) )
                   => ( ! [A4: real] :
                          ( member_real(A4,S3)
                         => ( aa_complex_complex(H,aa_real_complex(J,A4)) = aa_real_complex(G,A4) ) )
                     => ( groups713298508707869441omplex(G,S3) = groups3708469109370488835omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8440_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T3: set_complex,S3: set_VEBT_VEBT,I: fun_co4340422964966740447T_VEBT,J: fun_VE5569070535936029429omplex,T2: set_complex,G: fun_VE5569070535936029429omplex,H: fun_complex_complex] :
      ( finite5795047828879050333T_VEBT(S4)
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
       => ( ! [A4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(S3,S4)))
             => ( aa_complex_VEBT_VEBT(I,aa_VEBT_VEBT_complex(J,A4)) = A4 ) )
         => ( ! [A4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(S3,S4)))
               => pp(aa_set_complex_bool(member_complex(aa_VEBT_VEBT_complex(J,A4)),minus_811609699411566653omplex(T2,T3))) )
           => ( ! [B4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(T2,T3)))
                 => ( aa_VEBT_VEBT_complex(J,aa_complex_VEBT_VEBT(I,B4)) = B4 ) )
             => ( ! [B4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(T2,T3)))
                   => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_complex_VEBT_VEBT(I,B4)),minus_5127226145743854075T_VEBT(S3,S4))) )
               => ( ! [A4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),S4))
                     => ( aa_VEBT_VEBT_complex(G,A4) = one_one_complex ) )
                 => ( ! [B4: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B4),T3))
                       => ( aa_complex_complex(H,B4) = one_one_complex ) )
                   => ( ! [A4: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),S3))
                         => ( aa_complex_complex(H,aa_VEBT_VEBT_complex(J,A4)) = aa_VEBT_VEBT_complex(G,A4) ) )
                     => ( groups127312072573709053omplex(G,S3) = groups3708469109370488835omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8441_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_real,S3: 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,S4))
     => ( finite_finite_real(T3)
       => ( ! [A4: int] :
              ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S3,S4)))
             => ( aa_real_int(I,aa_int_real(J,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S3,S4)))
               => member_real(aa_int_real(J,A4),minus_minus_set_real(T2,T3)) )
           => ( ! [B4: real] :
                  ( member_real(B4,minus_minus_set_real(T2,T3))
                 => ( aa_int_real(J,aa_real_int(I,B4)) = B4 ) )
             => ( ! [B4: real] :
                    ( member_real(B4,minus_minus_set_real(T2,T3))
                   => pp(aa_set_int_bool(member_int(aa_real_int(I,B4)),minus_minus_set_int(S3,S4))) )
               => ( ! [A4: int] :
                      ( pp(aa_set_int_bool(member_int(A4),S4))
                     => ( aa_int_complex(G,A4) = one_one_complex ) )
                 => ( ! [B4: real] :
                        ( member_real(B4,T3)
                       => ( aa_real_complex(H,B4) = one_one_complex ) )
                   => ( ! [A4: int] :
                          ( pp(aa_set_int_bool(member_int(A4),S3))
                         => ( aa_real_complex(H,aa_int_real(J,A4)) = aa_int_complex(G,A4) ) )
                     => ( groups7440179247065528705omplex(G,S3) = groups713298508707869441omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8442_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_VEBT_VEBT,S3: set_int,I: fun_VEBT_VEBT_int,J: fun_int_VEBT_VEBT,T2: set_VEBT_VEBT,G: fun_int_complex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( finite5795047828879050333T_VEBT(T3)
       => ( ! [A4: int] :
              ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S3,S4)))
             => ( aa_VEBT_VEBT_int(I,aa_int_VEBT_VEBT(J,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S3,S4)))
               => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_int_VEBT_VEBT(J,A4)),minus_5127226145743854075T_VEBT(T2,T3))) )
           => ( ! [B4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(T2,T3)))
                 => ( aa_int_VEBT_VEBT(J,aa_VEBT_VEBT_int(I,B4)) = B4 ) )
             => ( ! [B4: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(T2,T3)))
                   => pp(aa_set_int_bool(member_int(aa_VEBT_VEBT_int(I,B4)),minus_minus_set_int(S3,S4))) )
               => ( ! [A4: int] :
                      ( pp(aa_set_int_bool(member_int(A4),S4))
                     => ( aa_int_complex(G,A4) = one_one_complex ) )
                 => ( ! [B4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),T3))
                       => ( aa_VEBT_VEBT_complex(H,B4) = one_one_complex ) )
                   => ( ! [A4: int] :
                          ( pp(aa_set_int_bool(member_int(A4),S3))
                         => ( aa_VEBT_VEBT_complex(H,aa_int_VEBT_VEBT(J,A4)) = aa_int_complex(G,A4) ) )
                     => ( groups7440179247065528705omplex(G,S3) = groups127312072573709053omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8443_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,N: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups129246275422532515t_real(aTP_Lamm_vl(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_8444_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups73079841787564623at_rat(aTP_Lamm_vm(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_8445_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups708209901874060359at_nat(aTP_Lamm_vn(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_8446_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups705719431365010083at_int(aTP_Lamm_vo(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_8447_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_complex] :
      ( finite_finite_real(A2)
     => ( groups713298508707869441omplex(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_vp(G)))) = groups713298508707869441omplex(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8448_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups7440179247065528705omplex(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_vq(G)))) = groups7440179247065528705omplex(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8449_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups3708469109370488835omplex(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_vr(G)))) = groups3708469109370488835omplex(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8450_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_real] :
      ( finite_finite_real(A2)
     => ( groups1681761925125756287l_real(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_vs(G)))) = groups1681761925125756287l_real(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8451_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups2316167850115554303t_real(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_vt(G)))) = groups2316167850115554303t_real(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8452_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups766887009212190081x_real(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_vu(G)))) = groups766887009212190081x_real(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8453_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_rat] :
      ( finite_finite_real(A2)
     => ( groups4061424788464935467al_rat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_vv(G)))) = groups4061424788464935467al_rat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8454_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups1072433553688619179nt_rat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_vw(G)))) = groups1072433553688619179nt_rat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8455_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_vx(G)))) = groups225925009352817453ex_rat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8456_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_nat] :
      ( finite_finite_real(A2)
     => ( groups4696554848551431203al_nat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_vy(G)))) = groups4696554848551431203al_nat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8457_exp__sum,axiom,
    ! [I5: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( aa_real_real(exp_real,groups8778361861064173332t_real(F,I5)) = groups2316167850115554303t_real(aTP_Lamm_vz(F),I5) ) ) ).

% exp_sum
tff(fact_8458_exp__sum,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( aa_real_real(exp_real,groups5808333547571424918x_real(F,I5)) = groups766887009212190081x_real(aTP_Lamm_wa(F),I5) ) ) ).

% exp_sum
tff(fact_8459_exp__sum,axiom,
    ! [I5: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( aa_complex_complex(exp_complex,groups2073611262835488442omplex(F,I5)) = groups6464643781859351333omplex(aTP_Lamm_wb(F),I5) ) ) ).

% exp_sum
tff(fact_8460_exp__sum,axiom,
    ! [I5: set_int,F: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( aa_complex_complex(exp_complex,groups3049146728041665814omplex(F,I5)) = groups7440179247065528705omplex(aTP_Lamm_wc(F),I5) ) ) ).

% exp_sum
tff(fact_8461_exp__sum,axiom,
    ! [I5: set_complex,F: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( aa_complex_complex(exp_complex,groups7754918857620584856omplex(F,I5)) = groups3708469109370488835omplex(aTP_Lamm_wd(F),I5) ) ) ).

% exp_sum
tff(fact_8462_exp__sum,axiom,
    ! [I5: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( aa_real_real(exp_real,groups6591440286371151544t_real(F,I5)) = groups129246275422532515t_real(aTP_Lamm_we(F),I5) ) ) ).

% exp_sum
tff(fact_8463_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ny(G),N),aa_nat_set_nat(set_ord_lessThan_nat,N)) = groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nat_diff_reindex
tff(fact_8464_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_wf(G),N),aa_nat_set_nat(set_ord_lessThan_nat,N)) = groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nat_diff_reindex
tff(fact_8465_Iic__subset__Iio__iff,axiom,
    ! [A: rat,B: rat] :
      ( 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_8466_Iic__subset__Iio__iff,axiom,
    ! [A: num,B: num] :
      ( ord_less_eq_set_num(set_ord_atMost_num(A),set_ord_lessThan_num(B))
    <=> pp(aa_num_bool(ord_less_num(A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_8467_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_8468_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_atMost_nat,A)),aa_nat_set_nat(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_8469_Iic__subset__Iio__iff,axiom,
    ! [A: real,B: real] :
      ( 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_8470_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_mf(G),N),M),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ).

% prod.atLeastAtMost_rev
tff(fact_8471_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_int,N: nat,M: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_wg(G),N),M),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ).

% prod.atLeastAtMost_rev
tff(fact_8472_prod_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))
       => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_wh(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_wi(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8473_prod_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P2))
       => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wj(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wk(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8474_prod_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P2))
       => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wl(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wm(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8475_prod_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P2))
       => ( groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wn(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wo(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8476_prod_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_int,H: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P2))
       => ( groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_wp(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_wq(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8477_less__1__prod2,axiom,
    ! [I5: set_real,I: real,F: fun_real_real] :
      ( finite_finite_real(I5)
     => ( member_real(I,I5)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(F,I)))
         => ( ! [I3: real] :
                ( member_real(I3,I5)
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_8478_less__1__prod2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(I5)
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_VEBT_VEBT_real(F,I)))
         => ( ! [I3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_VEBT_VEBT_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups2703838992350267259T_real(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_8479_less__1__prod2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( pp(aa_set_nat_bool(member_nat(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(F,I)))
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups129246275422532515t_real(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_8480_less__1__prod2,axiom,
    ! [I5: set_int,I: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( pp(aa_set_int_bool(member_int(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(F,I)))
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_8481_less__1__prod2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( pp(aa_set_complex_bool(member_complex(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_complex_real(F,I)))
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_complex_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups766887009212190081x_real(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_8482_less__1__prod2,axiom,
    ! [I5: set_real,I: real,F: fun_real_rat] :
      ( finite_finite_real(I5)
     => ( member_real(I,I5)
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_real_rat(F,I)))
         => ( ! [I3: real] :
                ( member_real(I3,I5)
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_real_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_8483_less__1__prod2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(I5)
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_VEBT_VEBT_rat(F,I)))
         => ( ! [I3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_VEBT_VEBT_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups5726676334696518183BT_rat(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_8484_less__1__prod2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( pp(aa_set_nat_bool(member_nat(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(F,I)))
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups73079841787564623at_rat(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_8485_less__1__prod2,axiom,
    ! [I5: set_int,I: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( pp(aa_set_int_bool(member_int(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(F,I)))
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_8486_less__1__prod2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( pp(aa_set_complex_bool(member_complex(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_complex_rat(F,I)))
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups225925009352817453ex_rat(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_8487_less__1__prod,axiom,
    ! [I5: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(I5)
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_VEBT_VEBT_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups2703838992350267259T_real(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_8488_less__1__prod,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_complex_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups766887009212190081x_real(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_8489_less__1__prod,axiom,
    ! [I5: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( pp(aa_set_nat_bool(member_nat(I3),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups129246275422532515t_real(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_8490_less__1__prod,axiom,
    ! [I5: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_8491_less__1__prod,axiom,
    ! [I5: set_real,F: fun_real_real] :
      ( finite_finite_real(I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( member_real(I3,I5)
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_8492_less__1__prod,axiom,
    ! [I5: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(I5)
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_VEBT_VEBT_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups5726676334696518183BT_rat(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_8493_less__1__prod,axiom,
    ! [I5: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_complex_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups225925009352817453ex_rat(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_8494_less__1__prod,axiom,
    ! [I5: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( pp(aa_set_nat_bool(member_nat(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups73079841787564623at_rat(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_8495_less__1__prod,axiom,
    ! [I5: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_8496_less__1__prod,axiom,
    ! [I5: set_real,F: fun_real_rat] :
      ( finite_finite_real(I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( member_real(I3,I5)
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_real_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_8497_prod_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))
       => ( groups2316167850115554303t_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups2316167850115554303t_real(G,minus_minus_set_int(A2,B2))),groups2316167850115554303t_real(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_8498_prod_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))
       => ( groups766887009212190081x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups766887009212190081x_real(G,minus_811609699411566653omplex(A2,B2))),groups766887009212190081x_real(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_8499_prod_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))
       => ( groups1072433553688619179nt_rat(G,A2) = aa_rat_rat(times_times_rat(groups1072433553688619179nt_rat(G,minus_minus_set_int(A2,B2))),groups1072433553688619179nt_rat(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_8500_prod_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))
       => ( groups225925009352817453ex_rat(G,A2) = aa_rat_rat(times_times_rat(groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A2,B2))),groups225925009352817453ex_rat(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_8501_prod_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))
       => ( groups1707563613775114915nt_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G,minus_minus_set_int(A2,B2))),groups1707563613775114915nt_nat(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_8502_prod_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))
       => ( groups861055069439313189ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A2,B2))),groups861055069439313189ex_nat(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_8503_prod_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))
       => ( groups858564598930262913ex_int(G,A2) = aa_int_int(times_times_int(groups858564598930262913ex_int(G,minus_811609699411566653omplex(A2,B2))),groups858564598930262913ex_int(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_8504_prod_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( groups129246275422532515t_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,minus_minus_set_nat(A2,B2))),groups129246275422532515t_real(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_8505_prod_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( groups73079841787564623at_rat(G,A2) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,minus_minus_set_nat(A2,B2))),groups73079841787564623at_rat(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_8506_prod_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( groups708209901874060359at_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,minus_minus_set_nat(A2,B2))),groups708209901874060359at_nat(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_8507_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A2,C4)
       => ( ord_less_eq_set_real(B2,C4)
         => ( ! [A4: real] :
                ( member_real(A4,minus_minus_set_real(C4,A2))
               => ( aa_real_nat(G,A4) = one_one_nat ) )
           => ( ! [B4: real] :
                  ( member_real(B4,minus_minus_set_real(C4,B2))
                 => ( aa_real_nat(H,B4) = one_one_nat ) )
             => ( ( groups4696554848551431203al_nat(G,A2) = groups4696554848551431203al_nat(H,B2) )
              <=> ( groups4696554848551431203al_nat(G,C4) = groups4696554848551431203al_nat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8508_prod_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,H: fun_VEBT_VEBT_nat] :
      ( finite5795047828879050333T_VEBT(C4)
     => ( ord_le4337996190870823476T_VEBT(A2,C4)
       => ( ord_le4337996190870823476T_VEBT(B2,C4)
         => ( ! [A4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_nat(G,A4) = one_one_nat ) )
           => ( ! [B4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_nat(H,B4) = one_one_nat ) )
             => ( ( groups6361806394783013919BT_nat(G,A2) = groups6361806394783013919BT_nat(H,B2) )
              <=> ( groups6361806394783013919BT_nat(G,C4) = groups6361806394783013919BT_nat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8509_prod_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A2)))
               => ( aa_int_nat(G,A4) = one_one_nat ) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C4,B2)))
                 => ( aa_int_nat(H,B4) = one_one_nat ) )
             => ( ( groups1707563613775114915nt_nat(G,A2) = groups1707563613775114915nt_nat(H,B2) )
              <=> ( groups1707563613775114915nt_nat(G,C4) = groups1707563613775114915nt_nat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8510_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A4: complex] :
                ( pp(aa_set_complex_bool(member_complex(A4),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_nat(G,A4) = one_one_nat ) )
           => ( ! [B4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_nat(H,B4) = one_one_nat ) )
             => ( ( groups861055069439313189ex_nat(G,A2) = groups861055069439313189ex_nat(H,B2) )
              <=> ( groups861055069439313189ex_nat(G,C4) = groups861055069439313189ex_nat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8511_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_int,H: fun_real_int] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A2,C4)
       => ( ord_less_eq_set_real(B2,C4)
         => ( ! [A4: real] :
                ( member_real(A4,minus_minus_set_real(C4,A2))
               => ( aa_real_int(G,A4) = one_one_int ) )
           => ( ! [B4: real] :
                  ( member_real(B4,minus_minus_set_real(C4,B2))
                 => ( aa_real_int(H,B4) = one_one_int ) )
             => ( ( groups4694064378042380927al_int(G,A2) = groups4694064378042380927al_int(H,B2) )
              <=> ( groups4694064378042380927al_int(G,C4) = groups4694064378042380927al_int(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8512_prod_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_int,H: fun_VEBT_VEBT_int] :
      ( finite5795047828879050333T_VEBT(C4)
     => ( ord_le4337996190870823476T_VEBT(A2,C4)
       => ( ord_le4337996190870823476T_VEBT(B2,C4)
         => ( ! [A4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_int(G,A4) = one_one_int ) )
           => ( ! [B4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B4),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_int(H,B4) = one_one_int ) )
             => ( ( groups6359315924273963643BT_int(G,A2) = groups6359315924273963643BT_int(H,B2) )
              <=> ( groups6359315924273963643BT_int(G,C4) = groups6359315924273963643BT_int(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8513_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A4: complex] :
                ( pp(aa_set_complex_bool(member_complex(A4),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_int(G,A4) = one_one_int ) )
           => ( ! [B4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_int(H,B4) = one_one_int ) )
             => ( ( groups858564598930262913ex_int(G,A2) = groups858564598930262913ex_int(H,B2) )
              <=> ( groups858564598930262913ex_int(G,C4) = groups858564598930262913ex_int(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8514_prod_Osame__carrier,axiom,
    ! [C4: set_nat,A2: set_nat,B2: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,C4))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),C4))
       => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),C4))
         => ( ! [A4: nat] :
                ( pp(aa_set_nat_bool(member_nat(A4),minus_minus_set_nat(C4,A2)))
               => ( aa_nat_complex(G,A4) = one_one_complex ) )
           => ( ! [B4: nat] :
                  ( pp(aa_set_nat_bool(member_nat(B4),minus_minus_set_nat(C4,B2)))
                 => ( aa_nat_complex(H,B4) = one_one_complex ) )
             => ( ( groups6464643781859351333omplex(G,A2) = groups6464643781859351333omplex(H,B2) )
              <=> ( groups6464643781859351333omplex(G,C4) = groups6464643781859351333omplex(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8515_prod_Osame__carrier,axiom,
    ! [C4: set_nat,A2: set_nat,B2: set_nat,G: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,C4))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),C4))
       => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),C4))
         => ( ! [A4: nat] :
                ( pp(aa_set_nat_bool(member_nat(A4),minus_minus_set_nat(C4,A2)))
               => ( aa_nat_real(G,A4) = one_one_real ) )
           => ( ! [B4: nat] :
                  ( pp(aa_set_nat_bool(member_nat(B4),minus_minus_set_nat(C4,B2)))
                 => ( aa_nat_real(H,B4) = one_one_real ) )
             => ( ( groups129246275422532515t_real(G,A2) = groups129246275422532515t_real(H,B2) )
              <=> ( groups129246275422532515t_real(G,C4) = groups129246275422532515t_real(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8516_prod_Osame__carrier,axiom,
    ! [C4: set_nat,A2: set_nat,B2: set_nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,C4))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),C4))
       => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),C4))
         => ( ! [A4: nat] :
                ( pp(aa_set_nat_bool(member_nat(A4),minus_minus_set_nat(C4,A2)))
               => ( aa_nat_rat(G,A4) = one_one_rat ) )
           => ( ! [B4: nat] :
                  ( pp(aa_set_nat_bool(member_nat(B4),minus_minus_set_nat(C4,B2)))
                 => ( aa_nat_rat(H,B4) = one_one_rat ) )
             => ( ( groups73079841787564623at_rat(G,A2) = groups73079841787564623at_rat(H,B2) )
              <=> ( groups73079841787564623at_rat(G,C4) = groups73079841787564623at_rat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8517_sum__choose__lower,axiom,
    ! [R: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_wr,R)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R),N))),N) ).

% sum_choose_lower
tff(fact_8518_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ws,N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_8519_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ws,N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),one_one_nat)),M) ).

% choose_rising_sum(2)
tff(fact_8520_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),groups708209901874060359at_nat(aTP_Lamm_an,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,N)),M))) ) ) ).

% fact_eq_fact_times
tff(fact_8521_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_wt(M),N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),M) ) ) ).

% sum_choose_diagonal
tff(fact_8522_vandermonde,axiom,
    ! [M: nat,N: nat,R: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_wu(M),N),R)),aa_nat_set_nat(set_ord_atMost_nat,R)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),R) ).

% vandermonde
tff(fact_8523_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) = groups708209901874060359at_nat(aTP_Lamm_an,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) ) ).

% fact_div_fact
tff(fact_8524_choose__row__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% choose_row_sum
tff(fact_8525_binomial,axiom,
    ! [A: nat,B: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),N) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_wv(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial
tff(fact_8526_polynomial__product__nat,axiom,
    ! [M: nat,A: fun_nat_nat,N: nat,B: fun_nat_nat,X: nat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I3))
         => ( aa_nat_nat(A,I3) = 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_ww(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ww(B),X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_wy(A),B),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product_nat
tff(fact_8527_choose__square__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_wz,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N) ).

% choose_square_sum
tff(fact_8528_complex__inverse,axiom,
    ! [A: real,B: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(aa_rea2649881322057434177omplex(complex2,A),B)) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% complex_inverse
tff(fact_8529_binomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% binomial_r_part_sum
tff(fact_8530_choose__linear__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_xa,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ).

% choose_linear_sum
tff(fact_8531_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)),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_8532_of__nat__id,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = N ).

% of_nat_id
tff(fact_8533_cot__npi,axiom,
    ! [N: nat] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).

% cot_npi
tff(fact_8534_cot__periodic,axiom,
    ! [X: real] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(cot_real,X) ).

% cot_periodic
tff(fact_8535_real__scaleR__def,axiom,
    ! [A: real,X: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),X) ).

% real_scaleR_def
tff(fact_8536_complex__scaleR,axiom,
    ! [R: real,A: real,B: real] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,R),aa_real_complex(aa_rea2649881322057434177omplex(complex2,A),B)) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),B)) ).

% complex_scaleR
tff(fact_8537_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J))) = groups1705073143266064639nt_int(aTP_Lamm_ao,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)))) ).

% prod_int_plus_eq
tff(fact_8538_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),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_8539_tan__cot_H,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),X)) = aa_real_real(cot_real,X) ).

% tan_cot'
tff(fact_8540_i__even__power,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(imaginary_unit),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) ).

% i_even_power
tff(fact_8541_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,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,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_8542_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_xb,Y)) ).

% arctan_def
tff(fact_8543_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_8544_sinh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X)),aa_real_real(sinh_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% sinh_real_le_iff
tff(fact_8545_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_8546_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_8547_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% sinh_real_nonpos_iff
tff(fact_8548_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% sinh_real_nonneg_iff
tff(fact_8549_complex__i__mult__minus,axiom,
    ! [X: complex] : aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(imaginary_unit),X)) = aa_complex_complex(uminus1482373934393186551omplex,X) ).

% complex_i_mult_minus
tff(fact_8550_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_8551_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_8552_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_8553_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_8554_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_8555_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_8556_divide__i,axiom,
    ! [X: complex] : aa_complex_complex(divide1717551699836669952omplex(X),imaginary_unit) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)),X) ).

% divide_i
tff(fact_8557_i__squared,axiom,
    aa_complex_complex(times_times_complex(imaginary_unit),imaginary_unit) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% i_squared
tff(fact_8558_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_8559_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_8560_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_8561_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_8562_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_8563_divide__numeral__i,axiom,
    ! [Z: complex,N: num] : aa_complex_complex(divide1717551699836669952omplex(Z),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(N)),imaginary_unit)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z))),numera6690914467698888265omplex(N)) ).

% divide_numeral_i
tff(fact_8564_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_8565_power2__i,axiom,
    aa_nat_complex(power_power_complex(imaginary_unit),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power2_i
tff(fact_8566_sinh__le__cosh__real,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X))) ).

% sinh_le_cosh_real
tff(fact_8567_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_8568_complex__i__not__numeral,axiom,
    ! [W: num] : imaginary_unit != numera6690914467698888265omplex(W) ).

% complex_i_not_numeral
tff(fact_8569_log__def,axiom,
    ! [A: real,X: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,A)) ).

% log_def
tff(fact_8570_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_8571_cosh__real__nonpos__le__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_eq_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_8572_cosh__real__nonneg__le__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_eq_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_8573_cosh__real__nonneg,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,X))) ).

% cosh_real_nonneg
tff(fact_8574_cosh__real__ge__1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(cosh_real,X))) ).

% cosh_real_ge_1
tff(fact_8575_i__times__eq__iff,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa_complex_complex(times_times_complex(imaginary_unit),W) = Z )
    <=> ( W = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z)) ) ) ).

% i_times_eq_iff
tff(fact_8576_complex__i__not__neg__numeral,axiom,
    ! [W: num] : imaginary_unit != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% complex_i_not_neg_numeral
tff(fact_8577_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_8578_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_8579_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_8580_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,X)) = X ) ) ).

% arcosh_cosh_real
tff(fact_8581_ln__neg__is__const,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( aa_real_real(ln_ln_real,X) = the_real(aTP_Lamm_xc) ) ) ).

% ln_neg_is_const
tff(fact_8582_i__mult__Complex,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(aa_rea2649881322057434177omplex(complex2,A),B)) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(uminus_uminus_real,B)),A) ).

% i_mult_Complex
tff(fact_8583_Complex__mult__i,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(aa_real_complex(aa_rea2649881322057434177omplex(complex2,A),B)),imaginary_unit) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(uminus_uminus_real,B)),A) ).

% Complex_mult_i
tff(fact_8584_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_8585_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_8586_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_8587_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_8588_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_8589_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_8590_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_8591_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_8592_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_8593_log2__of__power__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
     => ( aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ).

% log2_of_power_eq
tff(fact_8594_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_8595_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_8596_arccos__def,axiom,
    ! [Y: real] : aa_real_real(arccos,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_xd,Y)) ).

% arccos_def
tff(fact_8597_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_8598_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,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).

% less_log2_of_power
tff(fact_8599_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,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).

% le_log2_of_power
tff(fact_8600_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,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_8601_pi__half,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) = the_real(aTP_Lamm_xe) ).

% pi_half
tff(fact_8602_pi__def,axiom,
    pi = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),the_real(aTP_Lamm_xe)) ).

% pi_def
tff(fact_8603_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))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_8604_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,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_8605_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,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,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_8606_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))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_8607_arcsin__def,axiom,
    ! [Y: real] : aa_real_real(arcsin,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_xf,Y)) ).

% arcsin_def
tff(fact_8608_Arg__minus__ii,axiom,
    aa_complex_real(arg,aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one))) ).

% Arg_minus_ii
tff(fact_8609_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(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_8610_Arg__ii,axiom,
    aa_complex_real(arg,imaginary_unit) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) ).

% Arg_ii
tff(fact_8611_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(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_8612_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_divide_eq_div_numeral
tff(fact_8613_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B)))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_8614_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_8615_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,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(plus_plus_int(aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,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_8616_cis__minus__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) = aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).

% cis_minus_pi_half
tff(fact_8617_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(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_8618_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_8619_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_8620_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_8621_powr__nonneg__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X)),zero_zero_real))
    <=> ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_8622_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_8623_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_8624_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),one_one_real) = X )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% powr_one_gt_zero_iff
tff(fact_8625_powr__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),one_one_real) = X ) ) ).

% powr_one
tff(fact_8626_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_8627_numeral__powr__numeral__real,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,numeral_numeral_real(M)),numeral_numeral_real(N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(M)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_powr_numeral_real
tff(fact_8628_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_8629_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_8630_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B)) ).

% floor_divide_eq_div_numeral
tff(fact_8631_powr__numeral,axiom,
    ! [X: real,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),numeral_numeral_real(N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,N)) ) ) ).

% powr_numeral
tff(fact_8632_cis__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) = imaginary_unit ).

% cis_pi_half
tff(fact_8633_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,B)) ).

% floor_one_divide_eq_div_numeral
tff(fact_8634_cis__2pi,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)) = one_one_complex ).

% cis_2pi
tff(fact_8635_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_8636_square__powr__half,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_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_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,X) ).

% square_powr_half
tff(fact_8637_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(B)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_8638_powr__powr,axiom,
    ! [X: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),B) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% powr_powr
tff(fact_8639_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_8640_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_8641_powr__ge__pzero,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(powr_real,X),Y))) ).

% powr_ge_pzero
tff(fact_8642_powr__mono2,axiom,
    ! [A: real,X: real,Y: 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,X),Y))
         => 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,Y),A))) ) ) ) ).

% powr_mono2
tff(fact_8643_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_8644_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_8645_powr__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,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))) ) ) ).

% powr_mono
tff(fact_8646_cis__mult,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(aa_real_complex(cis,A)),aa_real_complex(cis,B)) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ).

% cis_mult
tff(fact_8647_cis__divide,axiom,
    ! [A: real,B: real] : aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(cis,A)),aa_real_complex(cis,B)) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ).

% cis_divide
tff(fact_8648_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_8649_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_8650_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_8651_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_8652_ge__one__powr__ge__zero,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_eq_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(powr_real,X),A))) ) ) ).

% ge_one_powr_ge_zero
tff(fact_8653_powr__mono__both,axiom,
    ! [A: real,B: real,X: real,Y: 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),B))
       => ( 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,X),Y))
           => 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,Y),B))) ) ) ) ) ).

% powr_mono_both
tff(fact_8654_powr__le1,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,X),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),one_one_real)) ) ) ) ).

% powr_le1
tff(fact_8655_powr__divide,axiom,
    ! [X: real,Y: 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,zero_zero_real),Y))
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_divide
tff(fact_8656_powr__mult,axiom,
    ! [X: real,Y: 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,zero_zero_real),Y))
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_mult
tff(fact_8657_inverse__powr,axiom,
    ! [Y: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(inverse_inverse_real,Y)),A) = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ).

% inverse_powr
tff(fact_8658_divide__powr__uminus,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,C))) ).

% divide_powr_uminus
tff(fact_8659_log__base__powr,axiom,
    ! [A: real,B: real,X: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),B)),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),B) ) ) ).

% log_base_powr
tff(fact_8660_ln__powr,axiom,
    ! [X: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(ln_ln_real,X)) ) ) ).

% ln_powr
tff(fact_8661_log__powr,axiom,
    ! [X: real,B: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ).

% log_powr
tff(fact_8662_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(plus_plus_int(K),one_one_int))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_8663_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_8664_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_8665_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_8666_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_8667_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_8668_real__of__int__floor__add__one__gt,axiom,
    ! [R: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R))),one_one_real))) ).

% real_of_int_floor_add_one_gt
tff(fact_8669_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_8670_real__of__int__floor__add__one__ge,axiom,
    ! [R: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R))),one_one_real))) ).

% real_of_int_floor_add_one_ge
tff(fact_8671_real__of__int__floor__gt__diff__one,axiom,
    ! [R: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R)))) ).

% real_of_int_floor_gt_diff_one
tff(fact_8672_real__of__int__floor__ge__diff__one,axiom,
    ! [R: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R)))) ).

% real_of_int_floor_ge_diff_one
tff(fact_8673_DeMoivre,axiom,
    ! [A: real,N: nat] : aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),N) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),A)) ).

% DeMoivre
tff(fact_8674_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_8675_powr__mult__base,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,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Y)) ) ) ).

% powr_mult_base
tff(fact_8676_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_8677_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_8678_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_8679_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_8680_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_8681_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_int_real(ring_1_of_int_real,B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_real_int(archim6058952711729229775r_real,A)),B) ) ) ).

% floor_divide_real_eq_div
tff(fact_8682_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_8683_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_8684_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_8685_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_8686_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_8687_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_8688_powr__half__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_real_real(sqrt,X) ) ) ).

% powr_half_sqrt
tff(fact_8689_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,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_8690_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,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(plus_plus_int(aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,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_8691_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_xg(N),aa_nat_set_nat(set_ord_lessThan_nat,N)),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_cg,N)))) ) ).

% bij_betw_roots_unity
tff(fact_8692_exp__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,pi)),imaginary_unit)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% exp_pi_i
tff(fact_8693_exp__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,pi))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% exp_pi_i'
tff(fact_8694_exp__two__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_real_complex(real_V4546457046886955230omplex,pi))),imaginary_unit)) = one_one_complex ).

% exp_two_pi_i
tff(fact_8695_exp__two__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) = one_one_complex ).

% exp_two_pi_i'
tff(fact_8696_complex__exp__exists,axiom,
    ! [Z: complex] :
    ? [A4: complex,R3: real] : Z = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R3)),aa_complex_complex(exp_complex,A4)) ).

% complex_exp_exists
tff(fact_8697_complex__of__real__mult__Complex,axiom,
    ! [R: real,X: real,Y: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R)),aa_real_complex(aa_rea2649881322057434177omplex(complex2,X),Y)) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),Y)) ).

% complex_of_real_mult_Complex
tff(fact_8698_Complex__mult__complex__of__real,axiom,
    ! [X: real,Y: real,R: real] : aa_complex_complex(times_times_complex(aa_real_complex(aa_rea2649881322057434177omplex(complex2,X),Y)),aa_real_complex(real_V4546457046886955230omplex,R)) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),R)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),R)) ).

% Complex_mult_complex_of_real
tff(fact_8699_complex__of__real__add__Complex,axiom,
    ! [R: real,X: real,Y: real] : aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,R)),aa_real_complex(aa_rea2649881322057434177omplex(complex2,X),Y)) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R),X)),Y) ).

% complex_of_real_add_Complex
tff(fact_8700_Complex__add__complex__of__real,axiom,
    ! [X: real,Y: real,R: real] : aa_complex_complex(plus_plus_complex(aa_real_complex(aa_rea2649881322057434177omplex(complex2,X),Y)),aa_real_complex(real_V4546457046886955230omplex,R)) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),R)),Y) ).

% Complex_add_complex_of_real
tff(fact_8701_cis__conv__exp,axiom,
    ! [B: real] : aa_real_complex(cis,B) = aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B))) ).

% cis_conv_exp
tff(fact_8702_complex__of__real__i,axiom,
    ! [R: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R)),imaginary_unit) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,zero_zero_real),R) ).

% complex_of_real_i
tff(fact_8703_i__complex__of__real,axiom,
    ! [R: real] : aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,R)) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,zero_zero_real),R) ).

% i_complex_of_real
tff(fact_8704_Complex__eq,axiom,
    ! [A: real,B: real] : aa_real_complex(aa_rea2649881322057434177omplex(complex2,A),B) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,A)),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B))) ).

% Complex_eq
tff(fact_8705_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R3: real,A4: real] : Z = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R3)),aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A4))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A4))))) ).

% complex_split_polar
tff(fact_8706_cmod__unit__one,axiom,
    ! [A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A))))) = one_one_real ).

% cmod_unit_one
tff(fact_8707_cmod__complex__polar,axiom,
    ! [R: real,A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R)),aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A)))))) = aa_real_real(abs_abs_real,R) ).

% cmod_complex_polar
tff(fact_8708_csqrt__ii,axiom,
    aa_complex_complex(csqrt,imaginary_unit) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(one_one_complex),imaginary_unit)),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one))))) ).

% csqrt_ii
tff(fact_8709_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(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_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(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_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(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_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_8710_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_8711_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_8712_nat__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_num_int(numeral_numeral_int,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_numeral
tff(fact_8713_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).

% zless_nat_conj
tff(fact_8714_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_8715_sgn__mult__dvd__iff,axiom,
    ! [R: int,L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,R)),L)),K))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        & ( ( R = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_8716_mult__sgn__dvd__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(L),aa_int_int(sgn_sgn_int,R))),K))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        & ( ( R = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_8717_dvd__sgn__mult__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,R)),K)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( R = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_8718_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(times_times_int(K),aa_int_int(sgn_sgn_int,R))))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( R = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_8719_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_8720_diff__nat__numeral,axiom,
    ! [V: num,V3: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V3)) = aa_int_nat(nat2,aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V3))) ).

% diff_nat_numeral
tff(fact_8721_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = aa_int_nat(nat2,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_8722_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_8723_power2__csqrt,axiom,
    ! [Z: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(csqrt,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z ).

% power2_csqrt
tff(fact_8724_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_8725_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_8726_nat__numeral__diff__1,axiom,
    ! [V: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat) = aa_int_nat(nat2,aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,V)),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_8727_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_8728_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_8729_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_8730_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_8731_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_8732_nat__numeral__as__int,axiom,
    ! [X4: num] : aa_num_nat(numeral_numeral_nat,X4) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X4)) ).

% nat_numeral_as_int
tff(fact_8733_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_8734_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N2: nat,L3: int] : K != aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L3)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% int_sgnE
tff(fact_8735_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_8736_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_eq_sgn_abs
tff(fact_8737_unset__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),N) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% unset_bit_nat_def
tff(fact_8738_nat__mask__eq,axiom,
    ! [N: nat] : aa_int_nat(nat2,aa_nat_int(bit_se2000444600071755411sk_int,N)) = aa_nat_nat(bit_se2002935070580805687sk_nat,N) ).

% nat_mask_eq
tff(fact_8739_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).

% nat_mono_iff
tff(fact_8740_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_8741_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_8742_nat__int__add,axiom,
    ! [A: nat,B: nat] : aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) ).

% nat_int_add
tff(fact_8743_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
       => ( aa_int_int(sgn_sgn_int,aa_int_int(modulo_modulo_int(K),L)) = aa_int_int(sgn_sgn_int,L) ) ) ) ).

% sgn_mod
tff(fact_8744_int__minus,axiom,
    ! [N: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1314217659103216013at_int,M)))) ).

% int_minus
tff(fact_8745_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(times_times_int(W),Z))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,W))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Z))) ).

% nat_abs_mult_distrib
tff(fact_8746_nat__plus__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_plus_as_int
tff(fact_8747_nat__times__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_times_as_int
tff(fact_8748_real__nat__ceiling__ge,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X))))) ).

% real_nat_ceiling_ge
tff(fact_8749_nat__minus__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_minus_as_int
tff(fact_8750_nat__div__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_div_as_int
tff(fact_8751_nat__mod__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_mod_as_int
tff(fact_8752_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_8753_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).

% nat_less_eq_zless
tff(fact_8754_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),W))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ) ).

% nat_le_eq_zle
tff(fact_8755_split__nat,axiom,
    ! [P: fun_nat_bool,I: int] :
      ( pp(aa_nat_bool(P,aa_int_nat(nat2,I)))
    <=> ( ! [N5: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N5) )
           => pp(aa_nat_bool(P,N5)) )
        & ( 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_8756_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_8757_nat__add__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z5))
       => ( aa_int_nat(nat2,aa_int_int(plus_plus_int(Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_add_distrib
tff(fact_8758_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,K))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_8759_nat__mult__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_int_nat(nat2,aa_int_int(times_times_int(Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ).

% nat_mult_distrib
tff(fact_8760_Suc__as__int,axiom,
    ! [X4: nat] : aa_nat_nat(suc,X4) = aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,X4)),one_one_int)) ).

% Suc_as_int
tff(fact_8761_nat__diff__distrib_H,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_nat(nat2,aa_int_int(minus_minus_int(X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_8762_nat__diff__distrib,axiom,
    ! [Z5: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z5),Z))
       => ( aa_int_nat(nat2,aa_int_int(minus_minus_int(Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_diff_distrib
tff(fact_8763_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(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_8764_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_8765_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_8766_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(times_times_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_int_int(sgn_sgn_int,L))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_8767_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),N) ) ) ).

% nat_power_eq
tff(fact_8768_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_8769_nat__mod__distrib,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_nat(nat2,aa_int_int(modulo_modulo_int(X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_8770_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% div_abs_eq_div_nat
tff(fact_8771_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_8772_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_8773_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(modulo_modulo_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% mod_abs_eq_div_nat
tff(fact_8774_divide__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) )
      & ( ( L != zero_zero_int )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)))))) ) ) ) ) ) ).

% divide_int_def
tff(fact_8775_of__real__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,X)) = aa_complex_complex(csqrt,aa_real_complex(real_V4546457046886955230omplex,X)) ) ) ).

% of_real_sqrt
tff(fact_8776_modulo__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero_int )
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) )
      & ( ( L != zero_zero_int )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_int_int(abs_abs_int,L)),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))))) ) ) ) ) ) ).

% modulo_int_def
tff(fact_8777_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_8778_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_nat_nat(suc,aa_int_nat(nat2,Z)) = aa_int_nat(nat2,aa_int_int(plus_plus_int(one_one_int),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_8779_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),M))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,M))) ) ) ).

% nat_less_iff
tff(fact_8780_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int))
     => ( aa_int_nat(nat2,aa_int_int(times_times_int(Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z5))) ) ) ).

% nat_mult_distrib_neg
tff(fact_8781_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(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(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_8782_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_8783_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_8784_diff__nat__eq__if,axiom,
    ! [Z5: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) = aa_int_nat(nat2,Z) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(minus_minus_int(Z),Z5)),zero_zero_int)),zero_zero_nat),aa_int_nat(nat2,aa_int_int(minus_minus_int(Z),Z5))) ) ) ) ).

% diff_nat_eq_if
tff(fact_8785_eucl__rel__int__remainderI,axiom,
    ! [R: int,L: int,K: int,Q2: int] :
      ( ( aa_int_int(sgn_sgn_int,R) = 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,R)),aa_int_int(abs_abs_int,L)))
       => ( ( K = aa_int_int(plus_plus_int(aa_int_int(times_times_int(Q2),L)),R) )
         => eucl_rel_int(K,L,product_Pair_int_int(Q2,R)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_8786_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K2: int] :
            ( ( A1 = K2 )
            & ( A22 = zero_zero_int )
            & ( A32 = product_Pair_int_int(zero_zero_int,K2) ) )
        | ? [L4: int,K2: int,Q4: int] :
            ( ( A1 = K2 )
            & ( A22 = L4 )
            & ( A32 = product_Pair_int_int(Q4,zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K2 = aa_int_int(times_times_int(Q4),L4) ) )
        | ? [R5: int,L4: int,K2: int,Q4: int] :
            ( ( A1 = K2 )
            & ( A22 = L4 )
            & ( A32 = product_Pair_int_int(Q4,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)))
            & ( K2 = aa_int_int(plus_plus_int(aa_int_int(times_times_int(Q4),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_8787_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( eucl_rel_int(A1,A22,A32)
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != product_Pair_int_int(zero_zero_int,A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = product_Pair_int_int(Q3,zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(times_times_int(Q3),A22) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = 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(plus_plus_int(aa_int_int(times_times_int(Q3),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_8788_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)))),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_8789_even__nat__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(nat2,K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)) ) ) ).

% even_nat_iff
tff(fact_8790_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_8791_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_8792_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_int ) )
      & ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
            | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_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_8793_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_cg,N))),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_xh(C),N)))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_8794_and__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( ( K = zero_zero_int )
          | ( L = zero_zero_int ) )
       => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = zero_zero_int ) )
      & ( ~ ( ( K = zero_zero_int )
            | ( L = zero_zero_int ) )
       => ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = L ) )
          & ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
           => ( ( ( L = aa_int_int(uminus_uminus_int,one_one_int) )
               => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = K ) )
              & ( ( L != aa_int_int(uminus_uminus_int,one_one_int) )
               => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_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(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% and_int_unfold
tff(fact_8795_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sgn_sgn_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% sgn_le_0_iff
tff(fact_8796_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% zero_le_sgn_iff
tff(fact_8797_real__root__zero,axiom,
    ! [N: nat] : aa_real_real(aa_nat_fun_real_real(root,N),zero_zero_real) = zero_zero_real ).

% real_root_zero
tff(fact_8798_concat__bit__of__zero__2,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_concat_bit(N,K),zero_zero_int) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) ).

% concat_bit_of_zero_2
tff(fact_8799_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_8800_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_8801_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_8802_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se725231765392027082nd_int(K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% and_nonnegative_int_iff
tff(fact_8803_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_8804_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_8805_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_8806_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_8807_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_8808_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_8809_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_8810_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_8811_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_8812_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_8813_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_8814_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_8815_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_8816_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_8817_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_8818_and__minus__numerals_I2_J,axiom,
    ! [N: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = one_one_int ).

% and_minus_numerals(2)
tff(fact_8819_and__minus__numerals_I6_J,axiom,
    ! [N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),one_one_int) = one_one_int ).

% and_minus_numerals(6)
tff(fact_8820_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_8821_and__minus__numerals_I5_J,axiom,
    ! [N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_8822_and__minus__numerals_I1_J,axiom,
    ! [N: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_8823_nat__take__bit__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_int_nat(nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_8824_take__bit__nat__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)) ) ) ).

% take_bit_nat_eq
tff(fact_8825_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_8826_concat__bit__take__bit__eq,axiom,
    ! [N: nat,B: int] : bit_concat_bit(N,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),B)) = bit_concat_bit(N,B) ).

% concat_bit_take_bit_eq
tff(fact_8827_concat__bit__eq__iff,axiom,
    ! [N: nat,K: int,L: int,R: int,S: int] :
      ( ( aa_int_int(bit_concat_bit(N,K),L) = aa_int_int(bit_concat_bit(N,R),S) )
    <=> ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),R) )
        & ( L = S ) ) ) ).

% concat_bit_eq_iff
tff(fact_8828_real__root__mult__exp,axiom,
    ! [M: nat,N: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),X) = aa_real_real(aa_nat_fun_real_real(root,M),aa_real_real(aa_nat_fun_real_real(root,N),X)) ).

% real_root_mult_exp
tff(fact_8829_real__root__mult,axiom,
    ! [N: nat,X: real,Y: real] : aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_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_mult
tff(fact_8830_real__root__minus,axiom,
    ! [N: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) ).

% real_root_minus
tff(fact_8831_take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(times_times_int(aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),L))) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(times_times_int(K),L)) ).

% take_bit_mult
tff(fact_8832_take__bit__minus,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(uminus_uminus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K))) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(uminus_uminus_int,K)) ).

% take_bit_minus
tff(fact_8833_take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(minus_minus_int(aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),L))) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(minus_minus_int(K),L)) ).

% take_bit_diff
tff(fact_8834_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q2: 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),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),Q2))) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_8835_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_8836_real__root__commute,axiom,
    ! [M: nat,N: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,M),aa_real_real(aa_nat_fun_real_real(root,N),X)) = aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(aa_nat_fun_real_real(root,M),X)) ).

% real_root_commute
tff(fact_8837_real__root__divide,axiom,
    ! [N: nat,X: real,Y: real] : aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_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_divide
tff(fact_8838_real__root__inverse,axiom,
    ! [N: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,X)) = aa_real_real(inverse_inverse_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) ).

% real_root_inverse
tff(fact_8839_real__root__pos__pos__le,axiom,
    ! [X: real,N: nat] :
      ( 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_nat_fun_real_real(root,N),X))) ) ).

% real_root_pos_pos_le
tff(fact_8840_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),K)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K))) ) ).

% take_bit_tightened_less_eq_int
tff(fact_8841_AND__lower,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),aa_int_int(bit_se725231765392027082nd_int(X),Y))) ) ).

% AND_lower
tff(fact_8842_AND__upper1,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,aa_int_int(bit_se725231765392027082nd_int(X),Y)),X)) ) ).

% AND_upper1
tff(fact_8843_AND__upper2,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_int_int(bit_se725231765392027082nd_int(X),Y)),Y)) ) ).

% AND_upper2
tff(fact_8844_AND__upper1_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_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Z)) ) ) ).

% AND_upper1'
tff(fact_8845_AND__upper2_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_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),Z)) ) ) ).

% AND_upper2'
tff(fact_8846_take__bit__nonnegative,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K))) ).

% take_bit_nonnegative
tff(fact_8847_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_8848_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(aa_nat_fun_int_int(bit_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_8849_not__take__bit__negative,axiom,
    ! [N: nat,K: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),zero_zero_int)) ).

% not_take_bit_negative
tff(fact_8850_real__sgn__eq,axiom,
    ! [X: real] : aa_real_real(sgn_sgn_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(abs_abs_real,X)) ).

% real_sgn_eq
tff(fact_8851_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_8852_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_8853_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_8854_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_8855_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_8856_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_8857_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_8858_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_8859_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_8860_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_8861_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) != zero_zero_int )
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(minus_minus_int(K),one_one_int)) = aa_int_int(minus_minus_int(aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_8862_sgn__eq,axiom,
    ! [Z: complex] : aa_complex_complex(sgn_sgn_complex,Z) = aa_complex_complex(divide1717551699836669952omplex(Z),aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,Z))) ).

% sgn_eq
tff(fact_8863_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_8864_real__root__strict__decreasing,axiom,
    ! [N: nat,N3: 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),N3))
       => ( 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,N3),X)),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ) ).

% real_root_strict_decreasing
tff(fact_8865_sqrt__def,axiom,
    sqrt = aa_nat_fun_real_real(root,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% sqrt_def
tff(fact_8866_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_8867_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_8868_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L)) ) ) ).

% even_and_iff_int
tff(fact_8869_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) = aa_nat_int(bit_se2000444600071755411sk_int,N) )
    <=> ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(plus_plus_int(K),one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_8870_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_8871_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_8872_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_8873_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_8874_real__root__strict__increasing,axiom,
    ! [N: nat,N3: 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),N3))
       => ( 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,N3),X))) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_8875_real__root__decreasing,axiom,
    ! [N: nat,N3: 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),N3))
       => ( 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,N3),X)),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ) ).

% real_root_decreasing
tff(fact_8876_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_8877_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_8878_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_8879_odd__real__root__pow,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))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),N) = X ) ) ).

% odd_real_root_pow
tff(fact_8880_odd__real__root__unique,axiom,
    ! [N: nat,Y: real,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))
     => ( ( 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 ) ) ) ).

% odd_real_root_unique
tff(fact_8881_odd__real__root__power__cancel,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))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = X ) ) ).

% odd_real_root_power_cancel
tff(fact_8882_take__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% take_bit_nat_def
tff(fact_8883_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_8884_take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% take_bit_int_less_exp
tff(fact_8885_take__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) = aa_int_int(modulo_modulo_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% take_bit_int_def
tff(fact_8886_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_8887_real__root__increasing,axiom,
    ! [N: nat,N3: 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),N3))
       => ( 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,N3),X))) ) ) ) ) ).

% real_root_increasing
tff(fact_8888_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(times_times_int(aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_minus_bit0
tff(fact_8889_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_8890_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K)) ) ).

% take_bit_int_less_self_iff
tff(fact_8891_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_8892_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_8893_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_8894_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_8895_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) = K )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_8896_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(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_8897_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(times_times_int(aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_numeral_minus_bit0
tff(fact_8898_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) != aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) )
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(plus_plus_int(K),one_one_int)) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)) ) ) ).

% take_bit_incr_eq
tff(fact_8899_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_8900_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_8901_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(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),aa_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_8902_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(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(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K))) ) ).

% take_bit_int_greater_eq
tff(fact_8903_signed__take__bit__eq__take__bit__shift,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K) = aa_int_int(minus_minus_int(aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),aa_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_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_8904_and__int__rec,axiom,
    ! [K: int,L: int] : aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_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(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% and_int_rec
tff(fact_8905_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) = aa_nat_int(bit_se2000444600071755411sk_int,N) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),aa_int_int(plus_plus_int(K),one_one_int))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_8906_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(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(uminus_uminus_int,K)) = aa_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_8907_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,X)),pi)),numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(arctan,X)) ) ) ).

% arctan_inverse
tff(fact_8908_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_8909_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(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(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.elims
tff(fact_8910_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(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(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.simps
tff(fact_8911_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_8912_pred__numeral__inc,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,inc(K)) = aa_num_nat(numeral_numeral_nat,K) ).

% pred_numeral_inc
tff(fact_8913_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_8914_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_8915_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_8916_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_8917_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_8918_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_8919_num__induct,axiom,
    ! [P: fun_num_bool,X: num] :
      ( pp(aa_num_bool(P,one))
     => ( ! [X2: num] :
            ( pp(aa_num_bool(P,X2))
           => pp(aa_num_bool(P,inc(X2))) )
       => pp(aa_num_bool(P,X)) ) ) ).

% num_induct
tff(fact_8920_add__inc,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,X),inc(Y)) = inc(aa_num_num(aa_num_fun_num_num(plus_plus_num,X),Y)) ).

% add_inc
tff(fact_8921_inc_Osimps_I1_J,axiom,
    inc(one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_8922_inc_Osimps_I2_J,axiom,
    ! [X: num] : inc(aa_num_num(bit0,X)) = aa_num_num(bit1,X) ).

% inc.simps(2)
tff(fact_8923_inc_Osimps_I3_J,axiom,
    ! [X: num] : inc(aa_num_num(bit1,X)) = aa_num_num(bit0,inc(X)) ).

% inc.simps(3)
tff(fact_8924_add__One,axiom,
    ! [X: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,X),one) = inc(X) ).

% add_One
tff(fact_8925_inc__BitM__eq,axiom,
    ! [N: num] : inc(bitM(N)) = aa_num_num(bit0,N) ).

% inc_BitM_eq
tff(fact_8926_BitM__inc__eq,axiom,
    ! [N: num] : bitM(inc(N)) = aa_num_num(bit1,N) ).

% BitM_inc_eq
tff(fact_8927_and__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_int_nat(nat2,aa_int_int(bit_se725231765392027082nd_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% and_nat_def
tff(fact_8928_mult__inc,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(times_times_num,X),inc(Y)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(times_times_num,X),Y)),X) ).

% mult_inc
tff(fact_8929_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),aa_int_int(plus_plus_int(one_one_int),N)))
     => ( set_or1266510415728281911st_int(M,aa_int_int(plus_plus_int(one_one_int),N)) = insert_int(aa_int_int(plus_plus_int(one_one_int),N),set_or1266510415728281911st_int(M,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_8930_simp__from__to,axiom,
    ! [J: int,I: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( set_or1266510415728281911st_int(I,J) = bot_bot_set_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( set_or1266510415728281911st_int(I,J) = insert_int(I,set_or1266510415728281911st_int(aa_int_int(plus_plus_int(I),one_one_int),J)) ) ) ) ).

% simp_from_to
tff(fact_8931_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_8932_and__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_8933_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,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(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(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_8934_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X),Xa2) = Y )
     => ( accp_P1096762738010456898nt_int(bit_and_int_rel,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(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(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) )
           => ~ accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(X,Xa2)) ) ) ) ).

% and_int.pelims
tff(fact_8935_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(aa_com8943241093633526360l_bool(aTP_Lamm_xi,Z)) ) ) ) ).

% Arg_def
tff(fact_8936_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_8937_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,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(M),N) ) ) ).

% atLeastAtMost_insertL
tff(fact_8938_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)))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_8939_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))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),N) = insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_8940_lessThan__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(pred_numeral,K))) ).

% lessThan_nat_numeral
tff(fact_8941_atMost__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(numeral_numeral_nat,K),aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(pred_numeral,K))) ).

% atMost_nat_numeral
tff(fact_8942_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_8943_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(N),aa_nat_set_nat(nat_set_decode,Z)))
     => ( aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),Z)) = insert_nat(N,aa_nat_set_nat(nat_set_decode,Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_8944_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(A0,A1))
     => ( ! [K3: int,L3: int] :
            ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(K3,L3))
           => ( ( ~ ( pp(aa_set_int_bool(member_int(K3),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(aa_int_fun_int_int(divide_divide_int,K3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_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,K3),L3)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).

% and_int.pinduct
tff(fact_8945_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(A0,A1))
     => ( ! [I3: int,J2: int] :
            ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(I3,J2))
           => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I3),J2))
               => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(plus_plus_int(I3),one_one_int)),J2)) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P,I3),J2)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).

% upto.pinduct
tff(fact_8946_signed__take__bit__eq__take__bit__minus,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K) = aa_int_int(minus_minus_int(aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),K)),aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N))),aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_8947_or__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
          | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) )
       => ( aa_int_int(bit_se1409905431419307370or_int(K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) )
      & ( ~ ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
            | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) )
       => ( ( ( K = zero_zero_int )
           => ( aa_int_int(bit_se1409905431419307370or_int(K),L) = L ) )
          & ( ( K != zero_zero_int )
           => ( ( ( L = zero_zero_int )
               => ( aa_int_int(bit_se1409905431419307370or_int(K),L) = K ) )
              & ( ( L != zero_zero_int )
               => ( aa_int_int(bit_se1409905431419307370or_int(K),L) = aa_int_int(plus_plus_int(aa_int_int(ord_max_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% or_int_unfold
tff(fact_8948_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se1409905431419307370or_int(K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% or_nonnegative_int_iff
tff(fact_8949_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_8950_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)))
    <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_8951_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(aa_nat_fun_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_8952_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),N)) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_8953_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_nat_nat(suc,N)))
    <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),N)) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_8954_or__minus__numerals_I6_J,axiom,
    ! [N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))) ).

% or_minus_numerals(6)
tff(fact_8955_or__minus__numerals_I2_J,axiom,
    ! [N: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))) ).

% or_minus_numerals(2)
tff(fact_8956_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N: num] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),aa_num_nat(pred_numeral,N))) ) ).

% bit_minus_numeral_int(1)
tff(fact_8957_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N: num] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_num_nat(numeral_numeral_nat,N)))
    <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,N))) ) ).

% bit_minus_numeral_int(2)
tff(fact_8958_bit__or__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(bit_se1409905431419307370or_int(K),L)),N))
    <=> ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N))
        | pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,L),N)) ) ) ).

% bit_or_int_iff
tff(fact_8959_bit__and__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),N))
    <=> ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,L),N)) ) ) ).

% bit_and_int_iff
tff(fact_8960_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(bit_se1409905431419307370or_int(K),L))) ) ).

% or_greater_eq
tff(fact_8961_OR__lower,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,zero_zero_int),aa_int_int(bit_se1409905431419307370or_int(X),Y))) ) ) ).

% OR_lower
tff(fact_8962_plus__and__or,axiom,
    ! [X: int,Y: int] : aa_int_int(plus_plus_int(aa_int_int(bit_se725231765392027082nd_int(X),Y)),aa_int_int(bit_se1409905431419307370or_int(X),Y)) = aa_int_int(plus_plus_int(X),Y) ).

% plus_and_or
tff(fact_8963_pow_Osimps_I1_J,axiom,
    ! [X: num] : aa_num_num(aa_num_fun_num_num(pow,X),one) = X ).

% pow.simps(1)
tff(fact_8964_bit__not__int__iff_H,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,K)),one_one_int)),N))
    <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ).

% bit_not_int_iff'
tff(fact_8965_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),K))) ) ) ).

% bit_imp_take_bit_positive
tff(fact_8966_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_8967_signed__take__bit__eq__concat__bit,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K) = aa_int_int(bit_concat_bit(N,K),aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_8968_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_8969_bit__int__def,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ).

% bit_int_def
tff(fact_8970_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_8971_or__int__rec,axiom,
    ! [K: int,L: int] : aa_int_int(bit_se1409905431419307370or_int(K),L) = aa_int_int(plus_plus_int(aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fdisj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% or_int_rec
tff(fact_8972_set__bit__eq,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),K) = aa_int_int(plus_plus_int(K),aa_int_int(times_times_int(aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% set_bit_eq
tff(fact_8973_unset__bit__eq,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N),K) = aa_int_int(minus_minus_int(K),aa_int_int(times_times_int(aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% unset_bit_eq
tff(fact_8974_take__bit__Suc__from__most,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),K) = aa_int_int(plus_plus_int(aa_int_int(times_times_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_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)) ).

% take_bit_Suc_from_most
tff(fact_8975_or__minus__numerals_I1_J,axiom,
    ! [N: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),bitM(N)))) ).

% or_minus_numerals(1)
tff(fact_8976_or__minus__numerals_I5_J,axiom,
    ! [N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),bitM(N)))) ).

% or_minus_numerals(5)
tff(fact_8977_cis__multiple__2pi,axiom,
    ! [N: real] :
      ( member_real(N,ring_1_Ints_real)
     => ( aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),N)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_8978_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_8979_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_8980_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_8981_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_8982_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_8983_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_8984_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_8985_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_8986_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_8987_or__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),aa_num_num(bit0,N)))) ).

% or_minus_numerals(8)
tff(fact_8988_or__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),aa_num_num(bit0,N)))) ).

% or_minus_numerals(4)
tff(fact_8989_or__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),bitM(N)))) ).

% or_minus_numerals(7)
tff(fact_8990_or__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),bitM(N)))) ).

% or_minus_numerals(3)
tff(fact_8991_or__not__num__neg_Osimps_I1_J,axiom,
    aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),one) = one ).

% or_not_num_neg.simps(1)
tff(fact_8992_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_8993_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_8994_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,N)),one) = aa_num_num(bit0,one) ).

% or_not_num_neg.simps(4)
tff(fact_8995_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,N)),aa_num_num(bit1,M)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ).

% or_not_num_neg.simps(6)
tff(fact_8996_or__not__num__neg_Osimps_I7_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,N)),one) = one ).

% or_not_num_neg.simps(7)
tff(fact_8997_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit1,M)) = aa_num_num(bit1,M) ).

% or_not_num_neg.simps(3)
tff(fact_8998_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,N)),aa_num_num(bit0,M)) = bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ).

% or_not_num_neg.simps(5)
tff(fact_8999_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_9000_or__not__num__neg_Osimps_I9_J,axiom,
    ! [N: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,N)),aa_num_num(bit1,M)) = bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ).

% or_not_num_neg.simps(9)
tff(fact_9001_or__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_int_nat(nat2,aa_int_int(bit_se1409905431419307370or_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% or_nat_def
tff(fact_9002_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit0,M)) = aa_num_num(bit1,M) ).

% or_not_num_neg.simps(2)
tff(fact_9003_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,N)),aa_num_num(bit0,M)) = bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ).

% or_not_num_neg.simps(8)
tff(fact_9004_bit__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),N))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ) ).

% bit_nat_iff
tff(fact_9005_sin__times__pi__eq__0,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),pi)) = zero_zero_real )
    <=> member_real(X,ring_1_Ints_real) ) ).

% sin_times_pi_eq_0
tff(fact_9006_or__not__num__neg_Oelims,axiom,
    ! [X: num,Xa2: num,Y: num] :
      ( ( aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,X),Xa2) = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != one ) ) )
       => ( ( ( X = one )
           => ! [M3: num] :
                ( ( Xa2 = aa_num_num(bit0,M3) )
               => ( Y != aa_num_num(bit1,M3) ) ) )
         => ( ( ( X = one )
             => ! [M3: num] :
                  ( ( Xa2 = aa_num_num(bit1,M3) )
                 => ( Y != aa_num_num(bit1,M3) ) ) )
           => ( ( ? [N2: num] : X = aa_num_num(bit0,N2)
               => ( ( Xa2 = one )
                 => ( Y != aa_num_num(bit0,one) ) ) )
             => ( ! [N2: num] :
                    ( ( X = aa_num_num(bit0,N2) )
                   => ! [M3: num] :
                        ( ( Xa2 = aa_num_num(bit0,M3) )
                       => ( Y != bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M3)) ) ) )
               => ( ! [N2: num] :
                      ( ( X = aa_num_num(bit0,N2) )
                     => ! [M3: num] :
                          ( ( Xa2 = aa_num_num(bit1,M3) )
                         => ( Y != aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M3)) ) ) )
                 => ( ( ? [N2: num] : X = aa_num_num(bit1,N2)
                     => ( ( Xa2 = one )
                       => ( Y != one ) ) )
                   => ( ! [N2: num] :
                          ( ( X = aa_num_num(bit1,N2) )
                         => ! [M3: num] :
                              ( ( Xa2 = aa_num_num(bit0,M3) )
                             => ( Y != bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M3)) ) ) )
                     => ~ ! [N2: num] :
                            ( ( X = aa_num_num(bit1,N2) )
                           => ! [M3: num] :
                                ( ( Xa2 = aa_num_num(bit1,M3) )
                               => ( Y != bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M3)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_9007_bit__nat__def,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,M),N))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ) ).

% bit_nat_def
tff(fact_9008_sin__integer__2pi,axiom,
    ! [N: real] :
      ( member_real(N,ring_1_Ints_real)
     => ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),N)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_9009_cos__integer__2pi,axiom,
    ! [N: real] :
      ( member_real(N,ring_1_Ints_real)
     => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),N)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_9010_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_9011_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_9012_or__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(aa_bool_bool(fdisj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_9013_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_9014_xor__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_bool_bool(fequal_bool(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_9015_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_9016_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_9017_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_9018_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% push_bit_nonnegative_int_iff
tff(fact_9019_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(aa_nat_fun_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_9020_concat__bit__of__zero__1,axiom,
    ! [N: nat,L: int] : aa_int_int(bit_concat_bit(N,zero_zero_int),L) = aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),L) ).

% concat_bit_of_zero_1
tff(fact_9021_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se6526347334894502574or_int(K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% xor_nonnegative_int_iff
tff(fact_9022_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_9023_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_9024_bit__xor__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(bit_se6526347334894502574or_int(K),L)),N))
    <=> ~ ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N))
        <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,L),N)) ) ) ).

% bit_xor_int_iff
tff(fact_9025_flip__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,N),K) = aa_int_int(bit_se6526347334894502574or_int(K),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int)) ).

% flip_bit_int_def
tff(fact_9026_push__bit__nat__eq,axiom,
    ! [N: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),K)) ).

% push_bit_nat_eq
tff(fact_9027_XOR__lower,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,zero_zero_int),aa_int_int(bit_se6526347334894502574or_int(X),Y))) ) ) ).

% XOR_lower
tff(fact_9028_flip__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_9029_set__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% set_bit_nat_def
tff(fact_9030_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,M),K)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% bit_push_bit_iff_int
tff(fact_9031_xor__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_int_nat(nat2,aa_int_int(bit_se6526347334894502574or_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% xor_nat_def
tff(fact_9032_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: 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),Q2)),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,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9033_concat__bit__eq,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_concat_bit(N,K),L) = aa_int_int(plus_plus_int(aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),L)) ).

% concat_bit_eq
tff(fact_9034_concat__bit__def,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_concat_bit(N,K),L) = aa_int_int(bit_se1409905431419307370or_int(aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),L)) ).

% concat_bit_def
tff(fact_9035_set__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),K) = aa_int_int(bit_se1409905431419307370or_int(K),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int)) ).

% set_bit_int_def
tff(fact_9036_push__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),K) = aa_int_int(times_times_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% push_bit_int_def
tff(fact_9037_push__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% push_bit_nat_def
tff(fact_9038_push__bit__minus__one,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% push_bit_minus_one
tff(fact_9039_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_9040_xor__int__rec,axiom,
    ! [K: int,L: int] : aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(plus_plus_int(aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_bool_bool(fequal_bool(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L)))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% xor_int_rec
tff(fact_9041_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
       => ( aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(bit_ri7919022796975470100ot_int,L) ) )
      & ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
       => ( ( ( L = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(bit_ri7919022796975470100ot_int,K) ) )
          & ( ( L != aa_int_int(uminus_uminus_int,one_one_int) )
           => ( ( ( K = zero_zero_int )
               => ( aa_int_int(bit_se6526347334894502574or_int(K),L) = L ) )
              & ( ( K != zero_zero_int )
               => ( ( ( L = zero_zero_int )
                   => ( aa_int_int(bit_se6526347334894502574or_int(K),L) = K ) )
                  & ( ( L != zero_zero_int )
                   => ( aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
tff(fact_9042_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_an),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_9043_Cauchy__iff2,axiom,
    ! [X5: fun_nat_real] :
      ( topolo4055970368930404560y_real(X5)
    <=> ! [J3: nat] :
        ? [M8: nat] :
        ! [M7: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M8),M7))
         => ! [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(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(X5,M7)),aa_nat_real(X5,N5)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3))))) ) ) ) ).

% Cauchy_iff2
tff(fact_9044_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)))),aa_nat_set_nat(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_9045_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(L),U))) ).

% finite_atLeastLessThan
tff(fact_9046_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_9047_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_9048_or__minus__minus__numerals,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se725231765392027082nd_int(aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,N)),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_9049_and__minus__minus__numerals,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se1409905431419307370or_int(aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,N)),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_9050_bit__not__int__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),N))
    <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ).

% bit_not_int_iff
tff(fact_9051_all__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M7: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M7),N))
         => pp(aa_nat_bool(P,M7)) )
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
         => pp(aa_nat_bool(P,X3)) ) ) ).

% all_nat_less_eq
tff(fact_9052_ex__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M7: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M7),N))
          & pp(aa_nat_bool(P,M7)) )
    <=> ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
          & pp(aa_nat_bool(P,X3)) ) ) ).

% ex_nat_less_eq
tff(fact_9053_or__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(bit_se1409905431419307370or_int(K),L) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_int_int(bit_ri7919022796975470100ot_int,L))) ).

% or_int_def
tff(fact_9054_not__int__def,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,K)),one_one_int) ).

% not_int_def
tff(fact_9055_and__not__numerals_I1_J,axiom,
    aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = zero_zero_int ).

% and_not_numerals(1)
tff(fact_9056_or__not__numerals_I1_J,axiom,
    aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(1)
tff(fact_9057_unset__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N),K) = aa_int_int(bit_se725231765392027082nd_int(K),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int))) ).

% unset_bit_int_def
tff(fact_9058_xor__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_se725231765392027082nd_int(K),aa_int_int(bit_ri7919022796975470100ot_int,L))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,K)),L)) ).

% xor_int_def
tff(fact_9059_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N3: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(N3),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N3)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_9060_not__int__div__2,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% not_int_div_2
tff(fact_9061_even__not__iff__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,K)))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)) ) ).

% even_not_iff_int
tff(fact_9062_and__not__numerals_I2_J,axiom,
    ! [N: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = one_one_int ).

% and_not_numerals(2)
tff(fact_9063_and__not__numerals_I4_J,axiom,
    ! [M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).

% and_not_numerals(4)
tff(fact_9064_or__not__numerals_I2_J,axiom,
    ! [N: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) ).

% or_not_numerals(2)
tff(fact_9065_or__not__numerals_I4_J,axiom,
    ! [M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ).

% or_not_numerals(4)
tff(fact_9066_bit__minus__int__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,K)),N))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(minus_minus_int(K),one_one_int))),N)) ) ).

% bit_minus_int_iff
tff(fact_9067_numeral__or__not__num__eq,axiom,
    ! [M: num,N: num] : aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),N)) = aa_int_int(uminus_uminus_int,aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% numeral_or_not_num_eq
tff(fact_9068_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M))) ).

% int_numeral_not_or_num_neg
tff(fact_9069_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),N))) ).

% int_numeral_or_not_num_neg
tff(fact_9070_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N)) = insert_nat(N,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
tff(fact_9071_and__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),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)))) ).

% and_not_numerals(5)
tff(fact_9072_and__not__numerals_I7_J,axiom,
    ! [M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).

% and_not_numerals(7)
tff(fact_9073_or__not__numerals_I3_J,axiom,
    ! [N: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) ).

% or_not_numerals(3)
tff(fact_9074_and__not__numerals_I3_J,axiom,
    ! [N: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_9075_or__not__numerals_I7_J,axiom,
    ! [M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(7)
tff(fact_9076_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)))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(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)))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(numeral_numeral_nat,K)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
tff(fact_9077_and__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),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)))) ).

% and_not_numerals(9)
tff(fact_9078_and__not__numerals_I6_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),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)))) ).

% and_not_numerals(6)
tff(fact_9079_or__not__numerals_I6_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% or_not_numerals(6)
tff(fact_9080_or__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(5)
tff(fact_9081_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),N) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_9082_and__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),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))))) ).

% and_not_numerals(8)
tff(fact_9083_or__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(9)
tff(fact_9084_or__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(8)
tff(fact_9085_not__int__rec,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(plus_plus_int(aa_bool_int(zero_n2684676970156552555ol_int,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% not_int_rec
tff(fact_9086_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I3: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),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,I3)),aa_nat_nat(A,J2))) ) )
     => ( ! [I3: nat,J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),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,I3))) ) )
       => 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_td(A),B)),aa_nat_set_nat(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),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),aa_set_nat_nat(groups3542108847815614940at_nat(B),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_9087_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(L,aa_int_int(plus_plus_int(U),one_one_int)) = set_or1266510415728281911st_int(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_9088_valid__eq,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,T),D))
    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),D)) ) ).

% valid_eq
tff(fact_9089_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),D))
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,T),D)) ) ).

% valid_eq1
tff(fact_9090_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,T),D))
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),D)) ) ).

% valid_eq2
tff(fact_9091_Code__Target__Int_Opositive__def,axiom,
    code_Target_positive = numeral_numeral_int ).

% Code_Target_Int.positive_def
tff(fact_9092_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_gn(L),Qr) ).

% divmod_step_integer_def
tff(fact_9093_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,L) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,L) ).

% minus_integer_code(2)
tff(fact_9094_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : minus_8373710615458151222nteger(K,zero_z3403309356797280102nteger) = K ).

% minus_integer_code(1)
tff(fact_9095_times__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger),L) = zero_z3403309356797280102nteger ).

% times_integer_code(2)
tff(fact_9096_times__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% times_integer_code(1)
tff(fact_9097_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : plus_p5714425477246183910nteger(zero_z3403309356797280102nteger,L) = L ).

% plus_integer_code(2)
tff(fact_9098_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : plus_p5714425477246183910nteger(K,zero_z3403309356797280102nteger) = K ).

% plus_integer_code(1)
tff(fact_9099_divmod__integer_H__def,axiom,
    ! [M: num,N: num] : unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_num_Code_integer(numera6620942414471956472nteger,N)),modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_num_Code_integer(numera6620942414471956472nteger,N))) ).

% divmod_integer'_def
tff(fact_9100_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_9101_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,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_Code_integer)) ) ) ) ) ) ).

% integer_of_int_code
tff(fact_9102_Code__Numeral_Opositive__def,axiom,
    code_positive = numera6620942414471956472nteger ).

% Code_Numeral.positive_def
tff(fact_9103_modulo__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : modulo364778990260209775nteger(aa_int_Code_integer(code_integer_of_int,Xa2),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(modulo_modulo_int(Xa2),X)) ).

% modulo_integer.abs_eq
tff(fact_9104_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_9105_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_9106_divide__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : divide6298287555418463151nteger(aa_int_Code_integer(code_integer_of_int,Xa2),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa2),X)) ).

% divide_integer.abs_eq
tff(fact_9107_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_9108_plus__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : plus_p5714425477246183910nteger(aa_int_Code_integer(code_integer_of_int,Xa2),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(plus_plus_int(Xa2),X)) ).

% plus_integer.abs_eq
tff(fact_9109_times__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(times_times_int(Xa2),X)) ).

% times_integer.abs_eq
tff(fact_9110_minus__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : minus_8373710615458151222nteger(aa_int_Code_integer(code_integer_of_int,Xa2),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(minus_minus_int(Xa2),X)) ).

% minus_integer.abs_eq
tff(fact_9111_integer__of__num_I3_J,axiom,
    ! [N: num] : aa_num_Code_integer(code_integer_of_num,aa_num_num(bit1,N)) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_num_Code_integer(code_integer_of_num,N),aa_num_Code_integer(code_integer_of_num,N)),one_one_Code_integer) ).

% integer_of_num(3)
tff(fact_9112_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = produc5135875973608410785r_bool(divide6298287555418463151nteger(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),K))) ).

% bit_cut_integer_def
tff(fact_9113_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_9114_integer__of__num__def,axiom,
    code_integer_of_num = numera6620942414471956472nteger ).

% integer_of_num_def
tff(fact_9115_integer__of__num__triv_I1_J,axiom,
    aa_num_Code_integer(code_integer_of_num,one) = one_one_Code_integer ).

% integer_of_num_triv(1)
tff(fact_9116_integer__of__num_I2_J,axiom,
    ! [N: num] : aa_num_Code_integer(code_integer_of_num,aa_num_num(bit0,N)) = plus_p5714425477246183910nteger(aa_num_Code_integer(code_integer_of_num,N),aa_num_Code_integer(code_integer_of_num,N)) ).

% integer_of_num(2)
tff(fact_9117_integer__of__num__triv_I2_J,axiom,
    aa_num_Code_integer(code_integer_of_num,aa_num_num(bit0,one)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% integer_of_num_triv(2)
tff(fact_9118_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_xj(K),code_divmod_abs(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ) ) ).

% bit_cut_integer_code
tff(fact_9119_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(csqrt,Z)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% csqrt.simps(1)
tff(fact_9120_card__Collect__less__nat,axiom,
    ! [N: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bh,N))) = N ).

% card_Collect_less_nat
tff(fact_9121_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_atLeastLessThan
tff(fact_9122_card__Collect__le__nat,axiom,
    ! [N: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bg,N))) = aa_nat_nat(suc,N) ).

% card_Collect_le_nat
tff(fact_9123_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,U)),L) ).

% card_atLeastAtMost
tff(fact_9124_complex__Re__numeral,axiom,
    ! [V: num] : aa_complex_real(re,numera6690914467698888265omplex(V)) = numeral_numeral_real(V) ).

% complex_Re_numeral
tff(fact_9125_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or4662586982721622107an_int(L,U)) = aa_int_nat(nat2,aa_int_int(minus_minus_int(U),L)) ).

% card_atLeastLessThan_int
tff(fact_9126_Re__divide__of__nat,axiom,
    ! [Z: complex,N: nat] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),aa_nat_complex(semiri8010041392384452111omplex,N))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% Re_divide_of_nat
tff(fact_9127_Re__divide__of__real,axiom,
    ! [Z: complex,R: real] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),aa_real_complex(real_V4546457046886955230omplex,R))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),R) ).

% Re_divide_of_real
tff(fact_9128_Re__sgn,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(sgn_sgn_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(real_V1022390504157884413omplex,Z)) ).

% Re_sgn
tff(fact_9129_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or1266510415728281911st_int(L,U)) = aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(U),L)),one_one_int)) ).

% card_atLeastAtMost_int
tff(fact_9130_Re__divide__numeral,axiom,
    ! [Z: complex,W: num] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),numeral_numeral_real(W)) ).

% Re_divide_numeral
tff(fact_9131_complex__Re__le__cmod,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(re,X)),aa_complex_real(real_V1022390504157884413omplex,X))) ).

% complex_Re_le_cmod
tff(fact_9132_plus__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(plus_plus_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(re,X)),aa_complex_real(re,Y)) ).

% plus_complex.simps(1)
tff(fact_9133_scaleR__complex_Osimps_I1_J,axiom,
    ! [R: real,X: complex] : aa_complex_real(re,aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,R),X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_complex_real(re,X)) ).

% scaleR_complex.simps(1)
tff(fact_9134_minus__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(minus_minus_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(re,X)),aa_complex_real(re,Y)) ).

% minus_complex.simps(1)
tff(fact_9135_abs__Re__le__cmod,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,X))),aa_complex_real(real_V1022390504157884413omplex,X))) ).

% abs_Re_le_cmod
tff(fact_9136_Re__csqrt,axiom,
    ! [Z: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z)))) ).

% Re_csqrt
tff(fact_9137_card__less__Suc2,axiom,
    ! [M4: set_nat,I: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M4))
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_xk(M4),I))) = finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_xl(M4),I))) ) ) ).

% card_less_Suc2
tff(fact_9138_card__less__Suc,axiom,
    ! [M4: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M4))
     => ( aa_nat_nat(suc,finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_xk(M4),I)))) = finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_xl(M4),I))) ) ) ).

% card_less_Suc
tff(fact_9139_card__less,axiom,
    ! [M4: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M4))
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_xl(M4),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_9140_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),finite_card_nat(A2)))))
     => ( A2 = aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),finite_card_nat(A2))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_9141_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N3: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(N3),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_nat(N3)),N)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_9142_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_an),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),finite_card_nat(S3)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_an),S3))) ).

% card_sum_le_nat_sum
tff(fact_9143_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_xh(C),N))) = N ) ) ) ).

% card_nth_roots
tff(fact_9144_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_cg,N))) = N ) ) ).

% card_roots_unity_eq
tff(fact_9145_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),zero_zero_real))
    <=> ( aa_complex_real(re,Z) = aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Z)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_9146_cos__n__Re__cis__pow__n,axiom,
    ! [N: nat,A: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),A)) = aa_complex_real(re,aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),N)) ).

% cos_n_Re_cis_pow_n
tff(fact_9147_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_9148_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_xm(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_xn(L),code_divmod_abs(K,L)))) ) ) ) ) ) ) ) ).

% divmod_integer_code
tff(fact_9149_csqrt_Ocode,axiom,
    ! [Z: complex] : aa_complex_complex(csqrt,Z) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,aa_complex_real(im,Z)),zero_zero_real)),one_one_real),aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(aa_num_num(bit0,one)))))) ).

% csqrt.code
tff(fact_9150_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(csqrt,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,aa_complex_real(im,Z)),zero_zero_real)),one_one_real),aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% csqrt.simps(2)
tff(fact_9151_complex__Im__numeral,axiom,
    ! [V: num] : aa_complex_real(im,numera6690914467698888265omplex(V)) = zero_zero_real ).

% complex_Im_numeral
tff(fact_9152_Im__i__times,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(imaginary_unit),Z)) = aa_complex_real(re,Z) ).

% Im_i_times
tff(fact_9153_Im__divide__of__real,axiom,
    ! [Z: complex,R: real] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),aa_real_complex(real_V4546457046886955230omplex,R))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),R) ).

% Im_divide_of_real
tff(fact_9154_Im__sgn,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(sgn_sgn_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(real_V1022390504157884413omplex,Z)) ).

% Im_sgn
tff(fact_9155_Re__power__real,axiom,
    ! [X: complex,N: nat] :
      ( ( aa_complex_real(im,X) = zero_zero_real )
     => ( aa_complex_real(re,aa_nat_complex(power_power_complex(X),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),N) ) ) ).

% Re_power_real
tff(fact_9156_Re__i__times,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(imaginary_unit),Z)) = aa_real_real(uminus_uminus_real,aa_complex_real(im,Z)) ).

% Re_i_times
tff(fact_9157_Im__divide__numeral,axiom,
    ! [Z: complex,W: num] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),numeral_numeral_real(W)) ).

% Im_divide_numeral
tff(fact_9158_Im__divide__of__nat,axiom,
    ! [Z: complex,N: nat] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),aa_nat_complex(semiri8010041392384452111omplex,N))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% Im_divide_of_nat
tff(fact_9159_csqrt__of__real__nonneg,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(im,X) = zero_zero_real )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,X)))
       => ( aa_complex_complex(csqrt,X) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_complex_real(re,X))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_9160_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(im,X)),zero_zero_real))
        | ( ( aa_complex_real(im,X) = zero_zero_real )
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,X))) ) )
     => ( aa_complex_complex(csqrt,aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(csqrt,X)) ) ) ).

% csqrt_minus
tff(fact_9161_csqrt__of__real__nonpos,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(im,X) = zero_zero_real )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(re,X)),zero_zero_real))
       => ( aa_complex_complex(csqrt,X) = aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_real_real(abs_abs_real,aa_complex_real(re,X))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_9162_plus__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(plus_plus_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(im,X)),aa_complex_real(im,Y)) ).

% plus_complex.simps(2)
tff(fact_9163_scaleR__complex_Osimps_I2_J,axiom,
    ! [R: real,X: complex] : aa_complex_real(im,aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,R),X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_complex_real(im,X)) ).

% scaleR_complex.simps(2)
tff(fact_9164_minus__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(minus_minus_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(im,X)),aa_complex_real(im,Y)) ).

% minus_complex.simps(2)
tff(fact_9165_abs__Im__le__cmod,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,X))),aa_complex_real(real_V1022390504157884413omplex,X))) ).

% abs_Im_le_cmod
tff(fact_9166_times__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(re,Y))) ).

% times_complex.simps(2)
tff(fact_9167_cmod__Im__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_real(re,X) = aa_complex_real(re,Y) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,X))),aa_real_real(abs_abs_real,aa_complex_real(im,Y)))) ) ) ).

% cmod_Im_le_iff
tff(fact_9168_cmod__Re__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_real(im,X) = aa_complex_real(im,Y) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,X))),aa_real_real(abs_abs_real,aa_complex_real(re,Y)))) ) ) ).

% cmod_Re_le_iff
tff(fact_9169_times__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(im,Y))) ).

% times_complex.simps(1)
tff(fact_9170_plus__complex_Ocode,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(plus_plus_complex(X),Y) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(im,X)),aa_complex_real(im,Y))) ).

% plus_complex.code
tff(fact_9171_scaleR__complex_Ocode,axiom,
    ! [R: real,X: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,R),X) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_complex_real(re,X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_complex_real(im,X))) ).

% scaleR_complex.code
tff(fact_9172_minus__complex_Ocode,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(minus_minus_complex(X),Y) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(im,X)),aa_complex_real(im,Y))) ).

% minus_complex.code
tff(fact_9173_csqrt__principal,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z))))
      | ( ( aa_complex_real(re,aa_complex_complex(csqrt,Z)) = zero_zero_real )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(csqrt,Z)))) ) ) ).

% csqrt_principal
tff(fact_9174_cmod__le,axiom,
    ! [Z: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_complex_real(re,Z))),aa_real_real(abs_abs_real,aa_complex_real(im,Z))))) ).

% cmod_le
tff(fact_9175_sin__n__Im__cis__pow__n,axiom,
    ! [N: nat,A: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),A)) = aa_complex_real(im,aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),N)) ).

% sin_n_Im_cis_pow_n
tff(fact_9176_Re__exp,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(exp_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,aa_complex_real(re,Z))),aa_real_real(cos_real,aa_complex_real(im,Z))) ).

% Re_exp
tff(fact_9177_Im__exp,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(exp_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,aa_complex_real(re,Z))),aa_real_real(sin_real,aa_complex_real(im,Z))) ).

% Im_exp
tff(fact_9178_complex__eq,axiom,
    ! [A: complex] : A = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(re,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(im,A)))) ).

% complex_eq
tff(fact_9179_times__complex_Ocode,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(times_times_complex(X),Y) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(re,Y)))) ).

% times_complex.code
tff(fact_9180_exp__eq__polar,axiom,
    ! [Z: complex] : aa_complex_complex(exp_complex,Z) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(exp_real,aa_complex_real(re,Z)))),aa_real_complex(cis,aa_complex_real(im,Z))) ).

% exp_eq_polar
tff(fact_9181_cmod__power2,axiom,
    ! [Z: complex] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cmod_power2
tff(fact_9182_Im__power2,axiom,
    ! [X: complex] : aa_complex_real(im,aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_complex_real(re,X))),aa_complex_real(im,X)) ).

% Im_power2
tff(fact_9183_Re__power2,axiom,
    ! [X: complex] : aa_complex_real(re,aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% Re_power2
tff(fact_9184_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero_complex )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real ) ) ).

% complex_eq_0
tff(fact_9185_norm__complex__def,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% norm_complex_def
tff(fact_9186_inverse__complex_Osimps_I1_J,axiom,
    ! [X: complex] : aa_complex_real(re,aa_complex_complex(invers8013647133539491842omplex,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(1)
tff(fact_9187_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
    <=> 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,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% complex_neq_0
tff(fact_9188_Re__divide,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(X),Y)) = 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,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Re_divide
tff(fact_9189_csqrt__square,axiom,
    ! [B: complex] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,B)))
        | ( ( aa_complex_real(re,B) = zero_zero_real )
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,B))) ) )
     => ( aa_complex_complex(csqrt,aa_nat_complex(power_power_complex(B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = B ) ) ).

% csqrt_square
tff(fact_9190_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z )
     => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,W)))
          | ( ( aa_complex_real(re,W) = zero_zero_real )
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,W))) ) )
       => ( aa_complex_complex(csqrt,Z) = W ) ) ) ).

% csqrt_unique
tff(fact_9191_inverse__complex_Osimps_I2_J,axiom,
    ! [X: complex] : aa_complex_real(im,aa_complex_complex(invers8013647133539491842omplex,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(2)
tff(fact_9192_Im__divide,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(X),Y)) = 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,aa_complex_real(im,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Im_divide
tff(fact_9193_complex__abs__le__norm,axiom,
    ! [Z: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_complex_real(re,Z))),aa_real_real(abs_abs_real,aa_complex_real(im,Z)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),aa_complex_real(real_V1022390504157884413omplex,Z)))) ).

% complex_abs_le_norm
tff(fact_9194_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ) ) ).

% complex_unit_circle
tff(fact_9195_inverse__complex_Ocode,axiom,
    ! [X: complex] : aa_complex_complex(invers8013647133539491842omplex,X) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% inverse_complex.code
tff(fact_9196_Complex__divide,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(divide1717551699836669952omplex(X),Y) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,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,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,aa_complex_real(im,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% Complex_divide
tff(fact_9197_Im__Reals__divide,axiom,
    ! [R: complex,Z: complex] :
      ( pp(aa_set_complex_bool(member_complex(R),real_V2521375963428798218omplex))
     => ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(R),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_complex_real(re,R))),aa_complex_real(im,Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Im_Reals_divide
tff(fact_9198_Re__divide__Reals,axiom,
    ! [R: complex,Z: complex] :
      ( pp(aa_set_complex_bool(member_complex(R),real_V2521375963428798218omplex))
     => ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),R)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(re,R)) ) ) ).

% Re_divide_Reals
tff(fact_9199_imaginary__eq__real__iff,axiom,
    ! [Y: complex,X: complex] :
      ( pp(aa_set_complex_bool(member_complex(Y),real_V2521375963428798218omplex))
     => ( pp(aa_set_complex_bool(member_complex(X),real_V2521375963428798218omplex))
       => ( ( aa_complex_complex(times_times_complex(imaginary_unit),Y) = X )
        <=> ( ( X = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
tff(fact_9200_real__eq__imaginary__iff,axiom,
    ! [Y: complex,X: complex] :
      ( pp(aa_set_complex_bool(member_complex(Y),real_V2521375963428798218omplex))
     => ( pp(aa_set_complex_bool(member_complex(X),real_V2521375963428798218omplex))
       => ( ( X = aa_complex_complex(times_times_complex(imaginary_unit),Y) )
        <=> ( ( X = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
tff(fact_9201_Im__divide__Reals,axiom,
    ! [R: complex,Z: complex] :
      ( pp(aa_set_complex_bool(member_complex(R),real_V2521375963428798218omplex))
     => ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),R)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(re,R)) ) ) ).

% Im_divide_Reals
tff(fact_9202_Re__Reals__divide,axiom,
    ! [R: complex,Z: complex] :
      ( pp(aa_set_complex_bool(member_complex(R),real_V2521375963428798218omplex))
     => ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(R),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,R)),aa_complex_real(re,Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Re_Reals_divide
tff(fact_9203_complex__diff__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(minus_minus_complex(Z),aa_complex_complex(cnj,Z)) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_complex_real(im,Z)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_9204_complex__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_mult_cnj
tff(fact_9205_complex__cnj__mult,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(times_times_complex(X),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(cnj,X)),aa_complex_complex(cnj,Y)) ).

% complex_cnj_mult
tff(fact_9206_complex__cnj__divide,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(divide1717551699836669952omplex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cnj,X)),aa_complex_complex(cnj,Y)) ).

% complex_cnj_divide
tff(fact_9207_complex__cnj__add,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(cnj,X)),aa_complex_complex(cnj,Y)) ).

% complex_cnj_add
tff(fact_9208_complex__cnj__numeral,axiom,
    ! [W: num] : aa_complex_complex(cnj,numera6690914467698888265omplex(W)) = numera6690914467698888265omplex(W) ).

% complex_cnj_numeral
tff(fact_9209_complex__cnj__diff,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(minus_minus_complex(X),Y)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(cnj,X)),aa_complex_complex(cnj,Y)) ).

% complex_cnj_diff
tff(fact_9210_complex__cnj__neg__numeral,axiom,
    ! [W: num] : aa_complex_complex(cnj,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% complex_cnj_neg_numeral
tff(fact_9211_complex__In__mult__cnj__zero,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z))) = zero_zero_real ).

% complex_In_mult_cnj_zero
tff(fact_9212_Re__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)) = zero_zero_real )
    <=> ( aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))) = zero_zero_real ) ) ).

% Re_complex_div_eq_0
tff(fact_9213_Im__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)) = zero_zero_real )
    <=> ( aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))) = zero_zero_real ) ) ).

% Im_complex_div_eq_0
tff(fact_9214_complex__mod__sqrt__Re__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(sqrt,aa_complex_real(re,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)))) ).

% complex_mod_sqrt_Re_mult_cnj
tff(fact_9215_Re__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real)) ) ).

% Re_complex_div_lt_0
tff(fact_9216_Re__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) ) ).

% Re_complex_div_gt_0
tff(fact_9217_Re__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) ) ).

% Re_complex_div_ge_0
tff(fact_9218_Re__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real)) ) ).

% Re_complex_div_le_0
tff(fact_9219_Im__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real)) ) ).

% Im_complex_div_lt_0
tff(fact_9220_Im__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) ) ).

% Im_complex_div_gt_0
tff(fact_9221_Im__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) ) ).

% Im_complex_div_ge_0
tff(fact_9222_Im__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real)) ) ).

% Im_complex_div_le_0
tff(fact_9223_complex__mod__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% complex_mod_mult_cnj
tff(fact_9224_complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) )
      & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) ) ) ).

% complex_div_gt_0
tff(fact_9225_complex__norm__square,axiom,
    ! [Z: complex] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)) ).

% complex_norm_square
tff(fact_9226_complex__add__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_complex_real(re,Z))) ).

% complex_add_cnj
tff(fact_9227_complex__div__cnj,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))),aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_div_cnj
tff(fact_9228_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,W))),aa_complex_complex(times_times_complex(aa_complex_complex(cnj,Z)),W)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_complex_real(re,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,W))))) ).

% cnj_add_mult_eq_Re
tff(fact_9229_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))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( ( pp(A4)
                   => ( Y = some_nat(zero_zero_nat) ) )
                  & ( ~ pp(A4)
                   => ( ( pp(B4)
                       => ( Y = some_nat(one_one_nat) ) )
                      & ( ~ pp(B4)
                       => ( Y = none_nat ) ) ) ) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Leaf(A4,B4))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = none_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y = some_nat(Mi2) )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2))) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_9230_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))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( ( pp(B4)
                   => ( Y = some_nat(one_one_nat) ) )
                  & ( ~ pp(B4)
                   => ( ( pp(A4)
                       => ( Y = some_nat(zero_zero_nat) ) )
                      & ( ~ pp(A4)
                       => ( Y = none_nat ) ) ) ) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Leaf(A4,B4))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = none_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y = some_nat(Ma2) )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2))) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_9231_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 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) )
                   => ( pp(Y)
                     => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2))) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2) )
                     => ( ~ pp(Y)
                       => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_9232_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 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2))) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_9233_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,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2))) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_9234_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aTP_Lamm_xo),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_9235_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fTrue,aTP_Lamm_xp),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_9236_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_9237_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_xq,N)),M) ).

% max_Suc2
tff(fact_9238_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_xr,N)),M) ).

% max_Suc1
tff(fact_9239_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_an),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ).

% diff_Suc
tff(fact_9240_finite__enumerate,axiom,
    ! [S3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ? [R3: fun_nat_nat] :
          ( pp(aa_set_nat_bool(strict1292158309912662752at_nat(R3),aa_nat_set_nat(set_ord_lessThan_nat,finite_card_nat(S3))))
          & ! [N6: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N6),finite_card_nat(S3)))
             => pp(aa_set_nat_bool(member_nat(aa_nat_nat(R3,N6)),S3)) ) ) ) ).

% finite_enumerate
tff(fact_9241_floor__real__def,axiom,
    ! [X: real] : aa_real_int(archim6058952711729229775r_real,X) = the_int(aa_real_fun_int_bool(aTP_Lamm_xs,X)) ).

% floor_real_def
tff(fact_9242_floor__rat__def,axiom,
    ! [X: rat] : archim3151403230148437115or_rat(X) = the_int(aTP_Lamm_xt(X)) ).

% floor_rat_def
tff(fact_9243_abs__rat__def,axiom,
    ! [A: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( abs_abs_rat(A) = uminus_uminus_rat(A) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( abs_abs_rat(A) = A ) ) ) ).

% abs_rat_def
tff(fact_9244_sgn__rat__def,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_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))
           => ( sgn_sgn_rat(A) = one_one_rat ) )
          & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
           => ( sgn_sgn_rat(A) = uminus_uminus_rat(one_one_rat) ) ) ) ) ) ).

% sgn_rat_def
tff(fact_9245_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_9246_obtain__pos__sum,axiom,
    ! [R: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R))
     => ~ ! [S2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),S2))
           => ! [T4: rat] :
                ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),T4))
               => ( R != plus_plus_rat(S2,T4) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9247_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_an),Nat) ).

% pred_def
tff(fact_9248_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(inverse_inverse_rat(P2)) = produc4245557441103728435nt_int(aTP_Lamm_xu,quotient_of(P2)) ).

% rat_inverse_code
tff(fact_9249_normalize__negative,axiom,
    ! [Q2: int,P2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Q2),zero_zero_int))
     => ( normalize(product_Pair_int_int(P2,Q2)) = normalize(product_Pair_int_int(aa_int_int(uminus_uminus_int,P2),aa_int_int(uminus_uminus_int,Q2))) ) ) ).

% normalize_negative
tff(fact_9250_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_9251_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(numeral_numeral_rat(K)) = product_Pair_int_int(aa_num_int(numeral_numeral_int,K),one_one_int) ).

% quotient_of_number(3)
tff(fact_9252_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(uminus_uminus_rat(numeral_numeral_rat(K))) = 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_9253_divide__rat__def,axiom,
    ! [Q2: rat,R: rat] : divide_divide_rat(Q2,R) = aa_rat_rat(times_times_rat(Q2),inverse_inverse_rat(R)) ).

% divide_rat_def
tff(fact_9254_diff__rat__def,axiom,
    ! [Q2: rat,R: rat] : minus_minus_rat(Q2,R) = plus_plus_rat(Q2,uminus_uminus_rat(R)) ).

% diff_rat_def
tff(fact_9255_rat__times__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(times_times_rat(P2),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_xw(Q2),quotient_of(P2)) ).

% rat_times_code
tff(fact_9256_rat__divide__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(divide_divide_rat(P2,Q2)) = produc4245557441103728435nt_int(aTP_Lamm_xy(Q2),quotient_of(P2)) ).

% rat_divide_code
tff(fact_9257_quotient__of__div,axiom,
    ! [R: rat,N: int,D: int] :
      ( ( quotient_of(R) = product_Pair_int_int(N,D) )
     => ( R = 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_9258_rat__plus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(plus_plus_rat(P2,Q2)) = produc4245557441103728435nt_int(aTP_Lamm_ya(Q2),quotient_of(P2)) ).

% rat_plus_code
tff(fact_9259_rat__minus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(minus_minus_rat(P2,Q2)) = produc4245557441103728435nt_int(aTP_Lamm_yc(Q2),quotient_of(P2)) ).

% rat_minus_code
tff(fact_9260_quotient__of__denom__pos,axiom,
    ! [R: rat,P2: int,Q2: int] :
      ( ( quotient_of(R) = product_Pair_int_int(P2,Q2) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q2)) ) ).

% quotient_of_denom_pos
tff(fact_9261_normalize__denom__pos,axiom,
    ! [R: product_prod_int_int,P2: int,Q2: int] :
      ( ( normalize(R) = product_Pair_int_int(P2,Q2) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q2)) ) ).

% normalize_denom_pos
tff(fact_9262_normalize__crossproduct,axiom,
    ! [Q2: int,S: int,P2: int,R: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( normalize(product_Pair_int_int(P2,Q2)) = normalize(product_Pair_int_int(R,S)) )
         => ( aa_int_int(times_times_int(P2),S) = aa_int_int(times_times_int(R),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9263_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_9264_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_9265_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_9266_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_9267_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_9268_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(aa_nat_fun_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_9269_drop__bit__minus__one,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% drop_bit_minus_one
tff(fact_9270_fst__divmod__nat,axiom,
    ! [M: nat,N: nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,divmod_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ).

% fst_divmod_nat
tff(fact_9271_snd__divmod__nat,axiom,
    ! [M: nat,N: nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,divmod_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) ).

% snd_divmod_nat
tff(fact_9272_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_9273_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_numeral_minus_bit0
tff(fact_9274_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9275_drop__bit__push__bit__int,axiom,
    ! [M: nat,N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),K)) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),K)) ).

% drop_bit_push_bit_int
tff(fact_9276_drop__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),K) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% drop_bit_int_def
tff(fact_9277_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(product_Pair_int_int(one_one_int,aa_num_int(numeral_numeral_int,K))) = divide_divide_rat(one_one_rat,numeral_numeral_rat(K)) ).

% Frct_code_post(5)
tff(fact_9278_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(product_Pair_int_int(aa_num_int(numeral_numeral_int,K),aa_num_int(numeral_numeral_int,L))) = divide_divide_rat(numeral_numeral_rat(K),numeral_numeral_rat(L)) ).

% Frct_code_post(6)
tff(fact_9279_fst__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] : produc8508995932063986495nteger(code_divmod_integer(K,L)) = divide6298287555418463151nteger(K,L) ).

% fst_divmod_integer
tff(fact_9280_snd__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] : produc6174133586879617921nteger(code_divmod_integer(K,L)) = modulo364778990260209775nteger(K,L) ).

% snd_divmod_integer
tff(fact_9281_fst__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] : produc8508995932063986495nteger(code_divmod_abs(K,L)) = divide6298287555418463151nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L)) ).

% fst_divmod_abs
tff(fact_9282_snd__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] : produc6174133586879617921nteger(code_divmod_abs(K,L)) = modulo364778990260209775nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L)) ).

% snd_divmod_abs
tff(fact_9283_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_9284_drop__bit__nat__eq,axiom,
    ! [N: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),K)) ).

% drop_bit_nat_eq
tff(fact_9285_quotient__of__denom__pos_H,axiom,
    ! [R: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(quotient_of(R)))) ).

% quotient_of_denom_pos'
tff(fact_9286_drop__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),M) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% drop_bit_nat_def
tff(fact_9287_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(product_Pair_int_int(aa_num_int(numeral_numeral_int,K),one_one_int)) = numeral_numeral_rat(K) ).

% Frct_code_post(4)
tff(fact_9288_one__mod__minus__numeral,axiom,
    ! [N: num] : aa_int_int(modulo_modulo_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,N)),product_snd_int_int(unique5052692396658037445od_int(one,N)))) ).

% one_mod_minus_numeral
tff(fact_9289_minus__one__mod__numeral,axiom,
    ! [N: num] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,N)),product_snd_int_int(unique5052692396658037445od_int(one,N))) ).

% minus_one_mod_numeral
tff(fact_9290_minus__numeral__mod__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,N)),product_snd_int_int(unique5052692396658037445od_int(M,N))) ).

% minus_numeral_mod_numeral
tff(fact_9291_numeral__mod__minus__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,N)),product_snd_int_int(unique5052692396658037445od_int(M,N)))) ).

% numeral_mod_minus_numeral
tff(fact_9292_Divides_Oadjust__mod__def,axiom,
    ! [R: int,L: int] :
      ( ( ( R = zero_zero_int )
       => ( aa_int_int(adjust_mod(L),R) = zero_zero_int ) )
      & ( ( R != zero_zero_int )
       => ( aa_int_int(adjust_mod(L),R) = aa_int_int(minus_minus_int(L),R) ) ) ) ).

% Divides.adjust_mod_def
tff(fact_9293_bezw_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( bezw(X,Y) = product_Pair_int_int(one_one_int,zero_zero_int) ) )
      & ( ( Y != zero_zero_nat )
       => ( bezw(X,Y) = 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(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(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_9294_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa2) = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y = product_Pair_int_int(one_one_int,zero_zero_int) ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y = 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(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(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_9295_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) = 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(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(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_9296_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa2) = Y )
     => ( accp_P4275260045618599050at_nat(bezw_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y = product_Pair_int_int(one_one_int,zero_zero_int) ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y = 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(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(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))))) ) ) )
           => ~ accp_P4275260045618599050at_nat(bezw_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)) ) ) ) ).

% bezw.pelims
tff(fact_9297_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)))
       => ( normalize(P2) = product_Pair_int_int(aa_int_int(aa_int_fun_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(aa_int_fun_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 )
           => ( normalize(P2) = product_Pair_int_int(zero_zero_int,one_one_int) ) )
          & ( ( product_snd_int_int(P2) != zero_zero_int )
           => ( normalize(P2) = product_Pair_int_int(aa_int_int(aa_int_fun_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(aa_int_fun_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_9298_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_9299_gcd__neg__numeral__2__int,axiom,
    ! [X: int,N: num] : aa_int_int(gcd_gcd_int(X),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(gcd_gcd_int(X),aa_num_int(numeral_numeral_int,N)) ).

% gcd_neg_numeral_2_int
tff(fact_9300_gcd__neg__numeral__1__int,axiom,
    ! [N: num,X: int] : aa_int_int(gcd_gcd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),X) = aa_int_int(gcd_gcd_int(aa_num_int(numeral_numeral_int,N)),X) ).

% gcd_neg_numeral_1_int
tff(fact_9301_gcd__red__int,axiom,
    ! [X: int,Y: int] : aa_int_int(gcd_gcd_int(X),Y) = aa_int_int(gcd_gcd_int(Y),aa_int_int(modulo_modulo_int(X),Y)) ).

% gcd_red_int
tff(fact_9302_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U2: int,V2: int] : aa_int_int(plus_plus_int(aa_int_int(times_times_int(U2),X)),aa_int_int(times_times_int(V2),Y)) = aa_int_int(gcd_gcd_int(X),Y) ).

% bezout_int
tff(fact_9303_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,N: int] : aa_int_int(times_times_int(aa_int_int(abs_abs_int,K)),aa_int_int(gcd_gcd_int(M),N)) = aa_int_int(gcd_gcd_int(aa_int_int(times_times_int(K),M)),aa_int_int(times_times_int(K),N)) ).

% gcd_mult_distrib_int
tff(fact_9304_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_9305_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_9306_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_9307_gcd__code__int,axiom,
    ! [K: int,L: int] : aa_int_int(gcd_gcd_int(K),L) = aa_int_int(abs_abs_int,aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,L),zero_zero_int),K),aa_int_int(gcd_gcd_int(L),aa_int_int(modulo_modulo_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))))) ).

% gcd_code_int
tff(fact_9308_nat__descend__induct,axiom,
    ! [N: nat,P: fun_nat_bool,M: nat] :
      ( ! [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K3))
         => pp(aa_nat_bool(P,K3)) )
     => ( ! [K3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N))
           => ( ! [I2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),I2))
                 => pp(aa_nat_bool(P,I2)) )
             => pp(aa_nat_bool(P,K3)) ) )
       => pp(aa_nat_bool(P,M)) ) ) ).

% nat_descend_induct
tff(fact_9309_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(X,Xa2) = Y )
     => ( 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))) ) ) )
           => ~ accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_9310_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_yd(L),code_divmod_abs(K,L)))) ) ) ) ) ) ).

% divmod_integer_eq_cases
tff(fact_9311_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_9312_gcd__red__nat,axiom,
    ! [X: nat,Y: 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_red_nat
tff(fact_9313_gcd__mult__distrib__nat,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(gcd_gcd_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_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)) ).

% gcd_mult_distrib_nat
tff(fact_9314_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_9315_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_9316_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_9317_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_9318_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_9319_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_9320_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_9321_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X2: nat,Y2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = 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_9322_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X2: 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),X2)))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2)),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),X2)))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X2)),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_9323_gcd__code__integer,axiom,
    ! [K: code_integer,L: code_integer] : gcd_gcd_Code_integer(K,L) = abs_abs_Code_integer(if_Code_integer(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,L),zero_z3403309356797280102nteger),K,gcd_gcd_Code_integer(L,modulo364778990260209775nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L))))) ).

% gcd_code_integer
tff(fact_9324_bezw__aux,axiom,
    ! [X: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(product_fst_int_int(bezw(X,Y))),aa_nat_int(semiri1314217659103216013at_int,X))),aa_int_int(times_times_int(product_snd_int_int(bezw(X,Y))),aa_nat_int(semiri1314217659103216013at_int,Y))) ).

% bezw_aux
tff(fact_9325_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa2) = Y )
     => ( 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)) ) ) )
           => ~ accp_P4275260045618599050at_nat(gcd_nat_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)) ) ) ) ).

% gcd_nat.pelims
tff(fact_9326_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or5832277885323065728an_int(L,U)) = aa_int_nat(nat2,aa_int_int(minus_minus_int(U),aa_int_int(plus_plus_int(L),one_one_int))) ).

% card_greaterThanLessThan_int
tff(fact_9327_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(aa_int_int(plus_plus_int(L),one_one_int),U) = set_or5832277885323065728an_int(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_9328_Code__Numeral_Onegative__def,axiom,
    code_negative = comp_C3531382070062128313er_num(uminus1351360451143612070nteger,numera6620942414471956472nteger) ).

% Code_Numeral.negative_def
tff(fact_9329_Code__Target__Int_Onegative__def,axiom,
    code_Target_negative = comp_int_int_num(uminus_uminus_int,numeral_numeral_int) ).

% Code_Target_Int.negative_def
tff(fact_9330_xor__minus__numerals_I2_J,axiom,
    ! [K: int,N: num] : aa_int_int(bit_se6526347334894502574or_int(K),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se6526347334894502574or_int(K),neg_numeral_sub_int(N,one))) ).

% xor_minus_numerals(2)
tff(fact_9331_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_or5834768355832116004an_nat(L),U))) ).

% finite_greaterThanLessThan
tff(fact_9332_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),aa_nat_nat(suc,L)) ).

% card_greaterThanLessThan
tff(fact_9333_xor__minus__numerals_I1_J,axiom,
    ! [N: num,K: int] : aa_int_int(bit_se6526347334894502574or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),K) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se6526347334894502574or_int(neg_numeral_sub_int(N,one)),K)) ).

% xor_minus_numerals(1)
tff(fact_9334_sub__BitM__One__eq,axiom,
    ! [N: num] : neg_numeral_sub_int(bitM(N),one) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),neg_numeral_sub_int(N,one)) ).

% sub_BitM_One_eq
tff(fact_9335_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_bg,aTP_Lamm_bh) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9336_Suc__funpow,axiom,
    ! [N: nat] : compow_fun_nat_nat(N,suc) = aa_nat_fun_nat_nat(plus_plus_nat,N) ).

% Suc_funpow
tff(fact_9337_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_ye,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ) ) ) ) ).

% int_of_integer_code
tff(fact_9338_int__of__integer__numeral,axiom,
    ! [K: num] : code_int_of_integer(aa_num_Code_integer(numera6620942414471956472nteger,K)) = aa_num_int(numeral_numeral_int,K) ).

% int_of_integer_numeral
tff(fact_9339_plus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] : code_int_of_integer(plus_p5714425477246183910nteger(X,Xa2)) = aa_int_int(plus_plus_int(code_int_of_integer(X)),code_int_of_integer(Xa2)) ).

% plus_integer.rep_eq
tff(fact_9340_times__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] : code_int_of_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Xa2)) = aa_int_int(times_times_int(code_int_of_integer(X)),code_int_of_integer(Xa2)) ).

% times_integer.rep_eq
tff(fact_9341_minus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] : code_int_of_integer(minus_8373710615458151222nteger(X,Xa2)) = aa_int_int(minus_minus_int(code_int_of_integer(X)),code_int_of_integer(Xa2)) ).

% minus_integer.rep_eq
tff(fact_9342_divide__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] : code_int_of_integer(divide6298287555418463151nteger(X,Xa2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,code_int_of_integer(X)),code_int_of_integer(Xa2)) ).

% divide_integer.rep_eq
tff(fact_9343_modulo__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] : code_int_of_integer(modulo364778990260209775nteger(X,Xa2)) = aa_int_int(modulo_modulo_int(code_int_of_integer(X)),code_int_of_integer(Xa2)) ).

% modulo_integer.rep_eq
tff(fact_9344_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_9345_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_9346_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(times_times_int(abs_Integ(Xa2)),abs_Integ(X)) = abs_Integ(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_yg),Xa2),X)) ).

% times_int.abs_eq
tff(fact_9347_Gcd__remove0__nat,axiom,
    ! [M4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M4))
     => ( gcd_Gcd_nat(M4) = gcd_Gcd_nat(minus_minus_set_nat(M4,insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_9348_nat_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] : aa_int_nat(nat2,abs_Integ(X)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),X) ).

% nat.abs_eq
tff(fact_9349_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,abs_Integ(Xa2)),abs_Integ(X)))
    <=> pp(aa_Pro2232559814647311250t_bool(produc5023667602222594023t_bool(aTP_Lamm_yi,Xa2),X)) ) ).

% less_int.abs_eq
tff(fact_9350_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,abs_Integ(Xa2)),abs_Integ(X)))
    <=> pp(aa_Pro2232559814647311250t_bool(produc5023667602222594023t_bool(aTP_Lamm_yk,Xa2),X)) ) ).

% less_eq_int.abs_eq
tff(fact_9351_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(plus_plus_int(abs_Integ(Xa2)),abs_Integ(X)) = abs_Integ(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_ym),Xa2),X)) ).

% plus_int.abs_eq
tff(fact_9352_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(minus_minus_int(abs_Integ(Xa2)),abs_Integ(X)) = abs_Integ(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_yo),Xa2),X)) ).

% minus_int.abs_eq
tff(fact_9353_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)) = 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_9354_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] : aa_nat_num(num_of_nat,aa_num_nat(numeral_numeral_nat,Q2)) = Q2 ).

% num_of_nat_numeral_eq
tff(fact_9355_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9356_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_9357_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_9358_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_9359_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_9360_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(nth_nat(linord2614967742042102400et_nat(aa_nat_set_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_9361_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(produc5023667602222594023t_bool(aTP_Lamm_yk,rep_Integ(X)),rep_Integ(Xa2))) ) ).

% less_eq_int.rep_eq
tff(fact_9362_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(produc5023667602222594023t_bool(aTP_Lamm_yi,rep_Integ(X)),rep_Integ(Xa2))) ) ).

% less_int.rep_eq
tff(fact_9363_nat_Orep__eq,axiom,
    ! [X: int] : aa_int_nat(nat2,X) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),rep_Integ(X)) ).

% nat.rep_eq
tff(fact_9364_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(nth_nat(linord2614967742042102400et_nat(aa_nat_set_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_9365_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_yp) ).

% prod_encode_def
tff(fact_9366_pow_Osimps_I3_J,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(pow,X),aa_num_num(bit1,Y)) = aa_num_num(aa_num_fun_num_num(times_times_num,sqr(aa_num_num(aa_num_fun_num_num(pow,X),Y))),X) ).

% pow.simps(3)
tff(fact_9367_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_or6659071591806873216st_nat(L),U))) ).

% finite_greaterThanAtMost
tff(fact_9368_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_greaterThanAtMost
tff(fact_9369_sqr_Osimps_I2_J,axiom,
    ! [N: num] : sqr(aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(bit0,sqr(N))) ).

% sqr.simps(2)
tff(fact_9370_sqr_Osimps_I1_J,axiom,
    sqr(one) = one ).

% sqr.simps(1)
tff(fact_9371_sqr__conv__mult,axiom,
    ! [X: num] : sqr(X) = aa_num_num(aa_num_fun_num_num(times_times_num,X),X) ).

% sqr_conv_mult
tff(fact_9372_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ).

% le_prod_encode_2
tff(fact_9373_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ).

% le_prod_encode_1
tff(fact_9374_pow_Osimps_I2_J,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(pow,X),aa_num_num(bit0,Y)) = sqr(aa_num_num(aa_num_fun_num_num(pow,X),Y)) ).

% pow.simps(2)
tff(fact_9375_sqr_Osimps_I3_J,axiom,
    ! [N: num] : sqr(aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,sqr(N)),N))) ).

% sqr.simps(3)
tff(fact_9376_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,nat_prod_decode_aux(K,M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K)),M) ).

% prod_encode_prod_decode_aux
tff(fact_9377_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_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,Summary2: vEBT_VEBT] :
              ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
             => ( ( Deg2 = Xa2 )
                & ! [X2: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_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(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,Summary2),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,Summary2)))),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_yq(Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_9378_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or6656581121297822940st_int(L,U)) = aa_int_nat(nat2,aa_int_int(minus_minus_int(U),L)) ).

% card_greaterThanAtMost_int
tff(fact_9379_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1266510415728281911st_int(aa_int_int(plus_plus_int(L),one_one_int),U) = set_or6656581121297822940st_int(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_9380_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima2,Deg,TreeList),Summary)),Deg3))
    <=> ( ( Deg = Deg3 )
        & ! [X3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_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(aa_VEB4832260192069861672t_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_yq(Deg),TreeList),Summary))),Mima2)) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_9381_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_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,Summary2: vEBT_VEBT] :
              ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
             => ( 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(aa_VEB4832260192069861672t_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(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,Summary2),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,Summary2)))),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_yq(Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_9382_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_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,Summary2: vEBT_VEBT] :
              ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_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(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,Summary2),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,Summary2)))),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_yq(Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_9383_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_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,Summary2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2),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(aa_VEB4832260192069861672t_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(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,Summary2),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,Summary2)))),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_yq(Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_9384_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_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,Summary2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2),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(aa_VEB4832260192069861672t_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(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,Summary2),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,Summary2)))),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_yq(Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_9385_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_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,Summary2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
               => ( ( 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(aa_VEB4832260192069861672t_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(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,Summary2),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,Summary2)))),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_yq(Deg2),TreeList2),Summary2))),Mima)) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2),Xa2)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_9386_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = case_option_int_num(zero_zero_int,aTP_Lamm_yr(M),bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M),N)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_9387_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,bitM(N))) ).

% and_minus_numerals(3)
tff(fact_9388_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_9389_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_9390_take__bit__num__simps_I5_J,axiom,
    ! [R: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R),one) = some_num(one) ).

% take_bit_num_simps(5)
tff(fact_9391_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_ys,bit_take_bit_num(N,M)) ).

% take_bit_num_simps(3)
tff(fact_9392_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_9393_take__bit__num__simps_I6_J,axiom,
    ! [R: num,M: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R),aa_num_num(bit0,M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ys,bit_take_bit_num(aa_num_nat(pred_numeral,R),M)) ).

% take_bit_num_simps(6)
tff(fact_9394_take__bit__num__simps_I7_J,axiom,
    ! [R: num,M: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(aa_num_nat(pred_numeral,R),M))) ).

% take_bit_num_simps(7)
tff(fact_9395_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bit0,N))) ).

% and_minus_numerals(8)
tff(fact_9396_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bit0,N))) ).

% and_minus_numerals(4)
tff(fact_9397_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,bitM(N))) ).

% and_minus_numerals(7)
tff(fact_9398_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_yt(M),N) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_9399_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_yu,N) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_9400_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one,one) = none_num ).

% and_not_num.simps(1)
tff(fact_9401_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_9402_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_9403_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_bool,B: nat] :
      ( ? [X_13: nat] : pp(aa_nat_bool(P,X_13))
     => ( ! [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_9404_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_9405_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_9406_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_9407_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_yv(M),N) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_9408_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_9409_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N: num,Q2: num] :
      ( ( bit_and_not_num(M,N) = some_num(Q2) )
    <=> ( 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,Q2) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_9410_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_yw,bit_and_not_num(M,N)) ).

% and_not_num.simps(8)
tff(fact_9411_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_9412_int__numeral__and__not__num,axiom,
    ! [M: num,N: 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))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,N)) ).

% int_numeral_and_not_num
tff(fact_9413_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(N,M)) ).

% int_numeral_not_and_num
tff(fact_9414_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_9415_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_yx) ).

% Rats_eq_int_div_nat
tff(fact_9416_Rats__abs__iff,axiom,
    ! [X: real] :
      ( member_real(aa_real_real(abs_abs_real,X),field_5140801741446780682s_real)
    <=> member_real(X,field_5140801741446780682s_real) ) ).

% Rats_abs_iff
tff(fact_9417_Rats__no__top__le,axiom,
    ! [X: real] :
    ? [X2: real] :
      ( member_real(X2,field_5140801741446780682s_real)
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),X2)) ) ).

% Rats_no_top_le
tff(fact_9418_Rats__dense__in__real,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ? [X2: real] :
          ( member_real(X2,field_5140801741446780682s_real)
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y)) ) ) ).

% Rats_dense_in_real
tff(fact_9419_Rats__no__bot__less,axiom,
    ! [X: real] :
    ? [X2: real] :
      ( member_real(X2,field_5140801741446780682s_real)
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),X)) ) ).

% Rats_no_bot_less
tff(fact_9420_Rats__eq__int__div__int,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_yy) ).

% Rats_eq_int_div_int
tff(fact_9421_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 ) ) )
           => ( ! [M3: num] :
                  ( ( X = aa_num_num(bit0,M3) )
                 => ( ( Xa2 = one )
                   => ( Y != some_num(aa_num_num(bit0,M3)) ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa_num_num(bit0,M3) )
                   => ! [N2: num] :
                        ( ( Xa2 = aa_num_num(bit0,N2) )
                       => ( Y != map_option_num_num(bit0,bit_and_not_num(M3,N2)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa_num_num(bit0,M3) )
                     => ! [N2: num] :
                          ( ( Xa2 = aa_num_num(bit1,N2) )
                         => ( Y != map_option_num_num(bit0,bit_and_not_num(M3,N2)) ) ) )
                 => ( ! [M3: num] :
                        ( ( X = aa_num_num(bit1,M3) )
                       => ( ( Xa2 = one )
                         => ( Y != some_num(aa_num_num(bit0,M3)) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa_num_num(bit1,M3) )
                         => ! [N2: num] :
                              ( ( Xa2 = aa_num_num(bit0,N2) )
                             => ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_yw,bit_and_not_num(M3,N2)) ) ) )
                     => ~ ! [M3: num] :
                            ( ( X = aa_num_num(bit1,M3) )
                           => ! [N2: num] :
                                ( ( Xa2 = aa_num_num(bit1,N2) )
                               => ( Y != map_option_num_num(bit0,bit_and_not_num(M3,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_9422_and__not__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa_num_num(bit0,M),aa_num_num(bit0,N)) = map_option_num_num(bit0,bit_and_not_num(M,N)) ).

% and_not_num.simps(5)
tff(fact_9423_and__not__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa_num_num(bit0,M),aa_num_num(bit1,N)) = map_option_num_num(bit0,bit_and_not_num(M,N)) ).

% and_not_num.simps(6)
tff(fact_9424_and__not__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa_num_num(bit1,M),aa_num_num(bit1,N)) = map_option_num_num(bit0,bit_and_not_num(M,N)) ).

% and_not_num.simps(9)
tff(fact_9425_rat__less__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,P2),Q2))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_za(Q2)),quotient_of(P2))) ) ).

% rat_less_code
tff(fact_9426_rat__less__eq__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,P2),Q2))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_zc(Q2)),quotient_of(P2))) ) ).

% rat_less_eq_code
tff(fact_9427_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_zd(D))) ).

% int_ge_less_than2_def
tff(fact_9428_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_ze(D))) ).

% int_ge_less_than_def
tff(fact_9429_and__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( aa_num_option_num(aa_num6390622462783740633on_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) ) ) )
           => ( ( ? [M3: num] : X = aa_num_num(bit0,M3)
               => ( ( Xa2 = one )
                 => ( Y != none_num ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa_num_num(bit0,M3) )
                   => ! [N2: num] :
                        ( ( Xa2 = aa_num_num(bit0,N2) )
                       => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M3),N2)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa_num_num(bit0,M3) )
                     => ! [N2: num] :
                          ( ( Xa2 = aa_num_num(bit1,N2) )
                         => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M3),N2)) ) ) )
                 => ( ( ? [M3: num] : X = aa_num_num(bit1,M3)
                     => ( ( Xa2 = one )
                       => ( Y != some_num(one) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa_num_num(bit1,M3) )
                         => ! [N2: num] :
                              ( ( Xa2 = aa_num_num(bit0,N2) )
                             => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M3),N2)) ) ) )
                     => ~ ! [M3: num] :
                            ( ( X = aa_num_num(bit1,M3) )
                           => ! [N2: num] :
                                ( ( Xa2 = aa_num_num(bit1,N2) )
                               => ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_yw,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M3),N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_9430_xor__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( aa_num_option_num(aa_num6390622462783740633on_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)) ) ) )
           => ( ! [M3: num] :
                  ( ( X = aa_num_num(bit0,M3) )
                 => ( ( Xa2 = one )
                   => ( Y != some_num(aa_num_num(bit1,M3)) ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa_num_num(bit0,M3) )
                   => ! [N2: num] :
                        ( ( Xa2 = aa_num_num(bit0,N2) )
                       => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M3),N2)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa_num_num(bit0,M3) )
                     => ! [N2: num] :
                          ( ( Xa2 = aa_num_num(bit1,N2) )
                         => ( Y != some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M3),N2))) ) ) )
                 => ( ! [M3: num] :
                        ( ( X = aa_num_num(bit1,M3) )
                       => ( ( Xa2 = one )
                         => ( Y != some_num(aa_num_num(bit0,M3)) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa_num_num(bit1,M3) )
                         => ! [N2: num] :
                              ( ( Xa2 = aa_num_num(bit0,N2) )
                             => ( Y != some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M3),N2))) ) ) )
                     => ~ ! [M3: num] :
                            ( ( X = aa_num_num(bit1,M3) )
                           => ! [N2: num] :
                                ( ( Xa2 = aa_num_num(bit1,N2) )
                               => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M3),N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
tff(fact_9431_and__num_Osimps_I1_J,axiom,
    aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),one) = some_num(one) ).

% and_num.simps(1)
tff(fact_9432_xor__num_Osimps_I1_J,axiom,
    aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),one) = none_num ).

% xor_num.simps(1)
tff(fact_9433_xor__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),N)) ).

% xor_num.simps(5)
tff(fact_9434_and__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),N)) ).

% and_num.simps(5)
tff(fact_9435_and__num_Osimps_I3_J,axiom,
    ! [N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),aa_num_num(bit1,N)) = some_num(one) ).

% and_num.simps(3)
tff(fact_9436_and__num_Osimps_I7_J,axiom,
    ! [M: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,M)),one) = some_num(one) ).

% and_num.simps(7)
tff(fact_9437_and__num_Osimps_I2_J,axiom,
    ! [N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),aa_num_num(bit0,N)) = none_num ).

% and_num.simps(2)
tff(fact_9438_and__num_Osimps_I4_J,axiom,
    ! [M: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,M)),one) = none_num ).

% and_num.simps(4)
tff(fact_9439_and__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),N)) ).

% and_num.simps(8)
tff(fact_9440_and__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),N)) ).

% and_num.simps(6)
tff(fact_9441_xor__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),N)) ).

% xor_num.simps(9)
tff(fact_9442_xor__num_Osimps_I2_J,axiom,
    ! [N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),aa_num_num(bit0,N)) = some_num(aa_num_num(bit1,N)) ).

% xor_num.simps(2)
tff(fact_9443_xor__num_Osimps_I3_J,axiom,
    ! [N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),aa_num_num(bit1,N)) = some_num(aa_num_num(bit0,N)) ).

% xor_num.simps(3)
tff(fact_9444_xor__num_Osimps_I4_J,axiom,
    ! [M: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,M)),one) = some_num(aa_num_num(bit1,M)) ).

% xor_num.simps(4)
tff(fact_9445_xor__num_Osimps_I7_J,axiom,
    ! [M: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,M)),one) = some_num(aa_num_num(bit0,M)) ).

% xor_num.simps(7)
tff(fact_9446_and__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_yw,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),N)) ).

% and_num.simps(9)
tff(fact_9447_xor__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),N))) ).

% xor_num.simps(8)
tff(fact_9448_xor__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),N))) ).

% xor_num.simps(6)
tff(fact_9449_xor__num__dict,axiom,
    bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).

% xor_num_dict
tff(fact_9450_and__num__dict,axiom,
    bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).

% and_num_dict
tff(fact_9451_Bit__Operations_Otake__bit__num__code,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,M) = produc478579273971653890on_num(aTP_Lamm_zi,product_Pair_nat_num(N,M)) ).

% Bit_Operations.take_bit_num_code
tff(fact_9452_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(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_rat(fract(A),B)))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(fract(A),B)),aa_int_rat(ring_1_of_int_rat,aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int)))) ) ).

% rat_floor_lemma
tff(fact_9453_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_zj,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = aa_nat_set_nat(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_zj,C),aa_nat_set_nat(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_zj,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9454_bij__betw__Suc,axiom,
    ! [M4: set_nat,N3: set_nat] :
      ( pp(aa_set_nat_bool(bij_betw_nat_nat(suc,M4),N3))
    <=> ( image_nat_nat(suc,M4) = N3 ) ) ).

% bij_betw_Suc
tff(fact_9455_mult__rat,axiom,
    ! [A: int,B: int,C: int,D: int] : aa_rat_rat(times_times_rat(aa_int_rat(fract(A),B)),aa_int_rat(fract(C),D)) = aa_int_rat(fract(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) ).

% mult_rat
tff(fact_9456_divide__rat,axiom,
    ! [A: int,B: int,C: int,D: int] : divide_divide_rat(aa_int_rat(fract(A),B),aa_int_rat(fract(C),D)) = aa_int_rat(fract(aa_int_int(times_times_int(A),D)),aa_int_int(times_times_int(B),C)) ).

% divide_rat
tff(fact_9457_floor__Fract,axiom,
    ! [A: int,B: int] : archim3151403230148437115or_rat(aa_int_rat(fract(A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ).

% floor_Fract
tff(fact_9458_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(fract(A),B)),aa_int_rat(fract(C),D)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(aa_int_int(times_times_int(A),D)),aa_int_int(times_times_int(B),D))),aa_int_int(times_times_int(aa_int_int(times_times_int(C),B)),aa_int_int(times_times_int(B),D)))) ) ) ) ).

% less_rat
tff(fact_9459_add__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( plus_plus_rat(aa_int_rat(fract(A),B),aa_int_rat(fract(C),D)) = aa_int_rat(fract(aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),D)),aa_int_int(times_times_int(C),B))),aa_int_int(times_times_int(B),D)) ) ) ) ).

% add_rat
tff(fact_9460_le__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_eq_rat,aa_int_rat(fract(A),B)),aa_int_rat(fract(C),D)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(aa_int_int(times_times_int(A),D)),aa_int_int(times_times_int(B),D))),aa_int_int(times_times_int(aa_int_int(times_times_int(C),B)),aa_int_int(times_times_int(B),D)))) ) ) ) ).

% le_rat
tff(fact_9461_diff__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( minus_minus_rat(aa_int_rat(fract(A),B),aa_int_rat(fract(C),D)) = aa_int_rat(fract(aa_int_int(minus_minus_int(aa_int_int(times_times_int(A),D)),aa_int_int(times_times_int(C),B))),aa_int_int(times_times_int(B),D)) ) ) ) ).

% diff_rat
tff(fact_9462_sgn__rat,axiom,
    ! [A: int,B: int] : sgn_sgn_rat(aa_int_rat(fract(A),B)) = aa_int_rat(ring_1_of_int_rat,aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,B))) ).

% sgn_rat
tff(fact_9463_Rat__induct__pos,axiom,
    ! [P: fun_rat_bool,Q2: rat] :
      ( ! [A4: int,B4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
         => pp(aa_rat_bool(P,aa_int_rat(fract(A4),B4))) )
     => pp(aa_rat_bool(P,Q2)) ) ).

% Rat_induct_pos
tff(fact_9464_eq__rat_I1_J,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( aa_int_rat(fract(A),B) = aa_int_rat(fract(C),D) )
        <=> ( aa_int_int(times_times_int(A),D) = aa_int_int(times_times_int(C),B) ) ) ) ) ).

% eq_rat(1)
tff(fact_9465_mult__rat__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_rat(fract(aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) = aa_int_rat(fract(A),B) ) ) ).

% mult_rat_cancel
tff(fact_9466_Fract__coprime,axiom,
    ! [A: int,B: int] : aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(gcd_gcd_int(A),B))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),aa_int_int(gcd_gcd_int(A),B))) = aa_int_rat(fract(A),B) ).

% Fract_coprime
tff(fact_9467_Fract__of__int__quotient,axiom,
    ! [K: int,L: int] : aa_int_rat(fract(K),L) = divide_divide_rat(aa_int_rat(ring_1_of_int_rat,K),aa_int_rat(ring_1_of_int_rat,L)) ).

% Fract_of_int_quotient
tff(fact_9468_rat__number__collapse_I3_J,axiom,
    ! [W: num] : aa_int_rat(fract(aa_num_int(numeral_numeral_int,W)),one_one_int) = numeral_numeral_rat(W) ).

% rat_number_collapse(3)
tff(fact_9469_rat__number__expand_I3_J,axiom,
    ! [K: num] : numeral_numeral_rat(K) = aa_int_rat(fract(aa_num_int(numeral_numeral_int,K)),one_one_int) ).

% rat_number_expand(3)
tff(fact_9470_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(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_9471_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(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_9472_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(fract(A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ).

% one_less_Fract_iff
tff(fact_9473_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(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_9474_Fract__add__one,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero_int )
     => ( aa_int_rat(fract(aa_int_int(plus_plus_int(M),N)),N) = plus_plus_rat(aa_int_rat(fract(M),N),one_one_rat) ) ) ).

% Fract_add_one
tff(fact_9475_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(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_9476_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(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_9477_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(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_9478_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(fract(A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% one_le_Fract_iff
tff(fact_9479_rat__number__expand_I5_J,axiom,
    ! [K: num] : uminus_uminus_rat(numeral_numeral_rat(K)) = aa_int_rat(fract(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int) ).

% rat_number_expand(5)
tff(fact_9480_rat__number__collapse_I4_J,axiom,
    ! [W: num] : aa_int_rat(fract(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),one_one_int) = uminus_uminus_rat(numeral_numeral_rat(W)) ).

% rat_number_collapse(4)
tff(fact_9481_Inf__real__def,axiom,
    ! [X5: set_real] : comple4887499456419720421f_real(X5) = aa_real_real(uminus_uminus_real,comple1385675409528146559p_real(image_real_real(uminus_uminus_real,X5))) ).

% Inf_real_def
tff(fact_9482_infinite__UNIV__nat,axiom,
    ~ pp(aa_set_nat_bool(finite_finite_nat,top_top_set_nat)) ).

% infinite_UNIV_nat
tff(fact_9483_nat__not__finite,axiom,
    ~ pp(aa_set_nat_bool(finite_finite_nat,top_top_set_nat)) ).

% nat_not_finite
tff(fact_9484_suminf__eq__SUP__real,axiom,
    ! [X5: fun_nat_real] :
      ( summable_real(X5)
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(X5,I3)))
       => ( suminf_real(X5) = comple1385675409528146559p_real(image_nat_real(aTP_Lamm_zk(X5),top_top_set_nat)) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_9485_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S3: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S3))
    <=> ~ pp(aa_set_nat_bool(finite_finite_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),S3))) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_9486_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aTP_Lamm_zl(L),set_or4662586982721622107an_int(zero_zero_int,aa_int_int(minus_minus_int(U),L))) = set_or4662586982721622107an_int(L,U) ).

% image_add_int_atLeastLessThan
tff(fact_9487_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_zm,N),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N) ) ) ).

% range_mod
tff(fact_9488_card__UNIV__bool,axiom,
    finite_card_bool(top_top_set_bool) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% card_UNIV_bool
tff(fact_9489_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_9490_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_zn,N)),X) ) ) ) ).

% root_def
tff(fact_9491_card__UNIV__char,axiom,
    finite_card_char(top_top_set_char) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))) ).

% card_UNIV_char
tff(fact_9492_UNIV__char__of__nat,axiom,
    top_top_set_char = image_nat_char(unique3096191561947761185of_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% UNIV_char_of_nat
tff(fact_9493_nat__of__char__less__256,axiom,
    ! [C: char] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% nat_of_char_less_256
tff(fact_9494_range__nat__of__char,axiom,
    image_char_nat(comm_s629917340098488124ar_nat,top_top_set_char) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% range_nat_of_char
tff(fact_9495_integer__of__char__code,axiom,
    ! [B0: bool,B1: bool,B22: bool,B32: bool,B42: bool,B52: bool,B62: bool,B72: bool] : integer_of_char(char2(B0,B1,B22,B32,B42,B52,B62,B72)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(B72)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger(B62))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger(B52))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger(B42))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger(B32))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger(B22))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger(B1))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger(B0)) ).

% integer_of_char_code
tff(fact_9496_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(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I)),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9497_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(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I)),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9498_String_Ochar__of__ascii__of,axiom,
    ! [C: char] : aa_char_nat(comm_s629917340098488124ar_nat,ascii_of(C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_char_nat(comm_s629917340098488124ar_nat,C)) ).

% String.char_of_ascii_of
tff(fact_9499_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,K)),cons_nat(K,nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_9500_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,K)),cons_nat(aa_nat_nat(suc,K),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_9501_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(minus_minus_int(J),one_one_int),cons_int(J,Js)) ) ) ) ).

% upto_aux_rec
tff(fact_9502_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( upto(X,Xa2) = Y )
     => ( accp_P1096762738010456898nt_int(upto_rel,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(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 ) ) )
           => ~ accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(X,Xa2)) ) ) ) ).

% upto.pelims
tff(fact_9503_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( accp_P1096762738010456898nt_int(upto_rel,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(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_9504_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_9505_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_9506_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_9507_upto__single,axiom,
    ! [I: int] : upto(I,I) = cons_int(I,nil_int) ).

% upto_single
tff(fact_9508_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(plus_plus_int(I),aa_nat_int(semiri1314217659103216013at_int,K))),J))
     => ( aa_nat_int(nth_int(upto(I,J)),K) = aa_int_int(plus_plus_int(I),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_9509_length__upto,axiom,
    ! [I: int,J: int] : size_size_list_int(upto(I,J)) = aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(J),I)),one_one_int)) ).

% length_upto
tff(fact_9510_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(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_9511_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(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_9512_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(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_9513_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(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_9514_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_9515_upto__code,axiom,
    ! [I: int,J: int] : upto(I,J) = upto_aux(I,J,nil_int) ).

% upto_code
tff(fact_9516_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : set_or1266510415728281911st_int(I,J) = set_int2(upto(I,J)) ).

% atLeastAtMost_upto
tff(fact_9517_distinct__upto,axiom,
    ! [I: int,J: int] : pp(aa_list_int_bool(distinct_int,upto(I,J))) ).

% distinct_upto
tff(fact_9518_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(plus_plus_int(J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_9519_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(minus_minus_int(J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_9520_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or4662586982721622107an_int(I,J) = set_int2(upto(I,aa_int_int(minus_minus_int(J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_9521_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : set_or6656581121297822940st_int(I,J) = set_int2(upto(aa_int_int(plus_plus_int(I),one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_9522_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(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_9523_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(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_9524_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(plus_plus_int(I),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_9525_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(minus_minus_int(J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_9526_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5832277885323065728an_int(I,J) = set_int2(upto(aa_int_int(plus_plus_int(I),one_one_int),aa_int_int(minus_minus_int(J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_9527_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(minus_minus_int(J),one_one_int)),cons_int(J,upto(aa_int_int(plus_plus_int(J),one_one_int),K))) ) ) ) ).

% upto_split3
tff(fact_9528_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D3: 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))
             => ( D3 = 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))
               => ( D3 = 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))
               => ( D3 = 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),D3,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9529_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_9530_MVT2,axiom,
    ! [A: real,B: real,F: fun_real_real,F4: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),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_eq_real,X2),B))
             => has_fi5821293074295781190e_real(F,aa_real_real(F4,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
       => ? [Z4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),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(F4,Z4)) ) ) ) ) ).

% MVT2
tff(fact_9531_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,X: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3),S3)
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D2))
                   => 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_9532_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,X: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3),S3)
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D2))
                   => 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_9533_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3),S3)
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D2))
                   => 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_9534_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3),S3)
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D2))
                   => 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_9535_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))
       => ? [D2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
            & ! [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),D2))
                 => 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_9536_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))
       => ? [D2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
            & ! [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),D2))
                 => 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_9537_DERIV__isconst__all,axiom,
    ! [F: fun_real_real,X: real,Y: real] :
      ( ! [X2: real] : has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( aa_real_real(F,X) = aa_real_real(F,Y) ) ) ).

% DERIV_isconst_all
tff(fact_9538_DERIV__mirror,axiom,
    ! [F: fun_real_real,Y: real,X: real] :
      ( has_fi5821293074295781190e_real(F,Y,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,X),top_top_set_real))
    <=> has_fi5821293074295781190e_real(aTP_Lamm_zo(F),aa_real_real(uminus_uminus_real,Y),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_mirror
tff(fact_9539_DERIV__const__ratio__const2,axiom,
    ! [A: real,B: real,F: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X2: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X2,top_top_set_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) = K ) ) ) ).

% DERIV_const_ratio_const2
tff(fact_9540_DERIV__const__ratio__const,axiom,
    ! [A: real,B: real,F: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X2: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X2,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)),K) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_9541_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))
       => ? [D2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
            & ! [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),D2))
                 => 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_9542_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))
       => ? [D2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
            & ! [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),D2))
                 => 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_9543_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))
     => ( member_real(X,set_or1633881224788618240n_real(A,B))
       => ( member_real(Y,set_or1633881224788618240n_real(A,B))
         => ( ! [X2: real] :
                ( member_real(X2,set_or1633881224788618240n_real(A,B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X2,top_top_set_real)) )
           => ( aa_real_real(F,X) = aa_real_real(F,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_9544_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))
     => ( ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B))
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X2,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_9545_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))
     => ( ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B))
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X2,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_9546_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G: fun_real_real,G2: fun_real_real] :
      ( ! [X2: real] :
          ( member_real(X2,set_or1222579329274155063t_real(A,B))
         => has_fi5821293074295781190e_real(G,aa_real_real(G2,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) )
     => ( ! [X2: real] :
            ( member_real(X2,set_or1222579329274155063t_real(A,B))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(G2,X2))) )
       => ( 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(G,A)),aa_real_real(G,B))) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_9547_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),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_eq_real,X2),B))
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3)) ) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,A)),aa_real_real(F,B))) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_9548_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),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_eq_real,X2),B))
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),zero_zero_real)) ) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,B)),aa_real_real(F,A))) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_9549_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_9550_DERIV__const__average,axiom,
    ! [A: real,B: real,V: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X2: real] : has_fi5821293074295781190e_real(V,K,topolo2177554685111907308n_real(X2,top_top_set_real))
       => ( aa_real_real(V,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(V,A)),aa_real_real(V,B))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ) ).

% DERIV_const_average
tff(fact_9551_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_9552_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_9553_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_9554_DERIV__pow,axiom,
    ! [N: nat,X: real,S: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_zp,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_9555_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,M: real,X: real,N: nat] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_zq(G),N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(G,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))),M),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_9556_has__real__derivative__powr,axiom,
    ! [Z: real,R: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_zr,R),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Z),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_9557_DERIV__series_H,axiom,
    ! [F: fun_re4238438383274954274t_real,F4: fun_re4238438383274954274t_real,X0: real,A: real,B: real,L5: fun_nat_real] :
      ( ! [N2: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_zs(F),N2),aa_nat_real(aa_real_fun_nat_real(F4,X0),N2),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X2: real] :
            ( member_real(X2,set_or1633881224788618240n_real(A,B))
           => summable_real(aa_real_fun_nat_real(F,X2)) )
       => ( member_real(X0,set_or1633881224788618240n_real(A,B))
         => ( summable_real(aa_real_fun_nat_real(F4,X0))
           => ( summable_real(L5)
             => ( ! [N2: nat,X2: real,Y2: real] :
                    ( member_real(X2,set_or1633881224788618240n_real(A,B))
                   => ( member_real(Y2,set_or1633881224788618240n_real(A,B))
                     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(F,X2),N2)),aa_nat_real(aa_real_fun_nat_real(F,Y2),N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(L5,N2)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y2))))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_zt(F),suminf_real(aa_real_fun_nat_real(F4,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_9558_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_9559_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,M: real,X: real,R: 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_zu(G),R),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),M),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_9560_DERIV__powr,axiom,
    ! [G: fun_real_real,M: real,X: real,F: fun_real_real,R: 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,R,topolo2177554685111907308n_real(X,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_zv(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,R),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_9561_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))),numeral_numeral_real(aa_num_num(bit0,one))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_9562_DERIV__arctan,axiom,
    ! [X: real] : has_fi5821293074295781190e_real(arctan,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X,top_top_set_real)) ).

% DERIV_arctan
tff(fact_9563_arsinh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] : has_fi5821293074295781190e_real(arsinh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X,A2)) ).

% arsinh_real_has_field_derivative
tff(fact_9564_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D3: real] :
      ( ( X != zero_zero_real )
     => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( D3 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),numeral_numeral_real(aa_num_num(bit0,one))) ) )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
           => ( D3 = 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)))),numeral_numeral_real(aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D3,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_9565_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_9566_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_9567_DERIV__power__series_H,axiom,
    ! [R2: real,F: fun_nat_real,X0: real] :
      ( ! [X2: real] :
          ( member_real(X2,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R2),R2))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_zw(F),X2)) )
     => ( member_real(X0,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R2),R2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
         => has_fi5821293074295781190e_real(aTP_Lamm_zy(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_zw(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_9568_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_9569_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_9570_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_9571_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 )
     => ( ! [M3: nat,X2: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X2),topolo2177554685111907308n_real(X2,top_top_set_real))
       => ? [T4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),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_zz(Diff),X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T4)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_9572_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 )
        & ! [M3: nat,X2: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X2),topolo2177554685111907308n_real(X2,top_top_set_real)) )
     => ? [T4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),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_zz(Diff),X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T4)),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_9573_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_9574_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 )
         => ( ! [M3: nat,T4: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,H),T4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),zero_zero_real)) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
           => ? [T4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),T4))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),zero_zero_real))
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_rd(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T4)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_9575_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 )
       => ( ! [M3: nat,T4: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T4))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),H)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ? [T4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T4))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),H))
              & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_rd(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T4)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_9576_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 )
         => ( ! [M3: nat,T4: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),H)) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
           => ? [T4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T4))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),H))
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_rd(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T4)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_9577_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 )
         => ( ! [M3: nat,X2: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X2),topolo2177554685111907308n_real(X2,top_top_set_real))
           => ? [T4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T4)))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T4)),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_zz(Diff),X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T4)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_9578_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 )
     => ( ! [M3: nat,T4: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,X))) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
       => ? [T4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),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_zz(Diff),X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T4)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_9579_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 )
       => ( ! [M3: nat,T4: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T4))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T4),topolo2177554685111907308n_real(T4,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))
             => ? [T4: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),T4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),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_aaa(Diff),A),C),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T4)),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_9580_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 )
       => ( ! [M3: nat,T4: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T4))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T4),topolo2177554685111907308n_real(T4,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))
             => ? [T4: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),T4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),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_aaa(Diff),B),C),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T4)),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_9581_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 )
       => ( ! [M3: nat,T4: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T4))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T4),topolo2177554685111907308n_real(T4,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 )
                   => ? [T4: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T4))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),C)) ) )
                        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),T4))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),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_aab(Diff),C),X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T4)),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_9582_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B2: real] :
      ( ! [M3: nat,T4: real] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),H)) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
     => ( ( N = aa_nat_nat(suc,K) )
       => ! [M2: nat,T5: 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),T5))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T5),H)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_aad(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)),T5)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_aae(Diff),M2),T5),aa_nat_set_nat(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,T5),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(T5,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_9583_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_aaf,suminf_real(aa_real_fun_nat_real(aTP_Lamm_aag,X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_9584_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_9585_isCont__Lb__Ub,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),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_eq_real,X2),B)) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),F) )
       => ? [L6: real,M9: 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)) )
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L6),aa_real_real(F,X4)))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),M9)) ) )
            & ! [Y3: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L6),Y3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),M9)) )
               => ? [X2: real] :
                    ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B))
                    & ( aa_real_real(F,X2) = Y3 ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_9586_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))
            & ! [X4: real] :
                ( ( ( X4 != 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),X4))),R3)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),zero_zero_real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_9587_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))
            & ! [X4: real] :
                ( ( ( X4 != 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),X4))),R3)) )
               => ( aa_real_real(F,X4) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_9588_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))
            & ! [X4: real] :
                ( ( ( X4 != 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),X4))),R3)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,X4))) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_9589_isCont__real__sqrt,axiom,
    ! [X: real] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),sqrt) ).

% isCont_real_sqrt
tff(fact_9590_isCont__real__root,axiom,
    ! [X: real,N: nat] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),aa_nat_fun_real_real(root,N)) ).

% isCont_real_root
tff(fact_9591_continuous__frac,axiom,
    ! [X: real] :
      ( ~ member_real(X,ring_1_Ints_real)
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),archim2898591450579166408c_real) ) ).

% continuous_frac
tff(fact_9592_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))
       => ( ! [Z4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z4),B))
               => ( aa_real_real(G,aa_real_real(F,Z4)) = Z4 ) ) )
         => ( ! [Z4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z4),B))
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z4,top_top_set_real),F) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_9593_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_9594_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_aah,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))),top_top_set_real)) ).

% LIM_cos_div_sin
tff(fact_9595_continuous__floor,axiom,
    ! [X: real] :
      ( ~ member_real(X,ring_1_Ints_real)
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),comp_int_real_real(ring_1_of_int_real,archim6058952711729229775r_real)) ) ).

% continuous_floor
tff(fact_9596_DERIV__inverse__function,axiom,
    ! [F: fun_real_real,D3: real,G: fun_real_real,X: real,A: real,B: real] :
      ( has_fi5821293074295781190e_real(F,D3,topolo2177554685111907308n_real(aa_real_real(G,X),top_top_set_real))
     => ( ( D3 != 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,D3),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_9597_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_9598_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_9599_LIM__less__bound,axiom,
    ! [B: real,X: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),X))
     => ( ! [X2: real] :
            ( member_real(X2,set_or1633881224788618240n_real(B,X))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2))) )
       => ( 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_9600_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_9601_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))
     => ( ! [Z4: 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,Z4),X))),D))
           => ( aa_real_real(G,aa_real_real(F,Z4)) = Z4 ) )
       => ( ! [Z4: 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,Z4),X))),D))
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z4,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_9602_GMVT_H,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [Z4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z4),B))
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z4,top_top_set_real),F) ) )
       => ( ! [Z4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z4),B))
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z4,top_top_set_real),G) ) )
         => ( ! [Z4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),B))
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z4),topolo2177554685111907308n_real(Z4,top_top_set_real)) ) )
           => ( ! [Z4: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z4))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),B))
                   => has_fi5821293074295781190e_real(F,aa_real_real(F4,Z4),topolo2177554685111907308n_real(Z4,top_top_set_real)) ) )
             => ? [C2: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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,C2)) = 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(F4,C2)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_9603_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)))
         => ! [N6: nat] : member_real(suminf_real(aTP_Lamm_aai(A)),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_aai(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6))),groups6591440286371151544t_real(aTP_Lamm_aai(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_9604_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))
         => ! [N6: nat] : member_real(suminf_real(aTP_Lamm_aai(A)),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_aai(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)),one_one_nat))),groups6591440286371151544t_real(aTP_Lamm_aai(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_9605_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_ai,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_9606_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_9607_monoseq__convergent,axiom,
    ! [X5: fun_nat_real,B2: real] :
      ( topolo6980174941875973593q_real(X5)
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(X5,I3))),B2))
       => ~ ! [L6: real] : ~ filterlim_nat_real(X5,topolo2815343760600316023s_real(L6),at_top_nat) ) ) ).

% monoseq_convergent
tff(fact_9608_LIMSEQ__root,axiom,
    filterlim_nat_real(aTP_Lamm_aaj,topolo2815343760600316023s_real(one_one_real),at_top_nat) ).

% LIMSEQ_root
tff(fact_9609_nested__sequence__unique,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(F,aa_nat_nat(suc,N2))))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N2))),aa_nat_real(G,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G,N2)))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ij(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L3: real] :
                ( ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N6)),L3))
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L3),at_top_nat)
                & ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L3),aa_nat_real(G,N6)))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_9610_LIMSEQ__inverse__zero,axiom,
    ! [X5: 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(X5,N2))) )
     => filterlim_nat_real(aTP_Lamm_aak(X5),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_9611_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_aal,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_9612_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_aam,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_9613_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_aan,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_9614_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))
       => ( ! [E2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
             => ? [N6: 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,N6)),E2))) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_9615_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_9616_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_aao(X),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_9617_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_9618_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_9619_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_aap,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_9620_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_aaq,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_9621_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_aar,X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),at_top_nat) ).

% tendsto_exp_limit_sequentially
tff(fact_9622_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_aas,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_9623_summable__Leibniz_I1_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => summable_real(aTP_Lamm_aai(A)) ) ) ).

% summable_Leibniz(1)
tff(fact_9624_summable,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [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,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => summable_real(aTP_Lamm_aai(A)) ) ) ) ).

% summable
tff(fact_9625_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_aat(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K3: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_aau(Theta),K3),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_9626_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_aav(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K3: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_aau(Theta),K3),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_9627_summable__Leibniz_I4_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => filterlim_nat_real(aTP_Lamm_aaw(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_aai(A))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_9628_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_nk,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_9629_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,N: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [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,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_aai(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),suminf_real(aTP_Lamm_aai(A)))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_9630_summable__Leibniz_H_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [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,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => filterlim_nat_real(aTP_Lamm_aaw(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_aai(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_9631_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L3: real] :
              ( ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_aai(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)))),L3))
              & filterlim_nat_real(aTP_Lamm_aaw(A),topolo2815343760600316023s_real(L3),at_top_nat)
              & ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L3),groups6591440286371151544t_real(aTP_Lamm_aai(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)),one_one_nat)))))
              & filterlim_nat_real(aTP_Lamm_aax(A),topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_9632_summable__Leibniz_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => filterlim_nat_real(aTP_Lamm_aax(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_aai(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_9633_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,N: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [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,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(aTP_Lamm_aai(A))),groups6591440286371151544t_real(aTP_Lamm_aai(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_9634_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [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,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => filterlim_nat_real(aTP_Lamm_aax(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_aai(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_9635_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_aay(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_9636_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_9637_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_bool] :
      ( ! [X2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),X2))
         => pp(aa_nat_bool(P,X2)) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_9638_eventually__sequentially,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N9: nat] :
        ! [N5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),N5))
         => pp(aa_nat_bool(P,N5)) ) ) ).

% eventually_sequentially
tff(fact_9639_sequentially__offset,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_aay(P),K),at_top_nat) ) ).

% sequentially_offset
tff(fact_9640_real__bounded__linear,axiom,
    ! [F: fun_real_real] :
      ( real_V5970128139526366754l_real(F)
    <=> ? [C3: real] :
        ! [X3: real] : aa_real_real(F,X3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),C3) ) ).

% real_bounded_linear
tff(fact_9641_sqrt__at__top,axiom,
    filterlim_real_real(sqrt,at_top_real,at_top_real) ).

% sqrt_at_top
tff(fact_9642_lhopital__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F4: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(F),F4),topolo2177554685111907308n_real(A,top_top_set_real))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(G),G2),topolo2177554685111907308n_real(A,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aba(F4),G2),at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aba(F),G),at_top_real,topolo2177554685111907308n_real(A,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_top
tff(fact_9643_lhopital__left__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F4: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(F),F4),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(G),G2),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aba(F4),G2),at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aba(F),G),at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_9644_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_9645_lhospital__at__top__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,X: real] :
      ( filterlim_real_real(G,at_top_real,at_top_real)
     => ( eventually_real(aTP_Lamm_abc(G2),at_top_real)
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(F),F4),at_top_real)
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(G),G2),at_top_real)
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abd(G2),F4),topolo2815343760600316023s_real(X),at_top_real)
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abd(G),F),topolo2815343760600316023s_real(X),at_top_real) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_9646_lhopital__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( eventually_real(aTP_Lamm_abc(G2),topolo2177554685111907308n_real(X,top_top_set_real))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(F),F4),topolo2177554685111907308n_real(X,top_top_set_real))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(G),G2),topolo2177554685111907308n_real(X,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abd(G2),F4),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abd(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_9647_lhopital__left__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
     => ( eventually_real(aTP_Lamm_abc(G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(F),F4),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(G),G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abd(G2),F4),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abd(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_9648_ln__x__over__x__tendsto__0,axiom,
    filterlim_real_real(aTP_Lamm_abe,topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% ln_x_over_x_tendsto_0
tff(fact_9649_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim_real_real(aa_nat_fun_real_real(aTP_Lamm_abf,K),topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% tendsto_power_div_exp_0
tff(fact_9650_lhopital,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,top_top_set_real))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,top_top_set_real))
       => ( eventually_real(aTP_Lamm_abc(G),topolo2177554685111907308n_real(X,top_top_set_real))
         => ( eventually_real(aTP_Lamm_abc(G2),topolo2177554685111907308n_real(X,top_top_set_real))
           => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(F),F4),topolo2177554685111907308n_real(X,top_top_set_real))
             => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(G),G2),topolo2177554685111907308n_real(X,top_top_set_real))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abd(G2),F4),F3,topolo2177554685111907308n_real(X,top_top_set_real))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aba(F),G),F3,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_9651_lhopital__left,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
       => ( eventually_real(aTP_Lamm_abc(G),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
         => ( eventually_real(aTP_Lamm_abc(G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
           => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(F),F4),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
             => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(G),G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abd(G2),F4),F3,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aba(F),G),F3,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_9652_tendsto__exp__limit__at__top,axiom,
    ! [X: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_abg,X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),at_top_real) ).

% tendsto_exp_limit_at_top
tff(fact_9653_filterlim__tan__at__left,axiom,
    filterlim_real_real(tan_real,at_top_real,topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))),set_or5984915006950818249n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))) ).

% filterlim_tan_at_left
tff(fact_9654_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),X2))
         => ? [Y3: real] :
              ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X2,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_9655_tendsto__arctan__at__top,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),at_top_real) ).

% tendsto_arctan_at_top
tff(fact_9656_dist__real__def,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,X),Y) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) ).

% dist_real_def
tff(fact_9657_dist__complex__def,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(real_V3694042436643373181omplex(X),Y) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y)) ).

% dist_complex_def
tff(fact_9658_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_abh(N),F),at_top_real,F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_9659_lhopital__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F4: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(F),F4),topolo2177554685111907308n_real(A,top_top_set_real))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(G),G2),topolo2177554685111907308n_real(A,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aba(F4),G2),at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aba(F),G),at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_bot
tff(fact_9660_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B))
         => ? [Y3: real] :
              ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X2,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_9661_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_abh(N),F),at_bot_real,F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_9662_lhopital__left__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F4: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(F),F4),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(G),G2),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aba(F4),G2),at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aba(F),G),at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
tff(fact_9663_tendsto__arctan__at__bot,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),at_bot_real) ).

% tendsto_arctan_at_bot
tff(fact_9664_Bseq__realpow,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_eq_real,X),one_one_real))
       => bfun_nat_real(aa_real_fun_nat_real(power_power_real,X),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_9665_tendsto__exp__limit__at__right,axiom,
    ! [X: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_abi,X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% tendsto_exp_limit_at_right
tff(fact_9666_filterlim__tan__at__right,axiom,
    filterlim_real_real(tan_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))))) ).

% filterlim_tan_at_right
tff(fact_9667_eventually__at__right__to__0,axiom,
    ! [P: fun_real_bool,A: real] :
      ( eventually_real(P,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
    <=> eventually_real(aa_rea7809428808904475482l_bool(aTP_Lamm_abj(P),A),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_right_to_0
tff(fact_9668_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_9669_lhopital__right__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F4: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(F),F4),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(G),G2),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aba(F4),G2),at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aba(F),G),at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
tff(fact_9670_lhopital__right__0,axiom,
    ! [F0: fun_real_real,G0: fun_real_real,G2: fun_real_real,F4: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( filterlim_real_real(G0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aTP_Lamm_abc(G0),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aTP_Lamm_abc(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(F0),F4),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(G0),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abd(G2),F4),F3,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aba(F0),G0),F3,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_9671_lhopital__right,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
       => ( eventually_real(aTP_Lamm_abc(G),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
         => ( eventually_real(aTP_Lamm_abc(G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
           => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(F),F4),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
             => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(G),G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abd(G2),F4),F3,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aba(F),G),F3,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_9672_lhopital__right__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F4: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(F),F4),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(G),G2),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aba(F4),G2),at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aba(F),G),at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
tff(fact_9673_lhopital__right__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
     => ( eventually_real(aTP_Lamm_abc(G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(F),F4),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(G),G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abd(G2),F4),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abd(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_9674_lhopital__right__0__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,X: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( eventually_real(aTP_Lamm_abc(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(F),F4),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(G),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abd(G2),F4),topolo2815343760600316023s_real(X),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abd(G),F),topolo2815343760600316023s_real(X),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_9675_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_9676_atLeast__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atLeast_nat,K),insert_nat(K,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_9677_decseq__bounded,axiom,
    ! [X5: fun_nat_real,B2: real] :
      ( order_9091379641038594480t_real(X5)
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_nat_real(X5,I3)))
       => bfun_nat_real(X5,at_top_nat) ) ) ).

% decseq_bounded
tff(fact_9678_decseq__convergent,axiom,
    ! [X5: fun_nat_real,B2: real] :
      ( order_9091379641038594480t_real(X5)
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_nat_real(X5,I3)))
       => ~ ! [L6: real] :
              ( filterlim_nat_real(X5,topolo2815343760600316023s_real(L6),at_top_nat)
             => ~ ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L6),aa_nat_real(X5,I2))) ) ) ) ).

% decseq_convergent
tff(fact_9679_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))
     => ( ! [X2: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B)) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),F) )
       => ( ! [X2: real] :
              ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B)) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X2,top_top_set_real)) )
         => ( ! [X2: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B)) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),G) )
           => ( ! [X2: real] :
                  ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B)) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X2,top_top_set_real)) )
             => ? [G_c: real,F_c: real,C2: real] :
                  ( has_fi5821293074295781190e_real(G,G_c,topolo2177554685111907308n_real(C2,top_top_set_real))
                  & has_fi5821293074295781190e_real(F,F_c,topolo2177554685111907308n_real(C2,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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_9680_real__differentiableE,axiom,
    ! [F: fun_real_real,X: real,S: set_real] :
      ( differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X,S))
     => ~ ! [Df: real] : ~ has_fi5821293074295781190e_real(F,Df,topolo2177554685111907308n_real(X,S)) ) ).

% real_differentiableE
tff(fact_9681_real__differentiable__def,axiom,
    ! [F: fun_real_real,X: real,S: set_real] :
      ( differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X,S))
    <=> ? [D6: real] : has_fi5821293074295781190e_real(F,D6,topolo2177554685111907308n_real(X,S)) ) ).

% real_differentiable_def
tff(fact_9682_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)
       => ( ! [X2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B))
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
         => ? [L3: real,Z4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z4))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),B))
              & has_fi5821293074295781190e_real(F,L3,topolo2177554685111907308n_real(Z4,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_9683_continuous__on__arcosh_H,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A2,F)
     => ( ! [X2: real] :
            ( member_real(X2,A2)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,X2))) )
       => topolo5044208981011980120l_real(A2,aTP_Lamm_abk(F)) ) ) ).

% continuous_on_arcosh'
tff(fact_9684_continuous__image__closed__interval,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ? [C2: real,D2: real] :
            ( ( image_real_real(F,set_or1222579329274155063t_real(A,B)) = set_or1222579329274155063t_real(C2,D2) )
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),D2)) ) ) ) ).

% continuous_image_closed_interval
tff(fact_9685_Rolle__deriv,axiom,
    ! [A: real,B: real,F: fun_real_real,F4: 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)
         => ( ! [X2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B))
                 => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F4,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
           => ? [Z4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z4))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),B))
                & ! [X4: real] : aa_real_real(aa_rea7340605660300375871l_real(F4,Z4),X4) = zero_zero_real ) ) ) ) ) ).

% Rolle_deriv
tff(fact_9686_mvt,axiom,
    ! [A: real,B: real,F: fun_real_real,F4: fun_re7681423839302404990l_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B))
               => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F4,X2),topolo2177554685111907308n_real(X2,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(F4,Xi),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ) ) ) ) ) ) ).

% mvt
tff(fact_9687_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)
       => ( ! [X2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
         => ( aa_real_real(F,B) = aa_real_real(F,A) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_9688_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))
     => ( ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B))
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X2,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_9689_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))
     => ( ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B))
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X2,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_9690_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)
       => ( ! [X2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X2,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_9691_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)
         => ( ! [X2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B))
                 => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
           => ? [Z4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z4))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),B))
                & has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(Z4,top_top_set_real)) ) ) ) ) ) ).

% Rolle
tff(fact_9692_uniformity__complex__def,axiom,
    topolo896644834953643431omplex = comple8358262395181532106omplex(image_5971271580939081552omplex(aTP_Lamm_abm,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_complex_def
tff(fact_9693_uniformity__real__def,axiom,
    topolo1511823702728130853y_real = comple2936214249959783750l_real(image_2178119161166701260l_real(aTP_Lamm_abo,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_real_def
tff(fact_9694_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_9695_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_9696_incseq__bounded,axiom,
    ! [X5: fun_nat_real,B2: real] :
      ( order_mono_nat_real(X5)
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,I3)),B2))
       => bfun_nat_real(X5,at_top_nat) ) ) ).

% incseq_bounded
tff(fact_9697_incseq__convergent,axiom,
    ! [X5: fun_nat_real,B2: real] :
      ( order_mono_nat_real(X5)
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,I3)),B2))
       => ~ ! [L6: real] :
              ( filterlim_nat_real(X5,topolo2815343760600316023s_real(L6),at_top_nat)
             => ~ ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,I2)),L6)) ) ) ) ).

% incseq_convergent
tff(fact_9698_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_abp,K)) ) ).

% mono_ge2_power_minus_self
tff(fact_9699_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( ! [X2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X2)))
     => ( order_mono_nat_real(F)
       => ( order_5726023648592871131at_nat(G)
         => ( bfun_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_abq(F),G),at_top_nat)
          <=> bfun_nat_real(F,at_top_nat) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
tff(fact_9700_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_9701_infinite__enumerate,axiom,
    ! [S3: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ? [R3: fun_nat_nat] :
          ( order_5726023648592871131at_nat(R3)
          & ! [N6: nat] : pp(aa_set_nat_bool(member_nat(aa_nat_nat(R3,N6)),S3)) ) ) ).

% infinite_enumerate
tff(fact_9702_inj__sgn__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_zn,N),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_9703_log__inj,axiom,
    ! [B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => inj_on_real_real(aa_rea7340605660300375871l_real(log,B),set_or5849166863359141190n_real(zero_zero_real)) ) ).

% log_inj
tff(fact_9704_pos__deriv__imp__strict__mono,axiom,
    ! [F: fun_real_real,F4: fun_real_real] :
      ( ! [X2: real] : has_fi5821293074295781190e_real(F,aa_real_real(F4,X2),topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F4,X2)))
       => order_7092887310737990675l_real(F) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_9705_inj__Suc,axiom,
    ! [N3: set_nat] : pp(aa_set_nat_bool(inj_on_nat_nat(suc),N3)) ).

% inj_Suc
tff(fact_9706_inj__on__set__encode,axiom,
    inj_on_set_nat_nat(nat_set_encode,collect_set_nat(finite_finite_nat)) ).

% inj_on_set_encode
tff(fact_9707_inj__on__diff__nat,axiom,
    ! [N3: set_nat,K: nat] :
      ( ! [N2: nat] :
          ( pp(aa_set_nat_bool(member_nat(N2),N3))
         => 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_zj,K)),N3)) ) ).

% inj_on_diff_nat
tff(fact_9708_summable__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( pp(aa_set_nat_bool(inj_on_nat_nat(G),top_top_set_nat))
       => ( ! [X2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X2)))
         => summable_real(comp_nat_real_nat(F,G)) ) ) ) ).

% summable_reindex
tff(fact_9709_suminf__reindex__mono,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( pp(aa_set_nat_bool(inj_on_nat_nat(G),top_top_set_nat))
       => ( ! [X2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X2)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(comp_nat_real_nat(F,G))),suminf_real(F))) ) ) ) ).

% suminf_reindex_mono
tff(fact_9710_inj__on__char__of__nat,axiom,
    pp(aa_set_nat_bool(inj_on_nat_char(unique3096191561947761185of_nat),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))))) ).

% inj_on_char_of_nat
tff(fact_9711_suminf__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( pp(aa_set_nat_bool(inj_on_nat_nat(G),top_top_set_nat))
       => ( ! [X2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X2)))
         => ( ! [X2: nat] :
                ( ~ pp(aa_set_nat_bool(member_nat(X2),image_nat_nat(G,top_top_set_nat)))
               => ( aa_nat_real(F,X2) = zero_zero_real ) )
           => ( suminf_real(comp_nat_real_nat(F,G)) = suminf_real(F) ) ) ) ) ) ).

% suminf_reindex
tff(fact_9712_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_9713_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
tff(fact_9714_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))
     => ( aa_nat_set_nat(set_or4665077453230672383an_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = sup_sup_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(I),J),aa_nat_set_nat(set_or4665077453230672383an_nat(J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_9715_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(aa_real_fun_int_real(power_int_real,X),N) ) ) ) ).

% powr_real_of_int'
tff(fact_9716_positive__rat,axiom,
    ! [A: int,B: int] :
      ( pp(aa_rat_bool(positive,aa_int_rat(fract(A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(times_times_int(A),B))) ) ).

% positive_rat
tff(fact_9717_Rat_Opositive__mult,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(positive,X))
     => ( pp(aa_rat_bool(positive,Y))
       => pp(aa_rat_bool(positive,aa_rat_rat(times_times_rat(X),Y))) ) ) ).

% Rat.positive_mult
tff(fact_9718_Rat_Opositive__add,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(positive,X))
     => ( pp(aa_rat_bool(positive,Y))
       => pp(aa_rat_bool(positive,plus_plus_rat(X,Y))) ) ) ).

% Rat.positive_add
tff(fact_9719_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,minus_minus_rat(Y,X))) ) ).

% less_rat_def
tff(fact_9720_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(times_times_int(product_fst_int_int(rep_Rat(X))),product_snd_int_int(rep_Rat(X))))) ) ).

% Rat.positive.rep_eq
tff(fact_9721_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_9722_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_9723_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_9724_min__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),aa_num_nat(pred_numeral,K))) ).

% min_Suc_numeral
tff(fact_9725_min__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),N)) ).

% min_numeral_Suc
tff(fact_9726_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_9727_concat__bit__assoc__sym,axiom,
    ! [M: nat,N: nat,K: int,L: int,R: int] : aa_int_int(bit_concat_bit(M,aa_int_int(bit_concat_bit(N,K),L)),R) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N),K),aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N),L),R)) ).

% concat_bit_assoc_sym
tff(fact_9728_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q2: 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)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ).

% nat_mult_min_left
tff(fact_9729_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q2: 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),Q2)) = 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),Q2)) ).

% nat_mult_min_right
tff(fact_9730_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_9731_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),aa_int_int(bit_concat_bit(N,K),L)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N),K),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),L)) ).

% take_bit_concat_bit_eq
tff(fact_9732_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_abr,N)),M) ).

% min_Suc1
tff(fact_9733_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_abs,N)),M) ).

% min_Suc2
tff(fact_9734_min__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,zero_z5237406670263579293d_enat),Q2) = zero_z5237406670263579293d_enat ).

% min_enat_simps(3)
tff(fact_9735_min__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,Q2),zero_z5237406670263579293d_enat) = zero_z5237406670263579293d_enat ).

% min_enat_simps(2)
tff(fact_9736_inf__enat__def,axiom,
    inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).

% inf_enat_def
tff(fact_9737_num__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),one_one_Code_integer))
       => ( code_num_of_integer(K) = one ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),one_one_Code_integer))
       => ( code_num_of_integer(K) = produc7336495610019696514er_num(aTP_Lamm_abt,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ) ) ).

% num_of_integer_code
tff(fact_9738_card__le__Suc__Max,axiom,
    ! [S3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_nat(S3)),aa_nat_nat(suc,lattic8265883725875713057ax_nat(S3)))) ) ).

% card_le_Suc_Max
tff(fact_9739_divide__nat__def,axiom,
    ! [N: nat,M: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_abu(N),M))) ) ) ) ).

% divide_nat_def
tff(fact_9740_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) = lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_abv(N),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_9741_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))
         => ( gcd_Gcd_nat(M4) = lattic8265883725875713057ax_nat(comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_abw,M4))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_9742_rat__floor__code,axiom,
    ! [P2: rat] : archim3151403230148437115or_rat(P2) = produc8211389475949308722nt_int(divide_divide_int,quotient_of(P2)) ).

% rat_floor_code
tff(fact_9743_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : adjust_div(Qr) = produc8211389475949308722nt_int(aTP_Lamm_abx,Qr) ).

% Divides.adjust_div_def
tff(fact_9744_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
       => ( code_nat_of_integer(K) = zero_zero_nat ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
       => ( code_nat_of_integer(K) = produc1555791787009142072er_nat(aTP_Lamm_aby,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ) ) ).

% nat_of_integer_code
tff(fact_9745_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] : code_nat_of_integer(aa_num_Code_integer(numera6620942414471956472nteger,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_of_integer_code_post(3)
tff(fact_9746_remdups__upt,axiom,
    ! [M: nat,N: nat] : remdups_nat(upt(M,N)) = upt(M,N) ).

% remdups_upt
tff(fact_9747_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_9748_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_9749_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_9750_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_9751_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_9752_sorted__list__of__set__range,axiom,
    ! [M: nat,N: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = upt(M,N) ).

% sorted_list_of_set_range
tff(fact_9753_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_9754_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(nth_nat(upt(I,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).

% nth_upt
tff(fact_9755_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_9756_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_9757_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_9758_atMost__upto,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,N))) ).

% atMost_upto
tff(fact_9759_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
      ( ( cons_nat(M,cons_nat(N,Ns)) = upt(M,Q2) )
    <=> ( cons_nat(N,Ns) = upt(aa_nat_nat(suc,M),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_9760_atLeast__upt,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = set_nat2(upt(zero_zero_nat,N)) ).

% atLeast_upt
tff(fact_9761_atLeastLessThan__upt,axiom,
    ! [I: nat,J: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(I),J) = set_nat2(upt(I,J)) ).

% atLeastLessThan_upt
tff(fact_9762_greaterThanAtMost__upt,axiom,
    ! [N: nat,M: nat] : aa_nat_set_nat(set_or6659071591806873216st_nat(N),M) = set_nat2(upt(aa_nat_nat(suc,N),aa_nat_nat(suc,M))) ).

% greaterThanAtMost_upt
tff(fact_9763_atLeastAtMost__upt,axiom,
    ! [N: nat,M: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(N),M) = set_nat2(upt(N,aa_nat_nat(suc,M))) ).

% atLeastAtMost_upt
tff(fact_9764_greaterThanLessThan__upt,axiom,
    ! [N: nat,M: nat] : aa_nat_set_nat(set_or5834768355832116004an_nat(N),M) = set_nat2(upt(aa_nat_nat(suc,N),M)) ).

% greaterThanLessThan_upt
tff(fact_9765_distinct__upt,axiom,
    ! [I: nat,J: nat] : pp(aa_list_nat_bool(distinct_nat,upt(I,J))) ).

% distinct_upt
tff(fact_9766_map__decr__upt,axiom,
    ! [M: nat,N: nat] : map_nat_nat(aTP_Lamm_abz,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = upt(M,N) ).

% map_decr_upt
tff(fact_9767_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_9768_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_9769_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_9770_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_9771_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_9772_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_9773_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_9774_tl__upt,axiom,
    ! [M: nat,N: nat] : tl_nat(upt(M,N)) = upt(aa_nat_nat(suc,M),N) ).

% tl_upt
tff(fact_9775_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_an),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ) ) ).

% sum_list_upt
tff(fact_9776_card__length__sum__list__rec,axiom,
    ! [M: nat,N3: 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_acb(M),N3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_acc(M),N3)))),finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_acd(M),N3)))) ) ) ).

% card_length_sum_list_rec
tff(fact_9777_card__length__sum__list,axiom,
    ! [M: nat,N3: nat] : finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_acb(M),N3))) = 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,N3),M)),one_one_nat)),N3) ).

% card_length_sum_list
tff(fact_9778_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,aa_VEB5832953974749077948T_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_9779_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,aa_VEB5832953974749077948T_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_9780_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_9781_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_9782_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(nth_nat(Ns),I))) ) ) ).

% sorted_wrt_less_idx
tff(fact_9783_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_pi,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uu)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_1
tff(fact_9784_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_qr,Uu) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu)) ).

% ATP.lambda_2
tff(fact_9785_ATP_Olambda__3,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_xe,Uu))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Uu))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uu),numeral_numeral_real(aa_num_num(bit0,one))))
        & ( aa_real_real(cos_real,Uu) = zero_zero_real ) ) ) ).

% ATP.lambda_3
tff(fact_9786_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ph,Uu) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uu)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu)) ).

% ATP.lambda_4
tff(fact_9787_ATP_Olambda__5,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_aah,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,Uu)),aa_real_real(sin_real,Uu)) ).

% ATP.lambda_5
tff(fact_9788_ATP_Olambda__6,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_abe,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Uu)),Uu) ).

% ATP.lambda_6
tff(fact_9789_ATP_Olambda__7,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_aaj,Uu) = aa_real_real(aa_nat_fun_real_real(root,Uu),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_7
tff(fact_9790_ATP_Olambda__8,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_abz,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_8
tff(fact_9791_ATP_Olambda__9,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_rp,Uu) = aa_complex_complex(plus_plus_complex(Uu),one_one_complex) ).

% ATP.lambda_9
tff(fact_9792_ATP_Olambda__10,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_rn,Uu) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),one_one_real) ).

% ATP.lambda_10
tff(fact_9793_ATP_Olambda__11,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_rl,Uu) = plus_plus_rat(Uu,one_one_rat) ).

% ATP.lambda_11
tff(fact_9794_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ro,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_12
tff(fact_9795_ATP_Olambda__13,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_rm,Uu) = aa_int_int(plus_plus_int(Uu),one_one_int) ).

% ATP.lambda_13
tff(fact_9796_ATP_Olambda__14,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_aal,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_14
tff(fact_9797_ATP_Olambda__15,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_aaf,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ns,Uu)) ).

% ATP.lambda_15
tff(fact_9798_ATP_Olambda__16,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_abw,Uu) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gg,Uu)) ).

% ATP.lambda_16
tff(fact_9799_ATP_Olambda__17,axiom,
    ! [Uu: real] : aa_rea5428163979925208277l_real(aTP_Lamm_abo,Uu) = princi6114159922880469582l_real(collec3799799289383736868l_real(produc5257810830037637649l_bool(aTP_Lamm_abn(Uu)))) ).

% ATP.lambda_17
tff(fact_9800_ATP_Olambda__18,axiom,
    ! [Uu: real] : aa_rea5117059003799598681omplex(aTP_Lamm_abm,Uu) = princi3496590319149328850omplex(collec8663557070575231912omplex(produc9214369023355538061x_bool(aTP_Lamm_abl(Uu)))) ).

% ATP.lambda_18
tff(fact_9801_ATP_Olambda__19,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_yw,Uu) = some_num(aa_num_num(bit1,Uu)) ).

% ATP.lambda_19
tff(fact_9802_ATP_Olambda__20,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_ys,Uu) = some_num(aa_num_num(bit0,Uu)) ).

% ATP.lambda_20
tff(fact_9803_ATP_Olambda__21,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_ln,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_21
tff(fact_9804_ATP_Olambda__22,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_yx,Uu))
    <=> ? [I4: int,N5: 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,N5)) )
          & ( N5 != zero_zero_nat ) ) ) ).

% ATP.lambda_22
tff(fact_9805_ATP_Olambda__23,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_yy,Uu))
    <=> ? [I4: int,J3: int] :
          ( ( Uu = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I4)),aa_int_real(ring_1_of_int_real,J3)) )
          & ( J3 != zero_zero_int ) ) ) ).

% ATP.lambda_23
tff(fact_9806_ATP_Olambda__24,axiom,
    ! [Uu: nat] : aa_nat_option_num(aTP_Lamm_yu,Uu) = some_num(one) ).

% ATP.lambda_24
tff(fact_9807_ATP_Olambda__25,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_zh(Uu),Uua) = case_num_option_num(some_num(one),aTP_Lamm_zf(Uua),aTP_Lamm_zg(Uua),Uu) ).

% ATP.lambda_25
tff(fact_9808_ATP_Olambda__26,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_qs(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),zero_zero_real),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_26
tff(fact_9809_ATP_Olambda__27,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_ye,Uu),Uua) = aa_int_int(if_int(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Uu))),aa_int_int(plus_plus_int(aa_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_27
tff(fact_9810_ATP_Olambda__28,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_nj,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_28
tff(fact_9811_ATP_Olambda__29,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_nh,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,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,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Uu))),one_one_real)) ).

% ATP.lambda_29
tff(fact_9812_ATP_Olambda__30,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_nf,Uu),Uua) = if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uu)),plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uu)),one_one_rat)) ).

% ATP.lambda_30
tff(fact_9813_ATP_Olambda__31,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ni,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_31
tff(fact_9814_ATP_Olambda__32,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ng,Uu),Uua) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),aa_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(plus_plus_int(aa_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_32
tff(fact_9815_ATP_Olambda__33,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_abt,Uu),Uua) = aa_num_num(aa_num_fun_num_num(if_num(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger)),aa_num_num(aa_num_fun_num_num(plus_plus_num,code_num_of_integer(Uu)),code_num_of_integer(Uu))),aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,code_num_of_integer(Uu)),code_num_of_integer(Uu))),one)) ).

% ATP.lambda_33
tff(fact_9816_ATP_Olambda__34,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_aby,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,code_nat_of_integer(Uu)),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,code_nat_of_integer(Uu)),code_nat_of_integer(Uu))),one_one_nat)) ).

% ATP.lambda_34
tff(fact_9817_ATP_Olambda__35,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_xu,Uu),Uua) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),zero_zero_int),product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,Uu)),Uua),aa_int_int(abs_abs_int,Uu))) ).

% ATP.lambda_35
tff(fact_9818_ATP_Olambda__36,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_zi,Uu),Uua) = case_nat_option_num(none_num,aTP_Lamm_zh(Uua),Uu) ).

% ATP.lambda_36
tff(fact_9819_ATP_Olambda__37,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_zf(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ys,bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_37
tff(fact_9820_ATP_Olambda__38,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_yt(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ys,bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_38
tff(fact_9821_ATP_Olambda__39,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_nl,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_39
tff(fact_9822_ATP_Olambda__40,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rb,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_40
tff(fact_9823_ATP_Olambda__41,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_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_41
tff(fact_9824_ATP_Olambda__42,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_xf,Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
        & ( aa_real_real(sin_real,Uua) = Uu ) ) ) ).

% ATP.lambda_42
tff(fact_9825_ATP_Olambda__43,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_xb,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),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),numeral_numeral_real(aa_num_num(bit0,one)))))
        & ( aa_real_real(tan_real,Uua) = Uu ) ) ) ).

% ATP.lambda_43
tff(fact_9826_ATP_Olambda__44,axiom,
    ! [Uu: complex,Uua: real] :
      ( pp(aa_real_bool(aa_com8943241093633526360l_bool(aTP_Lamm_xi,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_44
tff(fact_9827_ATP_Olambda__45,axiom,
    ! [Uu: real,Uua: int] :
      ( pp(aa_int_bool(aa_real_fun_int_bool(aTP_Lamm_xs,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(plus_plus_int(Uua),one_one_int)))) ) ) ).

% ATP.lambda_45
tff(fact_9828_ATP_Olambda__46,axiom,
    ! [Uu: rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_xt(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(plus_plus_int(Uua),one_one_int)))) ) ) ).

% ATP.lambda_46
tff(fact_9829_ATP_Olambda__47,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ns,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_47
tff(fact_9830_ATP_Olambda__48,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aag,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_48
tff(fact_9831_ATP_Olambda__49,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aai(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_49
tff(fact_9832_ATP_Olambda__50,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_rr(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_50
tff(fact_9833_ATP_Olambda__51,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rs,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),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_51
tff(fact_9834_ATP_Olambda__52,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_rq(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Uu),Uua)),minus_minus_rat(divide_divide_rat(Uu,numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_52
tff(fact_9835_ATP_Olambda__53,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wz,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% ATP.lambda_53
tff(fact_9836_ATP_Olambda__54,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rk,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_54
tff(fact_9837_ATP_Olambda__55,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_abf,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu)),aa_real_real(exp_real,Uua)) ).

% ATP.lambda_55
tff(fact_9838_ATP_Olambda__56,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abp,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_56
tff(fact_9839_ATP_Olambda__57,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ws,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uu) ).

% ATP.lambda_57
tff(fact_9840_ATP_Olambda__58,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wr,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uua) ).

% ATP.lambda_58
tff(fact_9841_ATP_Olambda__59,axiom,
    ! [Uu: nat,Uua: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_cg,Uu),Uua))
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_59
tff(fact_9842_ATP_Olambda__60,axiom,
    ! [Uu: nat,Uua: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aTP_Lamm_cf,Uu),Uua))
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_60
tff(fact_9843_ATP_Olambda__61,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aar,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)))),Uua) ).

% ATP.lambda_61
tff(fact_9844_ATP_Olambda__62,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_abi,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),Uua))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Uua)) ).

% ATP.lambda_62
tff(fact_9845_ATP_Olambda__63,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_abg,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),Uua))),Uua) ).

% ATP.lambda_63
tff(fact_9846_ATP_Olambda__64,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_nk,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_64
tff(fact_9847_ATP_Olambda__65,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_xd,Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),pi))
        & ( aa_real_real(cos_real,Uua) = Uu ) ) ) ).

% ATP.lambda_65
tff(fact_9848_ATP_Olambda__66,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_vl(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_66
tff(fact_9849_ATP_Olambda__67,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_vm(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_67
tff(fact_9850_ATP_Olambda__68,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_vn(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_68
tff(fact_9851_ATP_Olambda__69,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_vo(Uu),Uua) = aa_int_int(times_times_int(aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_69
tff(fact_9852_ATP_Olambda__70,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_my(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_70
tff(fact_9853_ATP_Olambda__71,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mv(Uu),Uua) = plus_plus_rat(aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_71
tff(fact_9854_ATP_Olambda__72,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_mx(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_72
tff(fact_9855_ATP_Olambda__73,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_mw(Uu),Uua) = aa_int_int(plus_plus_int(aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_73
tff(fact_9856_ATP_Olambda__74,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mn(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_74
tff(fact_9857_ATP_Olambda__75,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ml(Uu),Uua) = minus_minus_rat(aa_nat_rat(Uu,aa_nat_nat(suc,Uua)),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_75
tff(fact_9858_ATP_Olambda__76,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_mm(Uu),Uua) = aa_int_int(minus_minus_int(aa_nat_int(Uu,aa_nat_nat(suc,Uua))),aa_nat_int(Uu,Uua)) ).

% ATP.lambda_76
tff(fact_9859_ATP_Olambda__77,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_lo(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_77
tff(fact_9860_ATP_Olambda__78,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lq(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_78
tff(fact_9861_ATP_Olambda__79,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lp(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ).

% ATP.lambda_79
tff(fact_9862_ATP_Olambda__80,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_qd(Uu),Uua) = aa_int_int(times_times_int(aa_nat_int(Uu,Uua)),aa_nat_int(power_power_int(zero_zero_int),Uua)) ).

% ATP.lambda_80
tff(fact_9863_ATP_Olambda__81,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mu(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_81
tff(fact_9864_ATP_Olambda__82,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ms(Uu),Uua) = minus_minus_rat(aa_nat_rat(Uu,Uua),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_82
tff(fact_9865_ATP_Olambda__83,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_mt(Uu),Uua) = aa_int_int(minus_minus_int(aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_83
tff(fact_9866_ATP_Olambda__84,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_mo(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_84
tff(fact_9867_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mr(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_85
tff(fact_9868_ATP_Olambda__86,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mp(Uu),Uua) = minus_minus_rat(aa_nat_rat(Uu,Uua),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_86
tff(fact_9869_ATP_Olambda__87,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_mq(Uu),Uua) = aa_int_int(minus_minus_int(aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_87
tff(fact_9870_ATP_Olambda__88,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_og(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_88
tff(fact_9871_ATP_Olambda__89,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_oe(Uu),Uua) = minus_minus_rat(aa_nat_rat(Uu,Uua),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_89
tff(fact_9872_ATP_Olambda__90,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_of(Uu),Uua) = aa_int_int(minus_minus_int(aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_90
tff(fact_9873_ATP_Olambda__91,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_kx(Uu),Uua))
    <=> ( aa_complex_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_91
tff(fact_9874_ATP_Olambda__92,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_la(Uu),Uua))
    <=> ( aa_complex_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_92
tff(fact_9875_ATP_Olambda__93,axiom,
    ! [Uu: fun_real_complex,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_kt(Uu),Uua))
    <=> ( aa_real_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_93
tff(fact_9876_ATP_Olambda__94,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_kv(Uu),Uua))
    <=> ( aa_real_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_94
tff(fact_9877_ATP_Olambda__95,axiom,
    ! [Uu: fun_real_rat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ky(Uu),Uua))
    <=> ( aa_real_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_95
tff(fact_9878_ATP_Olambda__96,axiom,
    ! [Uu: fun_real_nat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_lb(Uu),Uua))
    <=> ( aa_real_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_96
tff(fact_9879_ATP_Olambda__97,axiom,
    ! [Uu: fun_int_complex,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ku(Uu),Uua))
    <=> ( aa_int_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_97
tff(fact_9880_ATP_Olambda__98,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_kw(Uu),Uua))
    <=> ( aa_int_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_98
tff(fact_9881_ATP_Olambda__99,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_kz(Uu),Uua))
    <=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_99
tff(fact_9882_ATP_Olambda__100,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_lc(Uu),Uua))
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_100
tff(fact_9883_ATP_Olambda__101,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_vr(Uu),Uua))
    <=> ( aa_complex_complex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_101
tff(fact_9884_ATP_Olambda__102,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_vu(Uu),Uua))
    <=> ( aa_complex_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_102
tff(fact_9885_ATP_Olambda__103,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_vx(Uu),Uua))
    <=> ( aa_complex_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_103
tff(fact_9886_ATP_Olambda__104,axiom,
    ! [Uu: fun_real_complex,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_vp(Uu),Uua))
    <=> ( aa_real_complex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_104
tff(fact_9887_ATP_Olambda__105,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_vs(Uu),Uua))
    <=> ( aa_real_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_105
tff(fact_9888_ATP_Olambda__106,axiom,
    ! [Uu: fun_real_rat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_vv(Uu),Uua))
    <=> ( aa_real_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_106
tff(fact_9889_ATP_Olambda__107,axiom,
    ! [Uu: fun_real_nat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_vy(Uu),Uua))
    <=> ( aa_real_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_107
tff(fact_9890_ATP_Olambda__108,axiom,
    ! [Uu: fun_int_complex,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_vq(Uu),Uua))
    <=> ( aa_int_complex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_108
tff(fact_9891_ATP_Olambda__109,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_vt(Uu),Uua))
    <=> ( aa_int_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_109
tff(fact_9892_ATP_Olambda__110,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_vw(Uu),Uua))
    <=> ( aa_int_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_110
tff(fact_9893_ATP_Olambda__111,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aax(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_aai(Uu),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_111
tff(fact_9894_ATP_Olambda__112,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aaw(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_aai(Uu),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_112
tff(fact_9895_ATP_Olambda__113,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yp,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uu) ).

% ATP.lambda_113
tff(fact_9896_ATP_Olambda__114,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_rt(Uu),Uua) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri5044797733671781792omplex,Uua))),aa_nat_complex(power_power_complex(Uu),Uua)) ).

% ATP.lambda_114
tff(fact_9897_ATP_Olambda__115,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ru,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_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_115
tff(fact_9898_ATP_Olambda__116,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_pk(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_116
tff(fact_9899_ATP_Olambda__117,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pj,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_117
tff(fact_9900_ATP_Olambda__118,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aTP_Lamm_yr(Uu),Uua) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,Uu)),aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,Uu))),aa_num_int(numeral_numeral_int,Uua))) ).

% ATP.lambda_118
tff(fact_9901_ATP_Olambda__119,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_zn,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_119
tff(fact_9902_ATP_Olambda__120,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rj,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_120
tff(fact_9903_ATP_Olambda__121,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ra,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_121
tff(fact_9904_ATP_Olambda__122,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_rw(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),Uu) ).

% ATP.lambda_122
tff(fact_9905_ATP_Olambda__123,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_rx,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),Uu) ).

% ATP.lambda_123
tff(fact_9906_ATP_Olambda__124,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_rv(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu) ).

% ATP.lambda_124
tff(fact_9907_ATP_Olambda__125,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_gk,Uu),Uua) = produc1086072967326762835nteger(Uu,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),one_one_Code_integer)) ).

% ATP.lambda_125
tff(fact_9908_ATP_Olambda__126,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_gj,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_126
tff(fact_9909_ATP_Olambda__127,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gi,Uu),Uua) = product_Pair_int_int(Uu,aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)) ).

% ATP.lambda_127
tff(fact_9910_ATP_Olambda__128,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_hm,Uu),Uua) = produc1086072967326762835nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_128
tff(fact_9911_ATP_Olambda__129,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_hl,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_129
tff(fact_9912_ATP_Olambda__130,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_hk,Uu),Uua) = product_Pair_int_int(Uu,aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_130
tff(fact_9913_ATP_Olambda__131,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_sd(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_131
tff(fact_9914_ATP_Olambda__132,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sc,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)),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% ATP.lambda_132
tff(fact_9915_ATP_Olambda__133,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_sb(Uu),Uua) = plus_plus_rat(Uu,divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua),numeral_numeral_rat(aa_num_num(bit0,one)))) ).

% ATP.lambda_133
tff(fact_9916_ATP_Olambda__134,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xa,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)) ).

% ATP.lambda_134
tff(fact_9917_ATP_Olambda__135,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aas,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_135
tff(fact_9918_ATP_Olambda__136,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_abx,Uu),Uua) = aa_int_int(plus_plus_int(Uu),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,Uua),zero_zero_int)))) ).

% ATP.lambda_136
tff(fact_9919_ATP_Olambda__137,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aaq,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_137
tff(fact_9920_ATP_Olambda__138,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aan,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_138
tff(fact_9921_ATP_Olambda__139,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_zk(Uu),Uua) = groups6591440286371151544t_real(Uu,aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_139
tff(fact_9922_ATP_Olambda__140,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_um,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),Uu)) ) ).

% ATP.lambda_140
tff(fact_9923_ATP_Olambda__141,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( pp(aa_set_complex_bool(aTP_Lamm_bj(Uu),Uua))
    <=> pp(aa_set_complex_bool(ord_le211207098394363844omplex(Uua),Uu)) ) ).

% ATP.lambda_141
tff(fact_9924_ATP_Olambda__142,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_bk(Uu),Uua))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(Uua),Uu)) ) ).

% ATP.lambda_142
tff(fact_9925_ATP_Olambda__143,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( pp(aa_set_int_bool(aTP_Lamm_bi(Uu),Uua))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(Uua),Uu)) ) ).

% ATP.lambda_143
tff(fact_9926_ATP_Olambda__144,axiom,
    ! [Uu: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_un,Uu),Uua))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Uua),Uu)) ) ).

% ATP.lambda_144
tff(fact_9927_ATP_Olambda__145,axiom,
    ! [Uu: num,Uua: num] :
      ( pp(aa_num_bool(aTP_Lamm_uo(Uu),Uua))
    <=> pp(aa_num_bool(ord_less_eq_num(Uua),Uu)) ) ).

% ATP.lambda_145
tff(fact_9928_ATP_Olambda__146,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bg,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uu)) ) ).

% ATP.lambda_146
tff(fact_9929_ATP_Olambda__147,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_up,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uua),Uu)) ) ).

% ATP.lambda_147
tff(fact_9930_ATP_Olambda__148,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zm,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_148
tff(fact_9931_ATP_Olambda__149,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_nx,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uua),Uu)) ) ).

% ATP.lambda_149
tff(fact_9932_ATP_Olambda__150,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_nt(Uu),Uua))
    <=> pp(aa_set_nat_bool(ord_less_set_nat(Uua),Uu)) ) ).

% ATP.lambda_150
tff(fact_9933_ATP_Olambda__151,axiom,
    ! [Uu: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_nu,Uu),Uua))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Uua),Uu)) ) ).

% ATP.lambda_151
tff(fact_9934_ATP_Olambda__152,axiom,
    ! [Uu: num,Uua: num] :
      ( pp(aa_num_bool(aTP_Lamm_nv(Uu),Uua))
    <=> pp(aa_num_bool(ord_less_num(Uua),Uu)) ) ).

% ATP.lambda_152
tff(fact_9935_ATP_Olambda__153,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bh,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu)) ) ).

% ATP.lambda_153
tff(fact_9936_ATP_Olambda__154,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_nw,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uu)) ) ).

% ATP.lambda_154
tff(fact_9937_ATP_Olambda__155,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ag,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),Uu) ).

% ATP.lambda_155
tff(fact_9938_ATP_Olambda__156,axiom,
    ! [Uu: rat,Uua: rat] : aa_rat_rat(aTP_Lamm_ah(Uu),Uua) = aa_rat_rat(times_times_rat(Uua),Uu) ).

% ATP.lambda_156
tff(fact_9939_ATP_Olambda__157,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ai,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_157
tff(fact_9940_ATP_Olambda__158,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aTP_Lamm_aj(Uu),Uua) = aa_int_int(times_times_int(Uua),Uu) ).

% ATP.lambda_158
tff(fact_9941_ATP_Olambda__159,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zj,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_159
tff(fact_9942_ATP_Olambda__160,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_zp,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_160
tff(fact_9943_ATP_Olambda__161,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_161
tff(fact_9944_ATP_Olambda__162,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aTP_Lamm_zl(Uu),Uua) = aa_int_int(plus_plus_int(Uua),Uu) ).

% ATP.lambda_162
tff(fact_9945_ATP_Olambda__163,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_zr,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu) ).

% ATP.lambda_163
tff(fact_9946_ATP_Olambda__164,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(aTP_Lamm_gf,Uu),Uua))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,Uua),Uu)) ) ).

% ATP.lambda_164
tff(fact_9947_ATP_Olambda__165,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_gd,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,Uua),Uu)) ) ).

% ATP.lambda_165
tff(fact_9948_ATP_Olambda__166,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_gg,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uua),Uu)) ) ).

% ATP.lambda_166
tff(fact_9949_ATP_Olambda__167,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ge,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uua),Uu)) ) ).

% ATP.lambda_167
tff(fact_9950_ATP_Olambda__168,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aam,Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_168
tff(fact_9951_ATP_Olambda__169,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aTP_Lamm_a(Uu),Uua))
    <=> pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uua),Uu)) ) ).

% ATP.lambda_169
tff(fact_9952_ATP_Olambda__170,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_aa(Uu),Uua))
    <=> pp(aa_set_complex_bool(member_complex(Uua),Uu)) ) ).

% ATP.lambda_170
tff(fact_9953_ATP_Olambda__171,axiom,
    ! [Uu: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ab(Uu),Uua))
    <=> member_real(Uua,Uu) ) ).

% ATP.lambda_171
tff(fact_9954_ATP_Olambda__172,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_ac(Uu),Uua))
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_172
tff(fact_9955_ATP_Olambda__173,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_ad(Uu),Uua))
    <=> member_set_nat(Uua,Uu) ) ).

% ATP.lambda_173
tff(fact_9956_ATP_Olambda__174,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_ae(Uu),Uua))
    <=> pp(aa_set_nat_bool(member_nat(Uua),Uu)) ) ).

% ATP.lambda_174
tff(fact_9957_ATP_Olambda__175,axiom,
    ! [Uu: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_af(Uu),Uua))
    <=> pp(aa_set_int_bool(member_int(Uua),Uu)) ) ).

% ATP.lambda_175
tff(fact_9958_ATP_Olambda__176,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_pd(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_176
tff(fact_9959_ATP_Olambda__177,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_pc(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_177
tff(fact_9960_ATP_Olambda__178,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_zo(Uu),Uua) = aa_real_real(Uu,aa_real_real(uminus_uminus_real,Uua)) ).

% ATP.lambda_178
tff(fact_9961_ATP_Olambda__179,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mk(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_179
tff(fact_9962_ATP_Olambda__180,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mh(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_180
tff(fact_9963_ATP_Olambda__181,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_mj(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_181
tff(fact_9964_ATP_Olambda__182,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_mi(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_182
tff(fact_9965_ATP_Olambda__183,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_zg(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_183
tff(fact_9966_ATP_Olambda__184,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_yv(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu))) ).

% ATP.lambda_184
tff(fact_9967_ATP_Olambda__185,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_yo,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_yn(Uu),Uua)) ).

% ATP.lambda_185
tff(fact_9968_ATP_Olambda__186,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_ym,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_yl(Uu),Uua)) ).

% ATP.lambda_186
tff(fact_9969_ATP_Olambda__187,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_yk,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_yj(Uu),Uua)) ).

% ATP.lambda_187
tff(fact_9970_ATP_Olambda__188,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_yi,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_yh(Uu),Uua)) ).

% ATP.lambda_188
tff(fact_9971_ATP_Olambda__189,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_yg,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_yf(Uu),Uua)) ).

% ATP.lambda_189
tff(fact_9972_ATP_Olambda__190,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_zy(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_zx(Uu),Uua)) ).

% ATP.lambda_190
tff(fact_9973_ATP_Olambda__191,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_xg(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,numeral_numeral_real(aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu))) ).

% ATP.lambda_191
tff(fact_9974_ATP_Olambda__192,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_abc(Uu),Uua))
    <=> ( aa_real_real(Uu,Uua) != zero_zero_real ) ) ).

% ATP.lambda_192
tff(fact_9975_ATP_Olambda__193,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_gh,Uu),Uua))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ) ).

% ATP.lambda_193
tff(fact_9976_ATP_Olambda__194,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_rz(Uu),Uua) = times_times_complex(aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_194
tff(fact_9977_ATP_Olambda__195,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_sa(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_195
tff(fact_9978_ATP_Olambda__196,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_ry(Uu),Uua) = times_times_rat(minus_minus_rat(Uu,aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_196
tff(fact_9979_ATP_Olambda__197,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_sh(Uu),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_197
tff(fact_9980_ATP_Olambda__198,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_sg(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_198
tff(fact_9981_ATP_Olambda__199,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_se(Uu),Uua) = times_times_rat(plus_plus_rat(Uu,aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_199
tff(fact_9982_ATP_Olambda__200,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_si(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_200
tff(fact_9983_ATP_Olambda__201,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_sf(Uu),Uua) = times_times_int(aa_int_int(plus_plus_int(Uu),aa_nat_int(semiri1314217659103216013at_int,Uua))) ).

% ATP.lambda_201
tff(fact_9984_ATP_Olambda__202,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aap,Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_202
tff(fact_9985_ATP_Olambda__203,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abr,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_203
tff(fact_9986_ATP_Olambda__204,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abs,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_204
tff(fact_9987_ATP_Olambda__205,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xr,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_205
tff(fact_9988_ATP_Olambda__206,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xq,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_206
tff(fact_9989_ATP_Olambda__207,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] : aa_complex_real(aTP_Lamm_pf(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(Uu,Uua)) ).

% ATP.lambda_207
tff(fact_9990_ATP_Olambda__208,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_pe(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_208
tff(fact_9991_ATP_Olambda__209,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_pg(Uu),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_209
tff(fact_9992_ATP_Olambda__210,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aak(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_210
tff(fact_9993_ATP_Olambda__211,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_uq(Uu),Uua) = times_times_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_211
tff(fact_9994_ATP_Olambda__212,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_ur(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_212
tff(fact_9995_ATP_Olambda__213,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_us(Uu),Uua) = times_times_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_213
tff(fact_9996_ATP_Olambda__214,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_ut(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_214
tff(fact_9997_ATP_Olambda__215,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_uu(Uu),Uua) = times_times_int(aa_nat_int(Uu,Uua)) ).

% ATP.lambda_215
tff(fact_9998_ATP_Olambda__216,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_abk(Uu),Uua) = aa_real_real(arcosh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_216
tff(fact_9999_ATP_Olambda__217,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hj(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_217
tff(fact_10000_ATP_Olambda__218,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_hi(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_218
tff(fact_10001_ATP_Olambda__219,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] : aa_complex_complex(aTP_Lamm_wd(Uu),Uua) = aa_complex_complex(exp_complex,aa_complex_complex(Uu,Uua)) ).

% ATP.lambda_219
tff(fact_10002_ATP_Olambda__220,axiom,
    ! [Uu: fun_complex_real,Uua: complex] : aa_complex_real(aTP_Lamm_wa(Uu),Uua) = aa_real_real(exp_real,aa_complex_real(Uu,Uua)) ).

% ATP.lambda_220
tff(fact_10003_ATP_Olambda__221,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_wb(Uu),Uua) = aa_complex_complex(exp_complex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_221
tff(fact_10004_ATP_Olambda__222,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_we(Uu),Uua) = aa_real_real(exp_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_222
tff(fact_10005_ATP_Olambda__223,axiom,
    ! [Uu: fun_int_complex,Uua: int] : aa_int_complex(aTP_Lamm_wc(Uu),Uua) = aa_complex_complex(exp_complex,aa_int_complex(Uu,Uua)) ).

% ATP.lambda_223
tff(fact_10006_ATP_Olambda__224,axiom,
    ! [Uu: fun_int_real,Uua: int] : aa_int_real(aTP_Lamm_vz(Uu),Uua) = aa_real_real(exp_real,aa_int_real(Uu,Uua)) ).

% ATP.lambda_224
tff(fact_10007_ATP_Olambda__225,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aav(Uu),Uua) = aa_real_real(cos_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_225
tff(fact_10008_ATP_Olambda__226,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_zt(Uu),Uua) = suminf_real(aa_real_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_226
tff(fact_10009_ATP_Olambda__227,axiom,
    ! [Uu: list_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aTP_Lamm_as(Uu),Uua))
    <=> ? [I4: nat] :
          ( ( Uua = nth_VEBT_VEBT(Uu,I4) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Uu))) ) ) ).

% ATP.lambda_227
tff(fact_10010_ATP_Olambda__228,axiom,
    ! [Uu: list_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ap(Uu),Uua))
    <=> ? [I4: nat] :
          ( ( Uua = aa_nat_real(nth_real(Uu),I4) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_real(Uu))) ) ) ).

% ATP.lambda_228
tff(fact_10011_ATP_Olambda__229,axiom,
    ! [Uu: list_list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_aq(Uu),Uua))
    <=> ? [I4: nat] :
          ( ( Uua = nth_list_nat(Uu,I4) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s3023201423986296836st_nat(Uu))) ) ) ).

% ATP.lambda_229
tff(fact_10012_ATP_Olambda__230,axiom,
    ! [Uu: list_bool,Uua: bool] :
      ( pp(aa_bool_bool(aTP_Lamm_at(Uu),Uua))
    <=> ? [I4: nat] :
          ( ( pp(Uua)
          <=> pp(aa_nat_bool(nth_bool(Uu),I4)) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_bool(Uu))) ) ) ).

% ATP.lambda_230
tff(fact_10013_ATP_Olambda__231,axiom,
    ! [Uu: list_set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_ar(Uu),Uua))
    <=> ? [I4: nat] :
          ( ( Uua = aa_nat_set_nat(nth_set_nat(Uu),I4) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s3254054031482475050et_nat(Uu))) ) ) ).

% ATP.lambda_231
tff(fact_10014_ATP_Olambda__232,axiom,
    ! [Uu: list_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_au(Uu),Uua))
    <=> ? [I4: nat] :
          ( ( Uua = aa_nat_nat(nth_nat(Uu),I4) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Uu))) ) ) ).

% ATP.lambda_232
tff(fact_10015_ATP_Olambda__233,axiom,
    ! [Uu: list_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_av(Uu),Uua))
    <=> ? [I4: nat] :
          ( ( Uua = aa_nat_int(nth_int(Uu),I4) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Uu))) ) ) ).

% ATP.lambda_233
tff(fact_10016_ATP_Olambda__234,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_qt(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_234
tff(fact_10017_ATP_Olambda__235,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_pb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)),aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_235
tff(fact_10018_ATP_Olambda__236,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_gn(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Uu)),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,aa_num_Code_integer(numera6620942414471956472nteger,Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_236
tff(fact_10019_ATP_Olambda__237,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_gl(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_237
tff(fact_10020_ATP_Olambda__238,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gm(Uu),Uua),Uub) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),Uub),product_Pair_int_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int),aa_int_int(minus_minus_int(Uub),aa_num_int(numeral_numeral_int,Uu))),product_Pair_int_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_238
tff(fact_10021_ATP_Olambda__239,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_pr(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_239
tff(fact_10022_ATP_Olambda__240,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ps(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_240
tff(fact_10023_ATP_Olambda__241,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pt(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_241
tff(fact_10024_ATP_Olambda__242,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_pu(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_242
tff(fact_10025_ATP_Olambda__243,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun8982142231764154876omplex(aTP_Lamm_gp(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uu),Uub),aa_VEBT_VEBT_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_243
tff(fact_10026_ATP_Olambda__244,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_gt(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uu),Uub)),aa_VEBT_VEBT_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_244
tff(fact_10027_ATP_Olambda__245,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_gx(Uu),Uua),Uub) = if_rat(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uu),Uub),aa_VEBT_VEBT_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_245
tff(fact_10028_ATP_Olambda__246,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_gv(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_246
tff(fact_10029_ATP_Olambda__247,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_go(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_247
tff(fact_10030_ATP_Olambda__248,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gs(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_248
tff(fact_10031_ATP_Olambda__249,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_gw(Uu),Uua),Uub) = if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_249
tff(fact_10032_ATP_Olambda__250,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_gq(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_250
tff(fact_10033_ATP_Olambda__251,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_gr(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_251
tff(fact_10034_ATP_Olambda__252,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_gu(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_252
tff(fact_10035_ATP_Olambda__253,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun8982142231764154876omplex(aTP_Lamm_sk(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uu),Uub),aa_VEBT_VEBT_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_253
tff(fact_10036_ATP_Olambda__254,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_sp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uu),Uub)),aa_VEBT_VEBT_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_254
tff(fact_10037_ATP_Olambda__255,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_sn(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_255
tff(fact_10038_ATP_Olambda__256,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_ss(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_256
tff(fact_10039_ATP_Olambda__257,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_sj(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_257
tff(fact_10040_ATP_Olambda__258,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_so(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_258
tff(fact_10041_ATP_Olambda__259,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sl(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_259
tff(fact_10042_ATP_Olambda__260,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_sq(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_260
tff(fact_10043_ATP_Olambda__261,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_sm(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_261
tff(fact_10044_ATP_Olambda__262,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_sr(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_262
tff(fact_10045_ATP_Olambda__263,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun8982142231764154876omplex(aTP_Lamm_gz(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uub),Uu),aa_VEBT_VEBT_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_263
tff(fact_10046_ATP_Olambda__264,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_hd(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uub),Uu)),aa_VEBT_VEBT_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_264
tff(fact_10047_ATP_Olambda__265,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_hh(Uu),Uua),Uub) = if_rat(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uub),Uu),aa_VEBT_VEBT_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_265
tff(fact_10048_ATP_Olambda__266,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_hf(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_266
tff(fact_10049_ATP_Olambda__267,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_gy(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_267
tff(fact_10050_ATP_Olambda__268,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_hc(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_268
tff(fact_10051_ATP_Olambda__269,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_hg(Uu),Uua),Uub) = if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_269
tff(fact_10052_ATP_Olambda__270,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ha(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_270
tff(fact_10053_ATP_Olambda__271,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_hb(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_271
tff(fact_10054_ATP_Olambda__272,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_he(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_272
tff(fact_10055_ATP_Olambda__273,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun8982142231764154876omplex(aTP_Lamm_su(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uub),Uu),aa_VEBT_VEBT_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_273
tff(fact_10056_ATP_Olambda__274,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_sz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uub),Uu)),aa_VEBT_VEBT_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_274
tff(fact_10057_ATP_Olambda__275,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_sx(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_275
tff(fact_10058_ATP_Olambda__276,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_tc(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_276
tff(fact_10059_ATP_Olambda__277,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_st(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_277
tff(fact_10060_ATP_Olambda__278,axiom,
    ! [Uu: 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(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu)),aa_real_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_278
tff(fact_10061_ATP_Olambda__279,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sv(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_279
tff(fact_10062_ATP_Olambda__280,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ta(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_280
tff(fact_10063_ATP_Olambda__281,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_sw(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_281
tff(fact_10064_ATP_Olambda__282,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_tb(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_282
tff(fact_10065_ATP_Olambda__283,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_xn(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_283
tff(fact_10066_ATP_Olambda__284,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_yd(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_284
tff(fact_10067_ATP_Olambda__285,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_xm(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_285
tff(fact_10068_ATP_Olambda__286,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_pn(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(Uu,Uub),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_286
tff(fact_10069_ATP_Olambda__287,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_po(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_287
tff(fact_10070_ATP_Olambda__288,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pp(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_288
tff(fact_10071_ATP_Olambda__289,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_pq(Uu),Uua),Uub) = aa_int_int(if_int(aa_nat_bool(Uu,Uub),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_289
tff(fact_10072_ATP_Olambda__290,axiom,
    ! [Uu: fun_VE5569070535936029429omplex,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun6792494735270171281omplex(aTP_Lamm_kf(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_VEBT_VEBT_bool(Uua,Uub),aa_VEBT_VEBT_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_290
tff(fact_10073_ATP_Olambda__291,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun2072160908118572431T_real(aTP_Lamm_km(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(Uua,Uub)),aa_VEBT_VEBT_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_291
tff(fact_10074_ATP_Olambda__292,axiom,
    ! [Uu: fun_VEBT_VEBT_rat,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] : aa_VEBT_VEBT_rat(aa_fun3193099451372820155BT_rat(aTP_Lamm_kr(Uu),Uua),Uub) = if_rat(aa_VEBT_VEBT_bool(Uua,Uub),aa_VEBT_VEBT_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_292
tff(fact_10075_ATP_Olambda__293,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_293
tff(fact_10076_ATP_Olambda__294,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_bool,Uub: real] : aa_real_complex(aa_fun6813637976755515871omplex(aTP_Lamm_kh(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(Uua,Uub),aa_real_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_294
tff(fact_10077_ATP_Olambda__295,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_295
tff(fact_10078_ATP_Olambda__296,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_bool,Uub: real] : aa_real_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_ks(Uu),Uua),Uub) = if_rat(aa_real_bool(Uua,Uub),aa_real_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_296
tff(fact_10079_ATP_Olambda__297,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_bool,Uub: nat] : aa_nat_complex(aa_fun8588439219086927967omplex(aTP_Lamm_kj(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(Uua,Uub),aa_nat_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_297
tff(fact_10080_ATP_Olambda__298,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_bool,Uub: int] : aa_int_complex(aa_fun3606824691168040799omplex(aTP_Lamm_kl(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(Uua,Uub),aa_int_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_298
tff(fact_10081_ATP_Olambda__299,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_299
tff(fact_10082_ATP_Olambda__300,axiom,
    ! [Uu: fun_VE5569070535936029429omplex,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun6792494735270171281omplex(aTP_Lamm_uv(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_VEBT_VEBT_bool(Uua,Uub),aa_VEBT_VEBT_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_300
tff(fact_10083_ATP_Olambda__301,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun2072160908118572431T_real(aTP_Lamm_va(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(Uua,Uub)),aa_VEBT_VEBT_real(Uu,Uub)),one_one_real) ).

% ATP.lambda_301
tff(fact_10084_ATP_Olambda__302,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_bool,Uub: complex] : aa_complex_complex(aa_fun1034237174562290143omplex(aTP_Lamm_uz(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_complex_bool(Uua,Uub),aa_complex_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_302
tff(fact_10085_ATP_Olambda__303,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_bool,Uub: complex] : aa_complex_real(aa_fun2902006202505214429x_real(aTP_Lamm_ve(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_303
tff(fact_10086_ATP_Olambda__304,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_bool,Uub: real] : aa_real_complex(aa_fun6813637976755515871omplex(aTP_Lamm_uw(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(Uua,Uub),aa_real_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_304
tff(fact_10087_ATP_Olambda__305,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_bool,Uub: real] : aa_real_real(aa_fun6348708079608492509l_real(aTP_Lamm_vb(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_305
tff(fact_10088_ATP_Olambda__306,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_bool,Uub: nat] : aa_nat_complex(aa_fun8588439219086927967omplex(aTP_Lamm_ux(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(Uua,Uub),aa_nat_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_306
tff(fact_10089_ATP_Olambda__307,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_bool,Uub: nat] : aa_nat_real(aa_fun1750223370215434333t_real(aTP_Lamm_vc(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_307
tff(fact_10090_ATP_Olambda__308,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_bool,Uub: int] : aa_int_complex(aa_fun3606824691168040799omplex(aTP_Lamm_uy(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(Uua,Uub),aa_int_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_308
tff(fact_10091_ATP_Olambda__309,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_bool,Uub: int] : aa_int_real(aa_fun6974987029372589917t_real(aTP_Lamm_vd(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_309
tff(fact_10092_ATP_Olambda__310,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_aaz(Uu),Uua),Uub))
    <=> has_fi5821293074295781190e_real(Uu,aa_real_real(Uua,Uub),topolo2177554685111907308n_real(Uub,top_top_set_real)) ) ).

% ATP.lambda_310
tff(fact_10093_ATP_Olambda__311,axiom,
    ! [Uu: fun_VE2006399705683203104omplex,Uua: complex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_com2862003112617944841omplex(aTP_Lamm_jf(Uu),Uua),Uub) = aa_complex_complex(aa_VEB5608662635949521omplex(Uu,Uub),Uua) ).

% ATP.lambda_311
tff(fact_10094_ATP_Olambda__312,axiom,
    ! [Uu: fun_VE597728521808591204at_nat,Uua: nat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_nat7776146579971162249BT_nat(aTP_Lamm_jw(Uu),Uua),Uub) = aa_nat_nat(aa_VEB1122092716774696213at_nat(Uu,Uub),Uua) ).

% ATP.lambda_312
tff(fact_10095_ATP_Olambda__313,axiom,
    ! [Uu: fun_VE7331017413731640256at_int,Uua: nat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_nat3598295560461965541BT_int(aTP_Lamm_tx(Uu),Uua),Uub) = aa_nat_int(aa_VEB6167613734120275313at_int(Uu,Uub),Uua) ).

% ATP.lambda_313
tff(fact_10096_ATP_Olambda__314,axiom,
    ! [Uu: fun_VE294566951569886236nt_int,Uua: int,Uub: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_int1343382402272391561BT_int(aTP_Lamm_io(Uu),Uua),Uub) = aa_int_int(aa_VEB7166967446021329869nt_int(Uu,Uub),Uua) ).

% ATP.lambda_314
tff(fact_10097_ATP_Olambda__315,axiom,
    ! [Uu: fun_co2286642472385812463l_bool,Uua: real,Uub: complex] :
      ( pp(aa_complex_bool(aa_rea5887071448072802392x_bool(aTP_Lamm_bs(Uu),Uua),Uub))
    <=> pp(aa_real_bool(aa_com8943241093633526360l_bool(Uu,Uub),Uua)) ) ).

% ATP.lambda_315
tff(fact_10098_ATP_Olambda__316,axiom,
    ! [Uu: fun_co2841405903858895691t_bool,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_ca(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_com5335841084339320244t_bool(Uu,Uub),Uua)) ) ).

% ATP.lambda_316
tff(fact_10099_ATP_Olambda__317,axiom,
    ! [Uu: fun_co636942507509095880at_nat,Uua: nat,Uub: complex] : aa_complex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_tt(Uu),Uua),Uub) = aa_nat_nat(aa_com6383720667567721865at_nat(Uu,Uub),Uua) ).

% ATP.lambda_317
tff(fact_10100_ATP_Olambda__318,axiom,
    ! [Uu: fun_co7370231399432144932at_int,Uua: nat,Uub: complex] : aa_complex_int(aa_nat8436805559129492965ex_int(aTP_Lamm_ud(Uu),Uua),Uub) = aa_nat_int(aa_com2205869648058525157at_int(Uu,Uub),Uua) ).

% ATP.lambda_318
tff(fact_10101_ATP_Olambda__319,axiom,
    ! [Uu: fun_co333780937270390912nt_int,Uua: int,Uub: complex] : aa_complex_int(aa_int6126987627036542017ex_int(aTP_Lamm_ja(Uu),Uua),Uub) = aa_int_int(aa_com3205223359959579713nt_int(Uu,Uub),Uua) ).

% ATP.lambda_319
tff(fact_10102_ATP_Olambda__320,axiom,
    ! [Uu: fun_re7208826350674946562omplex,Uua: complex,Uub: real] : aa_real_complex(aa_com4766338864777664963omplex(aTP_Lamm_jj(Uu),Uua),Uub) = aa_complex_complex(aa_rea2618301233319580611omplex(Uu,Uub),Uua) ).

% ATP.lambda_320
tff(fact_10103_ATP_Olambda__321,axiom,
    ! [Uu: fun_re1255401033344146673l_bool,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_bm(Uu),Uua),Uub))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(Uu,Uub),Uua)) ) ).

% ATP.lambda_321
tff(fact_10104_ATP_Olambda__322,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_zs(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_322
tff(fact_10105_ATP_Olambda__323,axiom,
    ! [Uu: fun_re9121486324101951821t_bool,Uua: nat,Uub: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aTP_Lamm_bu(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_real_fun_nat_bool(Uu,Uub),Uua)) ) ).

% ATP.lambda_323
tff(fact_10106_ATP_Olambda__324,axiom,
    ! [Uu: fun_real_fun_nat_nat,Uua: nat,Uub: real] : aa_real_nat(aa_nat_fun_real_nat(aTP_Lamm_ka(Uu),Uua),Uub) = aa_nat_nat(aa_real_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_324
tff(fact_10107_ATP_Olambda__325,axiom,
    ! [Uu: fun_real_fun_nat_int,Uua: nat,Uub: real] : aa_real_int(aa_nat_fun_real_int(aTP_Lamm_ua(Uu),Uua),Uub) = aa_nat_int(aa_real_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_325
tff(fact_10108_ATP_Olambda__326,axiom,
    ! [Uu: fun_re8661203372455305201t_bool,Uua: int,Uub: real] :
      ( pp(aa_real_bool(aa_int_fun_real_bool(aTP_Lamm_cc(Uu),Uua),Uub))
    <=> pp(aa_int_bool(aa_real_fun_int_bool(Uu,Uub),Uua)) ) ).

% ATP.lambda_326
tff(fact_10109_ATP_Olambda__327,axiom,
    ! [Uu: fun_real_fun_int_int,Uua: int,Uub: real] : aa_real_int(aa_int_fun_real_int(aTP_Lamm_is(Uu),Uua),Uub) = aa_int_int(aa_real_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_327
tff(fact_10110_ATP_Olambda__328,axiom,
    ! [Uu: fun_na1591161823130732966omplex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_jn(Uu),Uua),Uub) = aa_complex_complex(aa_nat3829984526574172007omplex(Uu,Uub),Uua) ).

% ATP.lambda_328
tff(fact_10111_ATP_Olambda__329,axiom,
    ! [Uu: fun_na6801867027684862029l_bool,Uua: real,Uub: nat] :
      ( pp(aa_nat_bool(aa_real_fun_nat_bool(aTP_Lamm_bo(Uu),Uua),Uub))
    <=> pp(aa_real_bool(aa_nat_fun_real_bool(Uu,Uub),Uua)) ) ).

% ATP.lambda_329
tff(fact_10112_ATP_Olambda__330,axiom,
    ! [Uu: fun_nat_fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bw(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(Uu,Uub),Uua)) ) ).

% ATP.lambda_330
tff(fact_10113_ATP_Olambda__331,axiom,
    ! [Uu: fun_nat_fun_int_bool,Uua: int,Uub: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(aTP_Lamm_ce(Uu),Uua),Uub))
    <=> pp(aa_int_bool(aa_nat_fun_int_bool(Uu,Uub),Uua)) ) ).

% ATP.lambda_331
tff(fact_10114_ATP_Olambda__332,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_iw(Uu),Uua),Uub) = aa_int_int(aa_nat_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_332
tff(fact_10115_ATP_Olambda__333,axiom,
    ! [Uu: fun_in1307110748760623106omplex,Uua: complex,Uub: int] : aa_int_complex(aa_com9156536935073639875omplex(aTP_Lamm_jr(Uu),Uua),Uub) = aa_complex_complex(aa_int3545933452204062147omplex(Uu,Uub),Uua) ).

% ATP.lambda_333
tff(fact_10116_ATP_Olambda__334,axiom,
    ! [Uu: fun_in3970322703178350321l_bool,Uua: real,Uub: int] :
      ( pp(aa_int_bool(aa_real_fun_int_bool(aTP_Lamm_bq(Uu),Uua),Uub))
    <=> pp(aa_real_bool(aa_int_fun_real_bool(Uu,Uub),Uua)) ) ).

% ATP.lambda_334
tff(fact_10117_ATP_Olambda__335,axiom,
    ! [Uu: fun_int_fun_nat_bool,Uua: nat,Uub: int] :
      ( pp(aa_int_bool(aa_nat_fun_int_bool(aTP_Lamm_by(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(Uu,Uub),Uua)) ) ).

% ATP.lambda_335
tff(fact_10118_ATP_Olambda__336,axiom,
    ! [Uu: fun_int_fun_nat_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_tp(Uu),Uua),Uub) = aa_nat_nat(aa_int_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_336
tff(fact_10119_ATP_Olambda__337,axiom,
    ! [Uu: nat,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_qk(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_337
tff(fact_10120_ATP_Olambda__338,axiom,
    ! [Uu: nat,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ql(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_338
tff(fact_10121_ATP_Olambda__339,axiom,
    ! [Uu: nat,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_qm(Uu),Uua),Uub) = aa_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_339
tff(fact_10122_ATP_Olambda__340,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod2825494507867081354r_bool(aa_Cod4981436349852597092r_bool(aTP_Lamm_xj(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_340
tff(fact_10123_ATP_Olambda__341,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_zc(Uu),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_zb(Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_341
tff(fact_10124_ATP_Olambda__342,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_za(Uu),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_yz(Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_342
tff(fact_10125_ATP_Olambda__343,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_yc(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_yb(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_343
tff(fact_10126_ATP_Olambda__344,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ya(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_xz(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_344
tff(fact_10127_ATP_Olambda__345,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_xy(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_xx(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_345
tff(fact_10128_ATP_Olambda__346,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_xw(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_xv(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_346
tff(fact_10129_ATP_Olambda__347,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zz(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_347
tff(fact_10130_ATP_Olambda__348,axiom,
    ! [Uu: real,Uua: fun_na5723442673449079972x_real,Uub: nat] : aa_nat_real(aa_fun5746040925223427547t_real(aTP_Lamm_rc(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_348
tff(fact_10131_ATP_Olambda__349,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_rd(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_349
tff(fact_10132_ATP_Olambda__350,axiom,
    ! [Uu: real,Uua: fun_nat_fun_rat_real,Uub: nat] : aa_nat_real(aa_fun507281639146497029t_real(aTP_Lamm_re(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_350
tff(fact_10133_ATP_Olambda__351,axiom,
    ! [Uu: real,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_rf(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_351
tff(fact_10134_ATP_Olambda__352,axiom,
    ! [Uu: real,Uua: fun_nat_fun_int_real,Uub: nat] : aa_nat_real(aa_fun241848180832411225t_real(aTP_Lamm_rg(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_352
tff(fact_10135_ATP_Olambda__353,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lr(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_353
tff(fact_10136_ATP_Olambda__354,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_lt(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_354
tff(fact_10137_ATP_Olambda__355,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ls(Uu),Uua),Uub) = divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(zero_zero_rat),Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_355
tff(fact_10138_ATP_Olambda__356,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zw(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_356
tff(fact_10139_ATP_Olambda__357,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_rh(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_357
tff(fact_10140_ATP_Olambda__358,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) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),one_one_nat) )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_358
tff(fact_10141_ATP_Olambda__359,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_qx(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_359
tff(fact_10142_ATP_Olambda__360,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qv(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_360
tff(fact_10143_ATP_Olambda__361,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) = Uu )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_361
tff(fact_10144_ATP_Olambda__362,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_bc(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_362
tff(fact_10145_ATP_Olambda__363,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_bb(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_363
tff(fact_10146_ATP_Olambda__364,axiom,
    ! [Uu: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_bd(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_364
tff(fact_10147_ATP_Olambda__365,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_bf(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Uub)),Uua)) ) ) ).

% ATP.lambda_365
tff(fact_10148_ATP_Olambda__366,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_be(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_366
tff(fact_10149_ATP_Olambda__367,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_ax(Uu),Uua),Uub))
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_367
tff(fact_10150_ATP_Olambda__368,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_aw(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uu))
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_368
tff(fact_10151_ATP_Olambda__369,axiom,
    ! [Uu: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_ay(Uu),Uua),Uub))
    <=> ( ord_less_eq_set_bool(set_bool2(Uub),Uu)
        & ( size_size_list_bool(Uub) = Uua ) ) ) ).

% ATP.lambda_369
tff(fact_10152_ATP_Olambda__370,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_ba(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Uub)),Uu))
        & ( size_size_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_370
tff(fact_10153_ATP_Olambda__371,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_az(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_371
tff(fact_10154_ATP_Olambda__372,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_acb(Uu),Uua),Uub))
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_372
tff(fact_10155_ATP_Olambda__373,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_xk(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_373
tff(fact_10156_ATP_Olambda__374,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_xl(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_374
tff(fact_10157_ATP_Olambda__375,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_qw(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_375
tff(fact_10158_ATP_Olambda__376,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qu(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_376
tff(fact_10159_ATP_Olambda__377,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qz(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_377
tff(fact_10160_ATP_Olambda__378,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_qy(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_378
tff(fact_10161_ATP_Olambda__379,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wt(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_379
tff(fact_10162_ATP_Olambda__380,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ze(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_380
tff(fact_10163_ATP_Olambda__381,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_zd(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_381
tff(fact_10164_ATP_Olambda__382,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_cv(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_382
tff(fact_10165_ATP_Olambda__383,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_cu(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_383
tff(fact_10166_ATP_Olambda__384,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ct(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_384
tff(fact_10167_ATP_Olambda__385,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_abv(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_385
tff(fact_10168_ATP_Olambda__386,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun4462450708475455754T_bool(aTP_Lamm_ke(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uub),Uu))
        & pp(aa_VEBT_VEBT_bool(Uua,Uub)) ) ) ).

% ATP.lambda_386
tff(fact_10169_ATP_Olambda__387,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_387
tff(fact_10170_ATP_Olambda__388,axiom,
    ! [Uu: set_real,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_kg(Uu),Uua),Uub))
    <=> ( member_real(Uub,Uu)
        & pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_388
tff(fact_10171_ATP_Olambda__389,axiom,
    ! [Uu: set_nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_ki(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_389
tff(fact_10172_ATP_Olambda__390,axiom,
    ! [Uu: set_int,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_kk(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_390
tff(fact_10173_ATP_Olambda__391,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun8092769222387074655T_bool(aTP_Lamm_fg(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uub),Uu))
        & ( aa_VEBT_VEBT_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_391
tff(fact_10174_ATP_Olambda__392,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun5605965666826184925T_bool(aTP_Lamm_fq(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uub),Uu))
        & ( aa_VEBT_VEBT_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_392
tff(fact_10175_ATP_Olambda__393,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aTP_Lamm_fo(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_393
tff(fact_10176_ATP_Olambda__394,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aTP_Lamm_fy(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_394
tff(fact_10177_ATP_Olambda__395,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aTP_Lamm_fi(Uu),Uua),Uub))
    <=> ( member_real(Uub,Uu)
        & ( aa_real_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_395
tff(fact_10178_ATP_Olambda__396,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_fs(Uu),Uua),Uub))
    <=> ( member_real(Uub,Uu)
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_396
tff(fact_10179_ATP_Olambda__397,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aTP_Lamm_fk(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_397
tff(fact_10180_ATP_Olambda__398,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_fu(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_398
tff(fact_10181_ATP_Olambda__399,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aTP_Lamm_fm(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_399
tff(fact_10182_ATP_Olambda__400,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_fw(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_400
tff(fact_10183_ATP_Olambda__401,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun8092769222387074655T_bool(aTP_Lamm_eh(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uub),Uu))
        & ( aa_VEBT_VEBT_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_401
tff(fact_10184_ATP_Olambda__402,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun5605965666826184925T_bool(aTP_Lamm_er(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uub),Uu))
        & ( aa_VEBT_VEBT_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_402
tff(fact_10185_ATP_Olambda__403,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aTP_Lamm_ep(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_403
tff(fact_10186_ATP_Olambda__404,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aTP_Lamm_ez(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_404
tff(fact_10187_ATP_Olambda__405,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aTP_Lamm_ej(Uu),Uua),Uub))
    <=> ( member_real(Uub,Uu)
        & ( aa_real_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_405
tff(fact_10188_ATP_Olambda__406,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_et(Uu),Uua),Uub))
    <=> ( member_real(Uub,Uu)
        & ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_406
tff(fact_10189_ATP_Olambda__407,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aTP_Lamm_el(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_407
tff(fact_10190_ATP_Olambda__408,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_ev(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_408
tff(fact_10191_ATP_Olambda__409,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aTP_Lamm_en(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_409
tff(fact_10192_ATP_Olambda__410,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_ex(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_410
tff(fact_10193_ATP_Olambda__411,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_set1728193350978918526T_bool(aTP_Lamm_ea(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uub),Uu))
        & ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uub),Uua)) ) ) ).

% ATP.lambda_411
tff(fact_10194_ATP_Olambda__412,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_set3741564773079358752x_bool(aTP_Lamm_eb(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ~ pp(aa_set_complex_bool(member_complex(Uub),Uua)) ) ) ).

% ATP.lambda_412
tff(fact_10195_ATP_Olambda__413,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_ec(Uu),Uua),Uub))
    <=> ( member_real(Uub,Uu)
        & ~ member_real(Uub,Uua) ) ) ).

% ATP.lambda_413
tff(fact_10196_ATP_Olambda__414,axiom,
    ! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_set257547205302447420t_bool(aTP_Lamm_ed(Uu),Uua),Uub))
    <=> ( member_list_nat(Uub,Uu)
        & ~ member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_414
tff(fact_10197_ATP_Olambda__415,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set9121396911425987952t_bool(aTP_Lamm_ee(Uu),Uua),Uub))
    <=> ( member_set_nat(Uub,Uu)
        & ~ member_set_nat(Uub,Uua) ) ) ).

% ATP.lambda_415
tff(fact_10198_ATP_Olambda__416,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_eg(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ~ pp(aa_set_nat_bool(member_nat(Uub),Uua)) ) ) ).

% ATP.lambda_416
tff(fact_10199_ATP_Olambda__417,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_ef(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ~ pp(aa_set_int_bool(member_int(Uub),Uua)) ) ) ).

% ATP.lambda_417
tff(fact_10200_ATP_Olambda__418,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_abu(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_418
tff(fact_10201_ATP_Olambda__419,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_abl(Uu),Uua),Uub))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V3694042436643373181omplex(Uua),Uub)),Uu)) ) ).

% ATP.lambda_419
tff(fact_10202_ATP_Olambda__420,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_abn(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_420
tff(fact_10203_ATP_Olambda__421,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_ga(Uu),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_421
tff(fact_10204_ATP_Olambda__422,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_xh(Uu),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_422
tff(fact_10205_ATP_Olambda__423,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_qg(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_423
tff(fact_10206_ATP_Olambda__424,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qh(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_424
tff(fact_10207_ATP_Olambda__425,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_aau(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_nat_int(Uua,Uub))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) ).

% ATP.lambda_425
tff(fact_10208_ATP_Olambda__426,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zx(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_426
tff(fact_10209_ATP_Olambda__427,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_qb(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_427
tff(fact_10210_ATP_Olambda__428,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pz(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_428
tff(fact_10211_ATP_Olambda__429,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ww(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_429
tff(fact_10212_ATP_Olambda__430,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_gb(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_430
tff(fact_10213_ATP_Olambda__431,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aba(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_431
tff(fact_10214_ATP_Olambda__432,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abd(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_432
tff(fact_10215_ATP_Olambda__433,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_td(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_433
tff(fact_10216_ATP_Olambda__434,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_te(Uu),Uua),Uub) = aa_int_int(times_times_int(aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_434
tff(fact_10217_ATP_Olambda__435,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_tf(Uu),Uua),Uub) = aa_int_int(times_times_int(aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_435
tff(fact_10218_ATP_Olambda__436,axiom,
    ! [Uu: fun_VE9219535225822561060nteger,Uua: fun_VE9219535225822561060nteger,Uub: vEBT_VEBT] : aa_VEB5935388676523804373nteger(aa_fun2400605388621188028nteger(aTP_Lamm_lf(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_VEB5935388676523804373nteger(Uua,Uub)),aa_VEB5935388676523804373nteger(Uu,Uub)) ).

% ATP.lambda_436
tff(fact_10219_ATP_Olambda__437,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_lj(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(Uua,Uub)),aa_VEBT_VEBT_real(Uu,Uub)) ).

% ATP.lambda_437
tff(fact_10220_ATP_Olambda__438,axiom,
    ! [Uu: fun_co8153928105989009288nteger,Uua: fun_co8153928105989009288nteger,Uub: complex] : aa_com4217951221296830281nteger(aa_fun5267417455630836558nteger(aTP_Lamm_lh(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_com4217951221296830281nteger(Uua,Uub)),aa_com4217951221296830281nteger(Uu,Uub)) ).

% ATP.lambda_438
tff(fact_10221_ATP_Olambda__439,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_ll(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uub)),aa_complex_real(Uu,Uub)) ).

% ATP.lambda_439
tff(fact_10222_ATP_Olambda__440,axiom,
    ! [Uu: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_le(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_real_Code_integer(Uua,Uub)),aa_real_Code_integer(Uu,Uub)) ).

% ATP.lambda_440
tff(fact_10223_ATP_Olambda__441,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_li(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_441
tff(fact_10224_ATP_Olambda__442,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_ld(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu,Uub)) ).

% ATP.lambda_442
tff(fact_10225_ATP_Olambda__443,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_lm(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu,Uub)) ).

% ATP.lambda_443
tff(fact_10226_ATP_Olambda__444,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_lg(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu,Uub)) ).

% ATP.lambda_444
tff(fact_10227_ATP_Olambda__445,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_lk(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uub)),aa_int_real(Uu,Uub)) ).

% ATP.lambda_445
tff(fact_10228_ATP_Olambda__446,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_ii(Uu),Uua),Uub) = aa_complex_complex(minus_minus_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_446
tff(fact_10229_ATP_Olambda__447,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ij(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_447
tff(fact_10230_ATP_Olambda__448,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ih(Uu),Uua),Uub) = aa_int_int(minus_minus_int(aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_448
tff(fact_10231_ATP_Olambda__449,axiom,
    ! [Uu: fun_VEBT_VEBT_nat,Uua: fun_VEBT_VEBT_nat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_fun8284231930576791164BT_nat(aTP_Lamm_lv(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_VEBT_VEBT_nat(Uua,Uub)),aa_VEBT_VEBT_nat(Uu,Uub)) ).

% ATP.lambda_449
tff(fact_10232_ATP_Olambda__450,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_lx(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_complex_nat(Uua,Uub)),aa_complex_nat(Uu,Uub)) ).

% ATP.lambda_450
tff(fact_10233_ATP_Olambda__451,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_lu(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_451
tff(fact_10234_ATP_Olambda__452,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_qn(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uua,Uub)),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_452
tff(fact_10235_ATP_Olambda__453,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ly(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_453
tff(fact_10236_ATP_Olambda__454,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_lw(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_454
tff(fact_10237_ATP_Olambda__455,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_ie(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_455
tff(fact_10238_ATP_Olambda__456,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ig(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_456
tff(fact_10239_ATP_Olambda__457,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_if(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_457
tff(fact_10240_ATP_Olambda__458,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_pv(Uu),Uua),Uub) = aa_int_int(plus_plus_int(aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_458
tff(fact_10241_ATP_Olambda__459,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_id(Uu),Uua),Uub) = aa_int_int(plus_plus_int(aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_459
tff(fact_10242_ATP_Olambda__460,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_460
tff(fact_10243_ATP_Olambda__461,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_cm(Uu),Uua),Uub))
    <=> ( pp(aa_complex_bool(Uu,Uub))
        | pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_461
tff(fact_10244_ATP_Olambda__462,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_ch(Uu),Uua),Uub))
    <=> ( pp(aa_real_bool(Uu,Uub))
        | pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_462
tff(fact_10245_ATP_Olambda__463,axiom,
    ! [Uu: fun_list_nat_bool,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_ci(Uu),Uua),Uub))
    <=> ( pp(aa_list_nat_bool(Uu,Uub))
        | pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_463
tff(fact_10246_ATP_Olambda__464,axiom,
    ! [Uu: fun_set_nat_bool,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_cj(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(Uu,Uub))
        | pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_464
tff(fact_10247_ATP_Olambda__465,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_ck(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        | pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_465
tff(fact_10248_ATP_Olambda__466,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_cl(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(Uu,Uub))
        | pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_466
tff(fact_10249_ATP_Olambda__467,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_cs(Uu),Uua),Uub))
    <=> ( pp(aa_complex_bool(Uu,Uub))
        & pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_467
tff(fact_10250_ATP_Olambda__468,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_cn(Uu),Uua),Uub))
    <=> ( pp(aa_real_bool(Uu,Uub))
        & pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_468
tff(fact_10251_ATP_Olambda__469,axiom,
    ! [Uu: fun_list_nat_bool,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_co(Uu),Uua),Uub))
    <=> ( pp(aa_list_nat_bool(Uu,Uub))
        & pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_469
tff(fact_10252_ATP_Olambda__470,axiom,
    ! [Uu: fun_set_nat_bool,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_cp(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(Uu,Uub))
        & pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_470
tff(fact_10253_ATP_Olambda__471,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_cq(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        & pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_471
tff(fact_10254_ATP_Olambda__472,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_cr(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(Uu,Uub))
        & pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_472
tff(fact_10255_ATP_Olambda__473,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_gc(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua)) ) ).

% ATP.lambda_473
tff(fact_10256_ATP_Olambda__474,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kd(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_474
tff(fact_10257_ATP_Olambda__475,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ul(Uu),Uua),Uub) = aa_int_int(modulo_modulo_int(aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_475
tff(fact_10258_ATP_Olambda__476,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_kc(Uu),Uua),Uub) = aa_int_int(modulo_modulo_int(aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_476
tff(fact_10259_ATP_Olambda__477,axiom,
    ! [Uu: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_ik(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(Uu,Uub)),Uua) ).

% ATP.lambda_477
tff(fact_10260_ATP_Olambda__478,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_pm(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_478
tff(fact_10261_ATP_Olambda__479,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_il(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_479
tff(fact_10262_ATP_Olambda__480,axiom,
    ! [Uu: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_hw(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(Uu,Uub)),Uua) ).

% ATP.lambda_480
tff(fact_10263_ATP_Olambda__481,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_hy(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_481
tff(fact_10264_ATP_Olambda__482,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hx(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_482
tff(fact_10265_ATP_Olambda__483,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_hv(Uu),Uua),Uub) = aa_int_int(times_times_int(aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_483
tff(fact_10266_ATP_Olambda__484,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_px(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uub)),Uu) ).

% ATP.lambda_484
tff(fact_10267_ATP_Olambda__485,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_py(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uub)),Uu) ).

% ATP.lambda_485
tff(fact_10268_ATP_Olambda__486,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_oc(Uu),Uua),Uub) = aa_complex_complex(minus_minus_complex(aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_486
tff(fact_10269_ATP_Olambda__487,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_od(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_487
tff(fact_10270_ATP_Olambda__488,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_oa(Uu),Uua),Uub) = minus_minus_rat(aa_nat_rat(Uu,Uub),Uua) ).

% ATP.lambda_488
tff(fact_10271_ATP_Olambda__489,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ob(Uu),Uua),Uub) = aa_int_int(minus_minus_int(aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_489
tff(fact_10272_ATP_Olambda__490,axiom,
    ! [Uu: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_zq(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_490
tff(fact_10273_ATP_Olambda__491,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tg(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_491
tff(fact_10274_ATP_Olambda__492,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_th(Uu),Uua),Uub) = aa_nat_int(power_power_int(aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_492
tff(fact_10275_ATP_Olambda__493,axiom,
    ! [Uu: fun_int_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_ti(Uu),Uua),Uub) = aa_nat_int(power_power_int(aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_493
tff(fact_10276_ATP_Olambda__494,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abh(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_494
tff(fact_10277_ATP_Olambda__495,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_zu(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_495
tff(fact_10278_ATP_Olambda__496,axiom,
    ! [Uu: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_qq(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_496
tff(fact_10279_ATP_Olambda__497,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_nd(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_497
tff(fact_10280_ATP_Olambda__498,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ne(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_498
tff(fact_10281_ATP_Olambda__499,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_nc(Uu),Uua),Uub) = plus_plus_rat(Uu,aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ).

% ATP.lambda_499
tff(fact_10282_ATP_Olambda__500,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nb(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_500
tff(fact_10283_ATP_Olambda__501,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_na(Uu),Uua),Uub) = aa_int_int(plus_plus_int(Uu),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,Uub)),Uua)) ).

% ATP.lambda_501
tff(fact_10284_ATP_Olambda__502,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ox(Uu),Uua),Uub) = aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_502
tff(fact_10285_ATP_Olambda__503,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_pa(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_503
tff(fact_10286_ATP_Olambda__504,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_oy(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_504
tff(fact_10287_ATP_Olambda__505,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_oz(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_505
tff(fact_10288_ATP_Olambda__506,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aao(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_506
tff(fact_10289_ATP_Olambda__507,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_mb(Uu),Uua),Uub) = aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_507
tff(fact_10290_ATP_Olambda__508,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_me(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_508
tff(fact_10291_ATP_Olambda__509,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_mc(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_509
tff(fact_10292_ATP_Olambda__510,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_md(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_510
tff(fact_10293_ATP_Olambda__511,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mz(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_511
tff(fact_10294_ATP_Olambda__512,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_abb(Uu),Uua),Uub))
    <=> member_real(Uub,set_or1633881224788618240n_real(Uu,Uua)) ) ).

% ATP.lambda_512
tff(fact_10295_ATP_Olambda__513,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_ia(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_513
tff(fact_10296_ATP_Olambda__514,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_pl(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_514
tff(fact_10297_ATP_Olambda__515,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ic(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_515
tff(fact_10298_ATP_Olambda__516,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ib(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_516
tff(fact_10299_ATP_Olambda__517,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_hz(Uu),Uua),Uub) = aa_int_int(times_times_int(Uu),aa_int_int(Uua,Uub)) ).

% ATP.lambda_517
tff(fact_10300_ATP_Olambda__518,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pw(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_518
tff(fact_10301_ATP_Olambda__519,axiom,
    ! [Uu: complex,Uua: fun_nat_nat,Uub: nat] : aa_nat_complex(aa_fun1958354139826840918omplex(aTP_Lamm_vg(Uu),Uua),Uub) = aa_nat_complex(power_power_complex(Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_519
tff(fact_10302_ATP_Olambda__520,axiom,
    ! [Uu: real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_vf(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_520
tff(fact_10303_ATP_Olambda__521,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_vh(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_521
tff(fact_10304_ATP_Olambda__522,axiom,
    ! [Uu: int,Uua: fun_nat_nat,Uub: nat] : aa_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_vi(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_522
tff(fact_10305_ATP_Olambda__523,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] : aa_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_vj(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_523
tff(fact_10306_ATP_Olambda__524,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_nz(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_524
tff(fact_10307_ATP_Olambda__525,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ny(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_525
tff(fact_10308_ATP_Olambda__526,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_wf(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_526
tff(fact_10309_ATP_Olambda__527,axiom,
    ! [Uu: fun_real_bool,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_abj(Uu),Uua),Uub))
    <=> pp(aa_real_bool(Uu,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uua))) ) ).

% ATP.lambda_527
tff(fact_10310_ATP_Olambda__528,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_qp(Uu),Uua),Uub) = aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_528
tff(fact_10311_ATP_Olambda__529,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ma(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_529
tff(fact_10312_ATP_Olambda__530,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_aay(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_530
tff(fact_10313_ATP_Olambda__531,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lz(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_531
tff(fact_10314_ATP_Olambda__532,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_vk(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_532
tff(fact_10315_ATP_Olambda__533,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qe(Uu),Uua),Uub) = aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_533
tff(fact_10316_ATP_Olambda__534,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_qf(Uu),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_534
tff(fact_10317_ATP_Olambda__535,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_abq(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_535
tff(fact_10318_ATP_Olambda__536,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_qc(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_536
tff(fact_10319_ATP_Olambda__537,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qa(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_537
tff(fact_10320_ATP_Olambda__538,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aat(Uu),Uua),Uub) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),Uua)) ).

% ATP.lambda_538
tff(fact_10321_ATP_Olambda__539,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun6713666175893136621x_bool(aTP_Lamm_dt(Uu),Uua),Uub))
    <=> ? [X3: real] :
          ( ( Uub = aa_real_complex(Uua,X3) )
          & pp(aa_real_bool(Uu,X3)) ) ) ).

% ATP.lambda_539
tff(fact_10322_ATP_Olambda__540,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_dq(Uu),Uua),Uub))
    <=> ? [X3: real] :
          ( ( Uub = aa_real_real(Uua,X3) )
          & pp(aa_real_bool(Uu,X3)) ) ) ).

% ATP.lambda_540
tff(fact_10323_ATP_Olambda__541,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3670751239728173737t_bool(aTP_Lamm_dr(Uu),Uua),Uub))
    <=> ? [X3: real] :
          ( ( Uub = aa_real_nat(Uua,X3) )
          & pp(aa_real_bool(Uu,X3)) ) ) ).

% ATP.lambda_541
tff(fact_10324_ATP_Olambda__542,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_int,Uub: int] :
      ( pp(aa_int_bool(aa_fun6938287985607813617t_bool(aTP_Lamm_ds(Uu),Uua),Uub))
    <=> ? [X3: real] :
          ( ( Uub = aa_real_int(Uua,X3) )
          & pp(aa_real_bool(Uu,X3)) ) ) ).

% ATP.lambda_542
tff(fact_10325_ATP_Olambda__543,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun678042768005816265x_bool(aTP_Lamm_dx(Uu),Uua),Uub))
    <=> ? [X3: nat] :
          ( ( Uub = aa_nat_complex(Uua,X3) )
          & pp(aa_nat_bool(Uu,X3)) ) ) ).

% ATP.lambda_543
tff(fact_10326_ATP_Olambda__544,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun2369234836118212941l_bool(aTP_Lamm_du(Uu),Uua),Uub))
    <=> ? [X3: nat] :
          ( ( Uub = aa_nat_real(Uua,X3) )
          & pp(aa_nat_bool(Uu,X3)) ) ) ).

% ATP.lambda_544
tff(fact_10327_ATP_Olambda__545,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1783405840085286277t_bool(aTP_Lamm_dv(Uu),Uua),Uub))
    <=> ? [X3: nat] :
          ( ( Uub = aa_nat_nat(Uua,X3) )
          & pp(aa_nat_bool(Uu,X3)) ) ) ).

% ATP.lambda_545
tff(fact_10328_ATP_Olambda__546,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_int,Uub: int] :
      ( pp(aa_int_bool(aa_fun5050942585964926157t_bool(aTP_Lamm_dw(Uu),Uua),Uub))
    <=> ? [X3: nat] :
          ( ( Uub = aa_nat_int(Uua,X3) )
          & pp(aa_nat_bool(Uu,X3)) ) ) ).

% ATP.lambda_546
tff(fact_10329_ATP_Olambda__547,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun1272151855474214385l_bool(aTP_Lamm_dy(Uu),Uua),Uub))
    <=> ? [X3: int] :
          ( ( Uub = aa_int_real(Uua,X3) )
          & pp(aa_int_bool(Uu,X3)) ) ) ).

% ATP.lambda_547
tff(fact_10330_ATP_Olambda__548,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun8751864718750488105t_bool(aTP_Lamm_dz(Uu),Uua),Uub))
    <=> ? [X3: int] :
          ( ( Uub = aa_int_nat(Uua,X3) )
          & pp(aa_int_bool(Uu,X3)) ) ) ).

% ATP.lambda_548
tff(fact_10331_ATP_Olambda__549,axiom,
    ! [Uu: fun_real_bool,Uua: fun_co2286642472385812463l_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun6039685132559610447x_bool(aTP_Lamm_br(Uu),Uua),Uub))
    <=> ? [Y4: real] :
          ( pp(aa_real_bool(Uu,Y4))
          & pp(aa_real_bool(aa_com8943241093633526360l_bool(Uua,Uub),Y4)) ) ) ).

% ATP.lambda_549
tff(fact_10332_ATP_Olambda__550,axiom,
    ! [Uu: fun_real_bool,Uua: fun_re1255401033344146673l_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun4964148113266127823l_bool(aTP_Lamm_bl(Uu),Uua),Uub))
    <=> ? [Y4: real] :
          ( pp(aa_real_bool(Uu,Y4))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(Uua,Uub),Y4)) ) ) ).

% ATP.lambda_550
tff(fact_10333_ATP_Olambda__551,axiom,
    ! [Uu: fun_real_bool,Uua: fun_na6801867027684862029l_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun7391232120346184911t_bool(aTP_Lamm_bn(Uu),Uua),Uub))
    <=> ? [Y4: real] :
          ( pp(aa_real_bool(Uu,Y4))
          & pp(aa_real_bool(aa_nat_fun_real_bool(Uua,Uub),Y4)) ) ) ).

% ATP.lambda_551
tff(fact_10334_ATP_Olambda__552,axiom,
    ! [Uu: fun_real_bool,Uua: fun_in3970322703178350321l_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun69789402421856207t_bool(aTP_Lamm_bp(Uu),Uua),Uub))
    <=> ? [Y4: real] :
          ( pp(aa_real_bool(Uu,Y4))
          & pp(aa_real_bool(aa_int_fun_real_bool(Uua,Uub),Y4)) ) ) ).

% ATP.lambda_552
tff(fact_10335_ATP_Olambda__553,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_co2841405903858895691t_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun6682930621474858099x_bool(aTP_Lamm_bz(Uu),Uua),Uub))
    <=> ? [Y4: nat] :
          ( pp(aa_nat_bool(Uu,Y4))
          & pp(aa_nat_bool(aa_com5335841084339320244t_bool(Uua,Uub),Y4)) ) ) ).

% ATP.lambda_553
tff(fact_10336_ATP_Olambda__554,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_re9121486324101951821t_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun4927090880566115059l_bool(aTP_Lamm_bt(Uu),Uua),Uub))
    <=> ? [Y4: nat] :
          ( pp(aa_nat_bool(Uu,Y4))
          & pp(aa_nat_bool(aa_real_fun_nat_bool(Uua,Uub),Y4)) ) ) ).

% ATP.lambda_554
tff(fact_10337_ATP_Olambda__555,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun8399658544460314611t_bool(aTP_Lamm_bv(Uu),Uua),Uub))
    <=> ? [Y4: nat] :
          ( pp(aa_nat_bool(Uu,Y4))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(Uua,Uub),Y4)) ) ) ).

% ATP.lambda_555
tff(fact_10338_ATP_Olambda__556,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_int_fun_nat_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun6484551404966387955t_bool(aTP_Lamm_bx(Uu),Uua),Uub))
    <=> ? [Y4: nat] :
          ( pp(aa_nat_bool(Uu,Y4))
          & pp(aa_nat_bool(aa_int_fun_nat_bool(Uua,Uub),Y4)) ) ) ).

% ATP.lambda_556
tff(fact_10339_ATP_Olambda__557,axiom,
    ! [Uu: fun_int_bool,Uua: fun_re8661203372455305201t_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun7352647192720469839l_bool(aTP_Lamm_cb(Uu),Uua),Uub))
    <=> ? [Y4: int] :
          ( pp(aa_int_bool(Uu,Y4))
          & pp(aa_int_bool(aa_real_fun_int_bool(Uua,Uub),Y4)) ) ) ).

% ATP.lambda_557
tff(fact_10340_ATP_Olambda__558,axiom,
    ! [Uu: fun_int_bool,Uua: fun_nat_fun_int_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun7626329022887090767t_bool(aTP_Lamm_cd(Uu),Uua),Uub))
    <=> ? [Y4: int] :
          ( pp(aa_int_bool(Uu,Y4))
          & pp(aa_int_bool(aa_nat_fun_int_bool(Uua,Uub),Y4)) ) ) ).

% ATP.lambda_558
tff(fact_10341_ATP_Olambda__559,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_wh(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),one_one_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_559
tff(fact_10342_ATP_Olambda__560,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wj(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_real(Uua,Uuc)),aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),one_one_real),aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_560
tff(fact_10343_ATP_Olambda__561,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wl(Uu),Uua),Uub),Uuc) = if_rat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_rat(Uua,Uuc),if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_rat,aa_nat_rat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_561
tff(fact_10344_ATP_Olambda__562,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wn(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_nat(Uua,Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),one_one_nat),aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_562
tff(fact_10345_ATP_Olambda__563,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_wp(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),one_one_int),aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_563
tff(fact_10346_ATP_Olambda__564,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_wi(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_564
tff(fact_10347_ATP_Olambda__565,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wk(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_565
tff(fact_10348_ATP_Olambda__566,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wm(Uu),Uua),Uub),Uuc) = 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_566
tff(fact_10349_ATP_Olambda__567,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wo(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_567
tff(fact_10350_ATP_Olambda__568,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_wq(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_568
tff(fact_10351_ATP_Olambda__569,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_qo(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_569
tff(fact_10352_ATP_Olambda__570,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_nr(Uu),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_nq(Uu),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_570
tff(fact_10353_ATP_Olambda__571,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_np(Uu),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_no(Uu),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_571
tff(fact_10354_ATP_Olambda__572,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nn(Uu),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nm(Uu),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_572
tff(fact_10355_ATP_Olambda__573,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: set_nat,Uuc: nat] : aa_nat_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_hu(Uu),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_ht(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_573
tff(fact_10356_ATP_Olambda__574,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: set_nat,Uuc: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_hs(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_hr(Uu),Uua),Uuc)),Uub) ).

% ATP.lambda_574
tff(fact_10357_ATP_Olambda__575,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: set_complex,Uuc: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_hq(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_hp(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_575
tff(fact_10358_ATP_Olambda__576,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: set_int,Uuc: int] : aa_int_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_ho(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_hn(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_576
tff(fact_10359_ATP_Olambda__577,axiom,
    ! [Uu: set_complex,Uua: fun_co7370231399432144932at_int,Uub: fun_co2841405903858895691t_bool,Uuc: nat] : aa_nat_int(aa_fun6041977265744554632at_int(aa_fun3475035881565545205at_int(aTP_Lamm_ue(Uu),Uua),Uub),Uuc) = groups858564598930262913ex_int(aa_nat8436805559129492965ex_int(aTP_Lamm_ud(Uua),Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_tu(Uu),Uub),Uuc))) ).

% ATP.lambda_577
tff(fact_10360_ATP_Olambda__578,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_int,Uub: fun_re9121486324101951821t_bool,Uuc: nat] : aa_nat_int(aa_fun1494899438858969354at_int(aa_fun6942549246588167161at_int(aTP_Lamm_ub(Uu),Uua),Uub),Uuc) = groups4694064378042380927al_int(aa_nat_fun_real_int(aTP_Lamm_ua(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_cx(Uu),Uub),Uuc))) ).

% ATP.lambda_578
tff(fact_10361_ATP_Olambda__579,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE7331017413731640256at_int,Uub: fun_VE3763956218935105199t_bool,Uuc: nat] : aa_nat_int(aa_fun1384843155787029862at_int(aa_fun6066863168309489065at_int(aTP_Lamm_ty(Uu),Uua),Uub),Uuc) = groups6359315924273963643BT_int(aa_nat3598295560461965541BT_int(aTP_Lamm_tx(Uua),Uuc),collect_VEBT_VEBT(aa_nat8029987864780243636T_bool(aa_fun4018725027000996448T_bool(aTP_Lamm_cw(Uu),Uub),Uuc))) ).

% ATP.lambda_579
tff(fact_10362_ATP_Olambda__580,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_tv(Uu),Uua),Uub),Uuc) = groups861055069439313189ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_tt(Uua),Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_tu(Uu),Uub),Uuc))) ).

% ATP.lambda_580
tff(fact_10363_ATP_Olambda__581,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_tq(Uu),Uua),Uub),Uuc) = groups1707563613775114915nt_nat(aa_nat_fun_int_nat(aTP_Lamm_tp(Uua),Uuc),collect_int(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_df(Uu),Uub),Uuc))) ).

% ATP.lambda_581
tff(fact_10364_ATP_Olambda__582,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_tm(Uu),Uua),Uub),Uuc) = groups4696554848551431203al_nat(aa_nat_fun_real_nat(aTP_Lamm_ka(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_cx(Uu),Uub),Uuc))) ).

% ATP.lambda_582
tff(fact_10365_ATP_Olambda__583,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE3763956218935105199t_bool,Uuc: nat] : aa_nat_nat(aa_fun5562694175296226570at_nat(aa_fun8318684123515291377at_nat(aTP_Lamm_tk(Uu),Uua),Uub),Uuc) = groups6361806394783013919BT_nat(aa_nat7776146579971162249BT_nat(aTP_Lamm_jw(Uua),Uuc),collect_VEBT_VEBT(aa_nat8029987864780243636T_bool(aa_fun4018725027000996448T_bool(aTP_Lamm_cw(Uu),Uub),Uuc))) ).

% ATP.lambda_583
tff(fact_10366_ATP_Olambda__584,axiom,
    ! [Uu: set_complex,Uua: fun_co333780937270390912nt_int,Uub: fun_co2381122952212249071t_bool,Uuc: int] : aa_int_int(aa_fun855801025254967176nt_int(aa_fun315996177946036377nt_int(aTP_Lamm_uk(Uu),Uua),Uub),Uuc) = groups858564598930262913ex_int(aa_int6126987627036542017ex_int(aTP_Lamm_ja(Uua),Uuc),collect_complex(aa_int5354780257279585880x_bool(aa_fun2352441141717865852x_bool(aTP_Lamm_jb(Uu),Uub),Uuc))) ).

% ATP.lambda_584
tff(fact_10367_ATP_Olambda__585,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_int_int,Uub: fun_re8661203372455305201t_bool,Uuc: int] : aa_int_int(aa_fun5532095235224157706nt_int(aa_fun4786959800906006429nt_int(aTP_Lamm_ui(Uu),Uua),Uub),Uuc) = groups4694064378042380927al_int(aa_int_fun_real_int(aTP_Lamm_is(Uua),Uuc),collect_real(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_cz(Uu),Uub),Uuc))) ).

% ATP.lambda_585
tff(fact_10368_ATP_Olambda__586,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE294566951569886236nt_int,Uub: fun_VE3303673267288458579t_bool,Uuc: int] : aa_int_int(aa_fun5422038952152218214nt_int(aa_fun3638225402958239493nt_int(aTP_Lamm_ug(Uu),Uua),Uub),Uuc) = groups6359315924273963643BT_int(aa_int1343382402272391561BT_int(aTP_Lamm_io(Uua),Uuc),collect_VEBT_VEBT(aa_int3199078826956362512T_bool(aa_fun4417922564241321824T_bool(aTP_Lamm_cy(Uu),Uub),Uuc))) ).

% ATP.lambda_586
tff(fact_10369_ATP_Olambda__587,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_kb(Uu),Uua),Uub),Uuc) = groups1935376822645274424al_nat(aa_nat_fun_real_nat(aTP_Lamm_ka(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_cx(Uu),Uub),Uuc))) ).

% ATP.lambda_587
tff(fact_10370_ATP_Olambda__588,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE3763956218935105199t_bool,Uuc: nat] : aa_nat_nat(aa_fun5562694175296226570at_nat(aa_fun8318684123515291377at_nat(aTP_Lamm_jx(Uu),Uua),Uub),Uuc) = groups771621172384141258BT_nat(aa_nat7776146579971162249BT_nat(aTP_Lamm_jw(Uua),Uuc),collect_VEBT_VEBT(aa_nat8029987864780243636T_bool(aa_fun4018725027000996448T_bool(aTP_Lamm_cw(Uu),Uub),Uuc))) ).

% ATP.lambda_588
tff(fact_10371_ATP_Olambda__589,axiom,
    ! [Uu: set_int,Uua: fun_in1307110748760623106omplex,Uub: fun_in5750378413879446255x_bool,Uuc: complex] : aa_complex_complex(aa_fun2686559102040213132omplex(aa_fun5422379709423809691omplex(aTP_Lamm_jt(Uu),Uua),Uub),Uuc) = groups3049146728041665814omplex(aa_com9156536935073639875omplex(aTP_Lamm_jr(Uua),Uuc),collect_int(aa_com2211550117870696024t_bool(aa_fun3087709329828230012t_bool(aTP_Lamm_js(Uu),Uub),Uuc))) ).

% ATP.lambda_589
tff(fact_10372_ATP_Olambda__590,axiom,
    ! [Uu: set_nat,Uua: fun_na1591161823130732966omplex,Uub: fun_na5941084329111710283x_bool,Uuc: complex] : aa_complex_complex(aa_fun1613615970879741416omplex(aa_fun7586814990830484307omplex(aTP_Lamm_jo(Uu),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_jn(Uua),Uuc),collect_nat(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_de(Uu),Uub),Uuc))) ).

% ATP.lambda_590
tff(fact_10373_ATP_Olambda__591,axiom,
    ! [Uu: set_real,Uua: fun_re7208826350674946562omplex,Uub: fun_re3010880487650781423x_bool,Uuc: complex] : aa_complex_complex(aa_fun8128191506076533516omplex(aa_fun840110154952133019omplex(aTP_Lamm_jk(Uu),Uua),Uub),Uuc) = groups5754745047067104278omplex(aa_com4766338864777664963omplex(aTP_Lamm_jj(Uua),Uuc),collect_real(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_db(Uu),Uub),Uuc))) ).

% ATP.lambda_591
tff(fact_10374_ATP_Olambda__592,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE2006399705683203104omplex,Uub: fun_VE2128610149277289041x_bool,Uuc: complex] : aa_complex_complex(aa_fun1095524714264483688omplex(aa_fun8876071380179997707omplex(aTP_Lamm_jg(Uu),Uua),Uub),Uuc) = groups1794756597179926696omplex(aa_com2862003112617944841omplex(aTP_Lamm_jf(Uua),Uuc),collect_VEBT_VEBT(aa_com4769948546659418770T_bool(aa_fun3341029659329701088T_bool(aTP_Lamm_da(Uu),Uub),Uuc))) ).

% ATP.lambda_592
tff(fact_10375_ATP_Olambda__593,axiom,
    ! [Uu: set_complex,Uua: fun_co333780937270390912nt_int,Uub: fun_co2381122952212249071t_bool,Uuc: int] : aa_int_int(aa_fun855801025254967176nt_int(aa_fun315996177946036377nt_int(aTP_Lamm_jc(Uu),Uua),Uub),Uuc) = groups5690904116761175830ex_int(aa_int6126987627036542017ex_int(aTP_Lamm_ja(Uua),Uuc),collect_complex(aa_int5354780257279585880x_bool(aa_fun2352441141717865852x_bool(aTP_Lamm_jb(Uu),Uub),Uuc))) ).

% ATP.lambda_593
tff(fact_10376_ATP_Olambda__594,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_int,Uub: fun_nat_fun_int_bool,Uuc: int] : aa_int_int(aa_fun4504283173926863078nt_int(aa_fun5512849559334105941nt_int(aTP_Lamm_ix(Uu),Uua),Uub),Uuc) = groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_iw(Uua),Uuc),collect_nat(aa_int_fun_nat_bool(aa_fun8184653438340336124t_bool(aTP_Lamm_dd(Uu),Uub),Uuc))) ).

% ATP.lambda_594
tff(fact_10377_ATP_Olambda__595,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_int_int,Uub: fun_re8661203372455305201t_bool,Uuc: int] : aa_int_int(aa_fun5532095235224157706nt_int(aa_fun4786959800906006429nt_int(aTP_Lamm_it(Uu),Uua),Uub),Uuc) = groups1932886352136224148al_int(aa_int_fun_real_int(aTP_Lamm_is(Uua),Uuc),collect_real(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_cz(Uu),Uub),Uuc))) ).

% ATP.lambda_595
tff(fact_10378_ATP_Olambda__596,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE294566951569886236nt_int,Uub: fun_VE3303673267288458579t_bool,Uuc: int] : aa_int_int(aa_fun5422038952152218214nt_int(aa_fun3638225402958239493nt_int(aTP_Lamm_ip(Uu),Uua),Uub),Uuc) = groups769130701875090982BT_int(aa_int1343382402272391561BT_int(aTP_Lamm_io(Uua),Uuc),collect_VEBT_VEBT(aa_int3199078826956362512T_bool(aa_fun4417922564241321824T_bool(aTP_Lamm_cy(Uu),Uub),Uuc))) ).

% ATP.lambda_596
tff(fact_10379_ATP_Olambda__597,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_wy(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_wx(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uub),Uuc)) ).

% ATP.lambda_597
tff(fact_10380_ATP_Olambda__598,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_aae(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_598
tff(fact_10381_ATP_Olambda__599,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_aac(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_599
tff(fact_10382_ATP_Olambda__600,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_aab(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_600
tff(fact_10383_ATP_Olambda__601,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_aaa(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_601
tff(fact_10384_ATP_Olambda__602,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_oh(Uu),Uua),Uub),Uuc) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uu),Uub)) ).

% ATP.lambda_602
tff(fact_10385_ATP_Olambda__603,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_on(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_603
tff(fact_10386_ATP_Olambda__604,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_oj(Uu),Uua),Uub),Uuc) = minus_minus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(plus_plus_rat(Uu,Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),Uuc)),aa_nat_rat(power_power_rat(Uu),Uub)) ).

% ATP.lambda_604
tff(fact_10387_ATP_Olambda__605,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ol(Uu),Uua),Uub),Uuc) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(plus_plus_int(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu),Uuc))),aa_nat_int(power_power_int(Uu),Uub)) ).

% ATP.lambda_605
tff(fact_10388_ATP_Olambda__606,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_yf(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_606
tff(fact_10389_ATP_Olambda__607,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_no(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_607
tff(fact_10390_ATP_Olambda__608,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_nq(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_608
tff(fact_10391_ATP_Olambda__609,axiom,
    ! [Uu: 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(times_times_rat(aa_nat_rat(power_power_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_609
tff(fact_10392_ATP_Olambda__610,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_wv(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_610
tff(fact_10393_ATP_Olambda__611,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ot(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_complex(power_power_complex(Uu),Uuc)) ).

% ATP.lambda_611
tff(fact_10394_ATP_Olambda__612,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ow(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)) ).

% ATP.lambda_612
tff(fact_10395_ATP_Olambda__613,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_ou(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_rat(power_power_rat(Uu),Uuc)) ).

% ATP.lambda_613
tff(fact_10396_ATP_Olambda__614,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_ov(Uu),Uua),Uub),Uuc) = aa_int_int(times_times_int(aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_int(power_power_int(Uu),Uuc)) ).

% ATP.lambda_614
tff(fact_10397_ATP_Olambda__615,axiom,
    ! [Uu: set_complex,Uua: fun_VE2128610149277289041x_bool,Uub: vEBT_VEBT,Uuc: complex] :
      ( pp(aa_complex_bool(aa_VEB8450443329232412106x_bool(aa_fun5298830385910147808x_bool(aTP_Lamm_jd(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_VEB8450443329232412106x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_615
tff(fact_10398_ATP_Olambda__616,axiom,
    ! [Uu: set_complex,Uua: fun_re3010880487650781423x_bool,Uub: real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_rea5887071448072802392x_bool(aa_fun9179849694055305084x_bool(aTP_Lamm_jh(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_616
tff(fact_10399_ATP_Olambda__617,axiom,
    ! [Uu: set_complex,Uua: fun_na5941084329111710283x_bool,Uub: nat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aa_fun2551091230351774588x_bool(aTP_Lamm_jl(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_617
tff(fact_10400_ATP_Olambda__618,axiom,
    ! [Uu: set_complex,Uua: fun_in5750378413879446255x_bool,Uub: int,Uuc: complex] :
      ( pp(aa_complex_bool(aa_int5354780257279585880x_bool(aa_fun6456964791495427196x_bool(aTP_Lamm_jp(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_618
tff(fact_10401_ATP_Olambda__619,axiom,
    ! [Uu: set_nat,Uua: fun_VE3763956218935105199t_bool,Uub: vEBT_VEBT,Uuc: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(aa_fun5431641998072558748t_bool(aTP_Lamm_ju(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_619
tff(fact_10402_ATP_Olambda__620,axiom,
    ! [Uu: set_nat,Uua: fun_co2841405903858895691t_bool,Uub: complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_tr(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_620
tff(fact_10403_ATP_Olambda__621,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_jy(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_621
tff(fact_10404_ATP_Olambda__622,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_tn(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_622
tff(fact_10405_ATP_Olambda__623,axiom,
    ! [Uu: set_int,Uua: fun_VE3303673267288458579t_bool,Uub: vEBT_VEBT,Uuc: int] :
      ( pp(aa_int_bool(aa_VEB1707969225601237452t_bool(aa_fun978093584635342820t_bool(aTP_Lamm_im(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_VEB1707969225601237452t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_623
tff(fact_10406_ATP_Olambda__624,axiom,
    ! [Uu: set_int,Uua: fun_co2381122952212249071t_bool,Uub: complex,Uuc: int] :
      ( pp(aa_int_bool(aa_com2211550117870696024t_bool(aa_fun8206557716905444476t_bool(aTP_Lamm_iy(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_com2211550117870696024t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_624
tff(fact_10407_ATP_Olambda__625,axiom,
    ! [Uu: set_int,Uua: fun_re8661203372455305201t_bool,Uub: real,Uuc: int] :
      ( pp(aa_int_bool(aa_real_fun_int_bool(aa_fun6227302519962330364t_bool(aTP_Lamm_iq(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_real_fun_int_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_625
tff(fact_10408_ATP_Olambda__626,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_iu(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_626
tff(fact_10409_ATP_Olambda__627,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE2128610149277289041x_bool,Uub: complex,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_com4769948546659418770T_bool(aa_fun3341029659329701088T_bool(aTP_Lamm_da(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uuc),Uu))
        & pp(aa_complex_bool(aa_VEB8450443329232412106x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_627
tff(fact_10410_ATP_Olambda__628,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE3763956218935105199t_bool,Uub: nat,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(aa_fun4018725027000996448T_bool(aTP_Lamm_cw(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uuc),Uu))
        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_628
tff(fact_10411_ATP_Olambda__629,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE3303673267288458579t_bool,Uub: int,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_int3199078826956362512T_bool(aa_fun4417922564241321824T_bool(aTP_Lamm_cy(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uuc),Uu))
        & pp(aa_int_bool(aa_VEB1707969225601237452t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_629
tff(fact_10412_ATP_Olambda__630,axiom,
    ! [Uu: set_complex,Uua: fun_co2841405903858895691t_bool,Uub: nat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_tu(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_630
tff(fact_10413_ATP_Olambda__631,axiom,
    ! [Uu: set_complex,Uua: fun_co2381122952212249071t_bool,Uub: int,Uuc: complex] :
      ( pp(aa_complex_bool(aa_int5354780257279585880x_bool(aa_fun2352441141717865852x_bool(aTP_Lamm_jb(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_int_bool(aa_com2211550117870696024t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_631
tff(fact_10414_ATP_Olambda__632,axiom,
    ! [Uu: set_real,Uua: fun_re3010880487650781423x_bool,Uub: complex,Uuc: real] :
      ( pp(aa_real_bool(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_db(Uu),Uua),Uub),Uuc))
    <=> ( member_real(Uuc,Uu)
        & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_632
tff(fact_10415_ATP_Olambda__633,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_cx(Uu),Uua),Uub),Uuc))
    <=> ( member_real(Uuc,Uu)
        & pp(aa_nat_bool(aa_real_fun_nat_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_633
tff(fact_10416_ATP_Olambda__634,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_cz(Uu),Uua),Uub),Uuc))
    <=> ( member_real(Uuc,Uu)
        & pp(aa_int_bool(aa_real_fun_int_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_634
tff(fact_10417_ATP_Olambda__635,axiom,
    ! [Uu: set_nat,Uua: fun_na5941084329111710283x_bool,Uub: complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_de(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_635
tff(fact_10418_ATP_Olambda__636,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_dc(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_636
tff(fact_10419_ATP_Olambda__637,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_dd(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_637
tff(fact_10420_ATP_Olambda__638,axiom,
    ! [Uu: set_int,Uua: fun_in5750378413879446255x_bool,Uub: complex,Uuc: int] :
      ( pp(aa_int_bool(aa_com2211550117870696024t_bool(aa_fun3087709329828230012t_bool(aTP_Lamm_js(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_638
tff(fact_10421_ATP_Olambda__639,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_df(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_639
tff(fact_10422_ATP_Olambda__640,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_oi(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu),Uuc)),aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_640
tff(fact_10423_ATP_Olambda__641,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_oo(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_641
tff(fact_10424_ATP_Olambda__642,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_ok(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu),Uuc)),minus_minus_rat(aa_nat_rat(power_power_rat(plus_plus_rat(Uu,Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)),aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_642
tff(fact_10425_ATP_Olambda__643,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_om(Uu),Uua),Uub),Uuc) = aa_int_int(times_times_int(aa_nat_int(power_power_int(Uu),Uuc)),aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_int_int(plus_plus_int(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_643
tff(fact_10426_ATP_Olambda__644,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_op(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu),Uuc)),aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_644
tff(fact_10427_ATP_Olambda__645,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_os(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_645
tff(fact_10428_ATP_Olambda__646,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_oq(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu),Uuc)),aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_646
tff(fact_10429_ATP_Olambda__647,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_or(Uu),Uua),Uub),Uuc) = aa_int_int(times_times_int(aa_nat_int(power_power_int(Uu),Uuc)),aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_647
tff(fact_10430_ATP_Olambda__648,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_wu(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_648
tff(fact_10431_ATP_Olambda__649,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_zb(Uu),Uua),Uub),Uuc))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(Uu),Uuc)),aa_int_int(times_times_int(Uua),Uub))) ) ).

% ATP.lambda_649
tff(fact_10432_ATP_Olambda__650,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_yj(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_650
tff(fact_10433_ATP_Olambda__651,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_yz(Uu),Uua),Uub),Uuc))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(Uu),Uuc)),aa_int_int(times_times_int(Uua),Uub))) ) ).

% ATP.lambda_651
tff(fact_10434_ATP_Olambda__652,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_yh(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_652
tff(fact_10435_ATP_Olambda__653,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_yl(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_653
tff(fact_10436_ATP_Olambda__654,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_yn(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_654
tff(fact_10437_ATP_Olambda__655,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: fun_VE5569070535936029429omplex,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun8092769222387074655T_bool(aa_fun4024526296178165685T_bool(aTP_Lamm_ei(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uuc),Uu))
        & ( aa_complex_complex(times_times_complex(aa_VEBT_VEBT_complex(Uua,Uuc)),aa_VEBT_VEBT_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_655
tff(fact_10438_ATP_Olambda__656,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun5605965666826184925T_bool(aa_fun4082248197810269365T_bool(aTP_Lamm_es(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(Uua,Uuc)),aa_VEBT_VEBT_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_656
tff(fact_10439_ATP_Olambda__657,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_eq(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_657
tff(fact_10440_ATP_Olambda__658,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_fa(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_658
tff(fact_10441_ATP_Olambda__659,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_ek(Uu),Uua),Uub),Uuc))
    <=> ( 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_659
tff(fact_10442_ATP_Olambda__660,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_eu(Uu),Uua),Uub),Uuc))
    <=> ( 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_660
tff(fact_10443_ATP_Olambda__661,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_em(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_661
tff(fact_10444_ATP_Olambda__662,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_ew(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_662
tff(fact_10445_ATP_Olambda__663,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_eo(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_663
tff(fact_10446_ATP_Olambda__664,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_ey(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_664
tff(fact_10447_ATP_Olambda__665,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: fun_VE5569070535936029429omplex,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun8092769222387074655T_bool(aa_fun4024526296178165685T_bool(aTP_Lamm_fh(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uuc),Uu))
        & ( aa_complex_complex(plus_plus_complex(aa_VEBT_VEBT_complex(Uua,Uuc)),aa_VEBT_VEBT_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_665
tff(fact_10448_ATP_Olambda__666,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun5605965666826184925T_bool(aa_fun4082248197810269365T_bool(aTP_Lamm_fr(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(Uua,Uuc)),aa_VEBT_VEBT_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_666
tff(fact_10449_ATP_Olambda__667,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_fp(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_667
tff(fact_10450_ATP_Olambda__668,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_fz(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_668
tff(fact_10451_ATP_Olambda__669,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_fj(Uu),Uua),Uub),Uuc))
    <=> ( 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_669
tff(fact_10452_ATP_Olambda__670,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_ft(Uu),Uua),Uub),Uuc))
    <=> ( 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_670
tff(fact_10453_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_fl(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_671
tff(fact_10454_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_fv(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_672
tff(fact_10455_ATP_Olambda__673,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_fn(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_673
tff(fact_10456_ATP_Olambda__674,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_fx(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_674
tff(fact_10457_ATP_Olambda__675,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_qi(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_675
tff(fact_10458_ATP_Olambda__676,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_qj(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_676
tff(fact_10459_ATP_Olambda__677,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_wx(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_677
tff(fact_10460_ATP_Olambda__678,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex,Uuc: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_hp(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uuc)) ).

% ATP.lambda_678
tff(fact_10461_ATP_Olambda__679,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_ht(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uuc)) ).

% ATP.lambda_679
tff(fact_10462_ATP_Olambda__680,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_hr(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uuc)) ).

% ATP.lambda_680
tff(fact_10463_ATP_Olambda__681,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_hn(Uu),Uua),Uub),Uuc) = aa_int_int(times_times_int(aa_int_int(Uu,Uub)),aa_int_int(Uua,Uuc)) ).

% ATP.lambda_681
tff(fact_10464_ATP_Olambda__682,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_ts(Uu),Uua),Uub),Uuc) = groups708209901874060359at_nat(aa_com6383720667567721865at_nat(Uua,Uuc),collect_nat(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_tr(Uu),Uub),Uuc))) ).

% ATP.lambda_682
tff(fact_10465_ATP_Olambda__683,axiom,
    ! [Uu: set_nat,Uua: fun_co7370231399432144932at_int,Uub: fun_co2841405903858895691t_bool,Uuc: complex] : aa_complex_int(aa_fun4872694516007147366ex_int(aa_fun2110580046701903059ex_int(aTP_Lamm_uc(Uu),Uua),Uub),Uuc) = groups705719431365010083at_int(aa_com2205869648058525157at_int(Uua,Uuc),collect_nat(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_tr(Uu),Uub),Uuc))) ).

% ATP.lambda_683
tff(fact_10466_ATP_Olambda__684,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_to(Uu),Uua),Uub),Uuc) = groups708209901874060359at_nat(aa_int_fun_nat_nat(Uua,Uuc),collect_nat(aa_int_fun_nat_bool(aa_fun3602880909931171836t_bool(aTP_Lamm_tn(Uu),Uub),Uuc))) ).

% ATP.lambda_684
tff(fact_10467_ATP_Olambda__685,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_tl(Uu),Uua),Uub),Uuc) = groups708209901874060359at_nat(aa_real_fun_nat_nat(Uua,Uuc),collect_nat(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_jy(Uu),Uub),Uuc))) ).

% ATP.lambda_685
tff(fact_10468_ATP_Olambda__686,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_int,Uub: fun_re9121486324101951821t_bool,Uuc: real] : aa_real_int(aa_fun1626300625876492518al_int(aa_fun1560246819506852821al_int(aTP_Lamm_tz(Uu),Uua),Uub),Uuc) = groups705719431365010083at_int(aa_real_fun_nat_int(Uua,Uuc),collect_nat(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_jy(Uu),Uub),Uuc))) ).

% ATP.lambda_686
tff(fact_10469_ATP_Olambda__687,axiom,
    ! [Uu: set_nat,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE3763956218935105199t_bool,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_fun1096289392377091562BT_nat(aa_fun7476968357888788569BT_nat(aTP_Lamm_tj(Uu),Uua),Uub),Uuc) = groups708209901874060359at_nat(aa_VEB1122092716774696213at_nat(Uua,Uuc),collect_nat(aa_VEB4832260192069861672t_bool(aa_fun5431641998072558748t_bool(aTP_Lamm_ju(Uu),Uub),Uuc))) ).

% ATP.lambda_687
tff(fact_10470_ATP_Olambda__688,axiom,
    ! [Uu: set_nat,Uua: fun_VE7331017413731640256at_int,Uub: fun_VE3763956218935105199t_bool,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_fun6141810409722670662BT_int(aa_fun6474440919679335769BT_int(aTP_Lamm_tw(Uu),Uua),Uub),Uuc) = groups705719431365010083at_int(aa_VEB6167613734120275313at_int(Uua,Uuc),collect_nat(aa_VEB4832260192069861672t_bool(aa_fun5431641998072558748t_bool(aTP_Lamm_ju(Uu),Uub),Uuc))) ).

% ATP.lambda_688
tff(fact_10471_ATP_Olambda__689,axiom,
    ! [Uu: set_int,Uua: fun_co333780937270390912nt_int,Uub: fun_co2381122952212249071t_bool,Uuc: complex] : aa_complex_int(aa_fun3700423334941611530ex_int(aa_fun3520494530470439451ex_int(aTP_Lamm_uj(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_com3205223359959579713nt_int(Uua,Uuc),collect_int(aa_com2211550117870696024t_bool(aa_fun8206557716905444476t_bool(aTP_Lamm_iy(Uu),Uub),Uuc))) ).

% ATP.lambda_689
tff(fact_10472_ATP_Olambda__690,axiom,
    ! [Uu: set_int,Uua: fun_real_fun_int_int,Uub: fun_re8661203372455305201t_bool,Uuc: real] : aa_real_int(aa_fun529217645232493962al_int(aa_fun5832317237790237469al_int(aTP_Lamm_uh(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_real_fun_int_int(Uua,Uuc),collect_int(aa_real_fun_int_bool(aa_fun6227302519962330364t_bool(aTP_Lamm_iq(Uu),Uub),Uuc))) ).

% ATP.lambda_690
tff(fact_10473_ATP_Olambda__691,axiom,
    ! [Uu: set_int,Uua: fun_VE294566951569886236nt_int,Uub: fun_VE3303673267288458579t_bool,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_fun4615758601980977314BT_int(aa_fun627869447348354137BT_int(aTP_Lamm_uf(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_VEB7166967446021329869nt_int(Uua,Uuc),collect_int(aa_VEB1707969225601237452t_bool(aa_fun978093584635342820t_bool(aTP_Lamm_im(Uu),Uub),Uuc))) ).

% ATP.lambda_691
tff(fact_10474_ATP_Olambda__692,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_jz(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_jy(Uu),Uub),Uuc))) ).

% ATP.lambda_692
tff(fact_10475_ATP_Olambda__693,axiom,
    ! [Uu: set_nat,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE3763956218935105199t_bool,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_fun1096289392377091562BT_nat(aa_fun7476968357888788569BT_nat(aTP_Lamm_jv(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_VEB1122092716774696213at_nat(Uua,Uuc)),collect_nat(aa_VEB4832260192069861672t_bool(aa_fun5431641998072558748t_bool(aTP_Lamm_ju(Uu),Uub),Uuc))) ).

% ATP.lambda_693
tff(fact_10476_ATP_Olambda__694,axiom,
    ! [Uu: set_complex,Uua: fun_in1307110748760623106omplex,Uub: fun_in5750378413879446255x_bool,Uuc: int] : aa_int_complex(aa_fun751309451324587018omplex(aa_fun3438240386181387033omplex(aTP_Lamm_jq(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_int3545933452204062147omplex(Uua,Uuc),collect_complex(aa_int5354780257279585880x_bool(aa_fun6456964791495427196x_bool(aTP_Lamm_jp(Uu),Uub),Uuc))) ).

% ATP.lambda_694
tff(fact_10477_ATP_Olambda__695,axiom,
    ! [Uu: set_complex,Uua: fun_na1591161823130732966omplex,Uub: fun_na5941084329111710283x_bool,Uuc: nat] : aa_nat_complex(aa_fun2700696402887535882omplex(aa_fun8019143150332516725omplex(aTP_Lamm_jm(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(Uua,Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun2551091230351774588x_bool(aTP_Lamm_jl(Uu),Uub),Uuc))) ).

% ATP.lambda_695
tff(fact_10478_ATP_Olambda__696,axiom,
    ! [Uu: set_complex,Uua: fun_re7208826350674946562omplex,Uub: fun_re3010880487650781423x_bool,Uuc: real] : aa_real_complex(aa_fun4873726854752153866omplex(aa_fun883667864959905433omplex(aTP_Lamm_ji(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_rea2618301233319580611omplex(Uua,Uuc),collect_complex(aa_rea5887071448072802392x_bool(aa_fun9179849694055305084x_bool(aTP_Lamm_jh(Uu),Uub),Uuc))) ).

% ATP.lambda_696
tff(fact_10479_ATP_Olambda__697,axiom,
    ! [Uu: set_complex,Uua: fun_VE2006399705683203104omplex,Uub: fun_VE2128610149277289041x_bool,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun6203732500129703078omplex(aa_fun7613540521737166425omplex(aTP_Lamm_je(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_VEB5608662635949521omplex(Uua,Uuc),collect_complex(aa_VEB8450443329232412106x_bool(aa_fun5298830385910147808x_bool(aTP_Lamm_jd(Uu),Uub),Uuc))) ).

% ATP.lambda_697
tff(fact_10480_ATP_Olambda__698,axiom,
    ! [Uu: set_int,Uua: fun_co333780937270390912nt_int,Uub: fun_co2381122952212249071t_bool,Uuc: complex] : aa_complex_int(aa_fun3700423334941611530ex_int(aa_fun3520494530470439451ex_int(aTP_Lamm_iz(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_com3205223359959579713nt_int(Uua,Uuc),collect_int(aa_com2211550117870696024t_bool(aa_fun8206557716905444476t_bool(aTP_Lamm_iy(Uu),Uub),Uuc))) ).

% ATP.lambda_698
tff(fact_10481_ATP_Olambda__699,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_int,Uub: fun_nat_fun_int_bool,Uuc: nat] : aa_nat_int(aa_fun3504929462025808522at_int(aa_fun3325927984641084153at_int(aTP_Lamm_iv(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_nat_fun_int_int(Uua,Uuc),collect_int(aa_nat_fun_int_bool(aa_fun7617171277304647676t_bool(aTP_Lamm_iu(Uu),Uub),Uuc))) ).

% ATP.lambda_699
tff(fact_10482_ATP_Olambda__700,axiom,
    ! [Uu: set_int,Uua: fun_real_fun_int_int,Uub: fun_re8661203372455305201t_bool,Uuc: real] : aa_real_int(aa_fun529217645232493962al_int(aa_fun5832317237790237469al_int(aTP_Lamm_ir(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_real_fun_int_int(Uua,Uuc),collect_int(aa_real_fun_int_bool(aa_fun6227302519962330364t_bool(aTP_Lamm_iq(Uu),Uub),Uuc))) ).

% ATP.lambda_700
tff(fact_10483_ATP_Olambda__701,axiom,
    ! [Uu: set_int,Uua: fun_VE294566951569886236nt_int,Uub: fun_VE3303673267288458579t_bool,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_fun4615758601980977314BT_int(aa_fun627869447348354137BT_int(aTP_Lamm_in(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_VEB7166967446021329869nt_int(Uua,Uuc),collect_int(aa_VEB1707969225601237452t_bool(aa_fun978093584635342820t_bool(aTP_Lamm_im(Uu),Uub),Uuc))) ).

% ATP.lambda_701
tff(fact_10484_ATP_Olambda__702,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_mg(Uu),Uua),Uub),Uuc) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_702
tff(fact_10485_ATP_Olambda__703,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_mf(Uu),Uua),Uub),Uuc) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_703
tff(fact_10486_ATP_Olambda__704,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_wg(Uu),Uua),Uub),Uuc) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_704
tff(fact_10487_ATP_Olambda__705,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_yb(Uu),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(minus_minus_int(aa_int_int(times_times_int(Uu),Uuc)),aa_int_int(times_times_int(Uub),Uua)),aa_int_int(times_times_int(Uua),Uuc))) ).

% ATP.lambda_705
tff(fact_10488_ATP_Olambda__706,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_xz(Uu),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(Uu),Uuc)),aa_int_int(times_times_int(Uub),Uua)),aa_int_int(times_times_int(Uua),Uuc))) ).

% ATP.lambda_706
tff(fact_10489_ATP_Olambda__707,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_xv(Uu),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(times_times_int(Uu),Uub),aa_int_int(times_times_int(Uua),Uuc))) ).

% ATP.lambda_707
tff(fact_10490_ATP_Olambda__708,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_xx(Uu),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(times_times_int(Uu),Uuc),aa_int_int(times_times_int(Uua),Uub))) ).

% ATP.lambda_708
tff(fact_10491_ATP_Olambda__709,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_bool,Uub: fun_re7812779276277946752omplex,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun8363404835654190852x_bool(aa_fun7464652376466389139x_bool(aTP_Lamm_dj(Uu),Uua),Uub),Uuc))
    <=> ? [X3: real,Y4: real] :
          ( ( Uuc = aa_real_complex(aa_rea2649881322057434177omplex(Uub,X3),Y4) )
          & pp(aa_real_bool(Uu,X3))
          & pp(aa_real_bool(Uua,Y4)) ) ) ).

% ATP.lambda_709
tff(fact_10492_ATP_Olambda__710,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_bool,Uub: fun_re7681423839302404990l_real,Uuc: real] :
      ( pp(aa_real_bool(aa_fun955784922888763656l_bool(aa_fun2902298374912565143l_bool(aTP_Lamm_dg(Uu),Uua),Uub),Uuc))
    <=> ? [X3: real,Y4: real] :
          ( ( Uuc = aa_real_real(aa_rea7340605660300375871l_real(Uub,X3),Y4) )
          & pp(aa_real_bool(Uu,X3))
          & pp(aa_real_bool(Uua,Y4)) ) ) ).

% ATP.lambda_710
tff(fact_10493_ATP_Olambda__711,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_bool,Uub: fun_re7428432522468557474al_nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun528418799539591104t_bool(aa_fun8259732966411813455t_bool(aTP_Lamm_dh(Uu),Uua),Uub),Uuc))
    <=> ? [X3: real,Y4: real] :
          ( ( Uuc = aa_real_nat(aa_real_fun_real_nat(Uub,X3),Y4) )
          & pp(aa_real_bool(Uu,X3))
          & pp(aa_real_bool(Uua,Y4)) ) ) ).

% ATP.lambda_711
tff(fact_10494_ATP_Olambda__712,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_bool,Uub: fun_re4938349377536830718al_int,Uuc: int] :
      ( pp(aa_int_bool(aa_fun1238366933954993416t_bool(aa_fun2410317078794417559t_bool(aTP_Lamm_di(Uu),Uua),Uub),Uuc))
    <=> ? [X3: real,Y4: real] :
          ( ( Uuc = aa_real_int(aa_real_fun_real_int(Uub,X3),Y4) )
          & pp(aa_real_bool(Uu,X3))
          & pp(aa_real_bool(Uua,Y4)) ) ) ).

% ATP.lambda_712
tff(fact_10495_ATP_Olambda__713,axiom,
    ! [Uu: fun_real_bool,Uua: fun_nat_bool,Uub: fun_re2084383175013312036omplex,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun9170867047750497504x_bool(aa_fun2493215948360222867x_bool(aTP_Lamm_dn(Uu),Uua),Uub),Uuc))
    <=> ? [X3: real,Y4: nat] :
          ( ( Uuc = aa_nat_complex(aa_rea1531280491672885477omplex(Uub,X3),Y4) )
          & pp(aa_real_bool(Uu,X3))
          & pp(aa_nat_bool(Uua,Y4)) ) ) ).

% ATP.lambda_713
tff(fact_10496_ATP_Olambda__714,axiom,
    ! [Uu: fun_real_bool,Uua: fun_nat_bool,Uub: fun_re4238438383274954274t_real,Uuc: real] :
      ( pp(aa_real_bool(aa_fun5008594764412271332l_bool(aa_fun7835077198776561559l_bool(aTP_Lamm_dk(Uu),Uua),Uub),Uuc))
    <=> ? [X3: real,Y4: nat] :
          ( ( Uuc = aa_nat_real(aa_real_fun_nat_real(Uub,X3),Y4) )
          & pp(aa_real_bool(Uu,X3))
          & pp(aa_nat_bool(Uua,Y4)) ) ) ).

% ATP.lambda_714
tff(fact_10497_ATP_Olambda__715,axiom,
    ! [Uu: fun_real_bool,Uua: fun_nat_bool,Uub: fun_real_fun_nat_nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun3685766209695167900t_bool(aa_fun2965812958138651215t_bool(aTP_Lamm_dl(Uu),Uua),Uub),Uuc))
    <=> ? [X3: real,Y4: nat] :
          ( ( Uuc = aa_nat_nat(aa_real_fun_nat_nat(Uub,X3),Y4) )
          & pp(aa_real_bool(Uu,X3))
          & pp(aa_nat_bool(Uua,Y4)) ) ) ).

% ATP.lambda_715
tff(fact_10498_ATP_Olambda__716,axiom,
    ! [Uu: fun_real_bool,Uua: fun_nat_bool,Uub: fun_real_fun_nat_int,Uuc: int] :
      ( pp(aa_int_bool(aa_fun4395714344110570212t_bool(aa_fun6339769107376031127t_bool(aTP_Lamm_dm(Uu),Uua),Uub),Uuc))
    <=> ? [X3: real,Y4: nat] :
          ( ( Uuc = aa_nat_int(aa_real_fun_nat_int(Uub,X3),Y4) )
          & pp(aa_real_bool(Uu,X3))
          & pp(aa_nat_bool(Uua,Y4)) ) ) ).

% ATP.lambda_716
tff(fact_10499_ATP_Olambda__717,axiom,
    ! [Uu: fun_real_bool,Uua: fun_int_bool,Uub: fun_re7457141977439324286t_real,Uuc: real] :
      ( pp(aa_real_bool(aa_fun6686107996194020744l_bool(aa_fun7698435294645330071l_bool(aTP_Lamm_do(Uu),Uua),Uub),Uuc))
    <=> ? [X3: real,Y4: int] :
          ( ( Uuc = aa_int_real(aa_real_fun_int_real(Uub,X3),Y4) )
          & pp(aa_real_bool(Uu,X3))
          & pp(aa_int_bool(Uua,Y4)) ) ) ).

% ATP.lambda_717
tff(fact_10500_ATP_Olambda__718,axiom,
    ! [Uu: fun_real_bool,Uua: fun_int_bool,Uub: fun_real_fun_int_nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun6723608294159301696t_bool(aa_fun543251173574902607t_bool(aTP_Lamm_dp(Uu),Uua),Uub),Uuc))
    <=> ? [X3: real,Y4: int] :
          ( ( Uuc = aa_int_nat(aa_real_fun_int_nat(Uub,X3),Y4) )
          & pp(aa_real_bool(Uu,X3))
          & pp(aa_int_bool(Uua,Y4)) ) ) ).

% ATP.lambda_718
tff(fact_10501_ATP_Olambda__719,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_aad(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_aac(Uua),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uub),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))))) ).

% ATP.lambda_719
tff(fact_10502_ATP_Olambda__720,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_yq(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_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(Uua,I4)),X_12))
            <=> 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_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_12)) ) )
        & ( ( 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_720
tff(fact_10503_ATP_Olambda__721,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_ak,Uu) = Uu ).

% ATP.lambda_721
tff(fact_10504_ATP_Olambda__722,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_al,Uu) = Uu ).

% ATP.lambda_722
tff(fact_10505_ATP_Olambda__723,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_am,Uu) = Uu ).

% ATP.lambda_723
tff(fact_10506_ATP_Olambda__724,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_an,Uu) = Uu ).

% ATP.lambda_724
tff(fact_10507_ATP_Olambda__725,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ao,Uu) = Uu ).

% ATP.lambda_725
tff(fact_10508_ATP_Olambda__726,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_fb,Uu) = zero_zero_complex ).

% ATP.lambda_726
tff(fact_10509_ATP_Olambda__727,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_fc,Uu) = zero_zero_real ).

% ATP.lambda_727
tff(fact_10510_ATP_Olambda__728,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_fd,Uu) = zero_zero_rat ).

% ATP.lambda_728
tff(fact_10511_ATP_Olambda__729,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_fe,Uu) = zero_zero_nat ).

% ATP.lambda_729
tff(fact_10512_ATP_Olambda__730,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ff,Uu) = zero_zero_int ).

% ATP.lambda_730
tff(fact_10513_ATP_Olambda__731,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_xc,Uu))
    <=> $false ) ).

% ATP.lambda_731
tff(fact_10514_ATP_Olambda__732,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_xp,Uu))
    <=> $false ) ).

% ATP.lambda_732
tff(fact_10515_ATP_Olambda__733,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_xo,Uu))
    <=> $true ) ).

% ATP.lambda_733

% Helper facts (62)
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__Num__Onum_T,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(if_num(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(if_num(fTrue),X),Y) = X ).

tff(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] : if_rat(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: 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__HOL__Obool_T,axiom,
    ! [X: bool,Y: bool] :
      ( ( X != Y )
      | pp(aa_bool_bool(fequal_bool(X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X: bool,Y: bool] :
      ( ~ pp(aa_bool_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__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] : aa_VEB5832953974749077948T_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] : aa_VEB5832953974749077948T_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__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( X != Y )
      | pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,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__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] : 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] : 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_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,R2: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_3745307990312268622T_VEBT(P,Q),R2) = aa_bool_bool(P,aa_VEBT_VEBT_bool(Q,R2)) ).

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,R2: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBS_6222329278419335549l_bool(P,Q),R2) = aa_bool_bool(aa_VEB8047906998784800507l_bool(P,R2),aa_VEBT_VEBT_bool(Q,R2)) ).

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,R2: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_2885587777813823575T_VEBT(P,Q),R2) = aa_fun_nat_bool_bool(P,aa_VEB4832260192069861672t_bool(Q,R2)) ).

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,R2: vEBT_VEBT] : aa_VEB8047906998784800507l_bool(cOMBB_3438363824644082366T_VEBT(P,Q),R2) = aa_boo8970175072276269406l_bool(P,aa_VEBT_VEBT_bool(Q,R2)) ).

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,R2: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBC_771164880915498576T_bool(P,Q),R2) = aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(P,R2),Q) ).

% Conjectures (1)
tff(conj_0,conjecture,
    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,succy),aa_nat_nat(aa_nat_fun_nat_nat(power_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))))))
    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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))))))
    & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,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(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = deg ) ) ).

%------------------------------------------------------------------------------